\(\int \frac {1}{(d+e x) (a+b x+c x^2)^{5/4}} \, dx\) [758]

Optimal result

Integrand size = 22, antiderivative size = 914 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=-\frac {4 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt [4]{a+b x+c x^2}}+\frac {\sqrt [4]{-b^2+4 a c} e^{3/2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \left (c d^2-b d e+a e^2\right )^{5/4} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt [4]{-b^2+4 a c} e^{3/2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} \left (c d^2-b d e+a e^2\right )^{5/4} \sqrt [4]{a+b x+c x^2}}+\frac {2 \sqrt {2} (2 c d-b e) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt {-b^2+4 a c} e (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} \left (c d^2-b d e+a e^2\right )^{3/2} (b+2 c x) \sqrt [4]{a+b x+c x^2}}+\frac {\sqrt {-b^2+4 a c} e (2 c d-b e) \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} \left (c d^2-b d e+a e^2\right )^{3/2} (b+2 c x) \sqrt [4]{a+b x+c x^2}} \] Output:

(-4*b*c*d+4*b^2*e-8*a*c*e-4*c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c* 
d^2)/(c*x^2+b*x+a)^(1/4)+(4*a*c-b^2)^(1/4)*e^(3/2)*(-c*(c*x^2+b*x+a)/(-4*a 
*c+b^2))^(1/4)*arctan(1/2*(4*a*c-b^2)^(1/4)*e^(1/2)*(1-(2*c*x+b)^2/(-4*a*c 
+b^2))^(1/4)*2^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4))/c^(1/4)/(a*e^2-b*d 
*e+c*d^2)^(5/4)/(c*x^2+b*x+a)^(1/4)-(4*a*c-b^2)^(1/4)*e^(3/2)*(-c*(c*x^2+b 
*x+a)/(-4*a*c+b^2))^(1/4)*arctanh(1/2*(4*a*c-b^2)^(1/4)*e^(1/2)*(1-(2*c*x+ 
b)^2/(-4*a*c+b^2))^(1/4)*2^(1/2)/c^(1/4)/(a*e^2-b*d*e+c*d^2)^(1/4))/c^(1/4 
)/(a*e^2-b*d*e+c*d^2)^(5/4)/(c*x^2+b*x+a)^(1/4)+2*2^(1/2)*(-b*e+2*c*d)*(-c 
*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticE(sin(1/2*arcsin((2*c*x+b)/(-4* 
a*c+b^2)^(1/2))),2^(1/2))/(-4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)/(c*x^2+b* 
x+a)^(1/4)-1/2*(4*a*c-b^2)^(1/2)*e*(-b*e+2*c*d)*((2*c*x+b)^2/(-4*a*c+b^2)) 
^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticPi((1-(2*c*x+b)^2/(-4 
*a*c+b^2))^(1/4),-1/2*(4*a*c-b^2)^(1/2)*e/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2 
),I)*2^(1/2)/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(3/2)/(2*c*x+b)/(c*x^2+b*x+a)^(1/ 
4)+1/2*(4*a*c-b^2)^(1/2)*e*(-b*e+2*c*d)*((2*c*x+b)^2/(-4*a*c+b^2))^(1/2)*( 
-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/4)*EllipticPi((1-(2*c*x+b)^2/(-4*a*c+b^2 
))^(1/4),1/2*(4*a*c-b^2)^(1/2)*e/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(1/2),I)*2^(1 
/2)/c^(1/2)/(a*e^2-b*d*e+c*d^2)^(3/2)/(2*c*x+b)/(c*x^2+b*x+a)^(1/4)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 12.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.20 \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=-\frac {\left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{5/4} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{5/4} \operatorname {AppellF1}\left (\frac {5}{2},\frac {5}{4},\frac {5}{4},\frac {7}{2},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{10 \sqrt {2} e (a+x (b+c x))^{5/4}} \] Input:

Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(5/4)),x]
 

Output:

-1/10*(((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*((e*(b + 
Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(5/4)*AppellF1[5/2, 5/4, 5/4, 7 
/2, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sq 
rt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])/(Sqrt[2]*e*(a + x*(b + c*x))^(5/4))
 

Rubi [A] (warning: unable to verify)

Time = 2.22 (sec) , antiderivative size = 1385, normalized size of antiderivative = 1.52, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.955, Rules used = {1165, 27, 1269, 1094, 834, 761, 1174, 1173, 25, 504, 310, 353, 73, 27, 827, 218, 221, 993, 1510, 1537, 412}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {4 \int -\frac {4 c^2 d^2+b^2 e^2-2 c e (b d+2 a e)+2 c e (2 c d-b e) x}{4 (d+e x) \sqrt [4]{c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {4 c^2 d^2+b^2 e^2-2 c e (b d+2 a e)+2 c e (2 c d-b e) x}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {e^2 \left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx+2 c (2 c d-b e) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {e^2 \left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx+\frac {8 c \sqrt {(b+2 c x)^2} (2 c d-b e) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{b+2 c x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {e^2 \left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx+\frac {8 c \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\sqrt {b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {8 c \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}+e^2 \left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1174

\(\displaystyle \frac {\frac {8 c \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}+\frac {e^2 \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{(d+e x) \sqrt [4]{-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{\sqrt [4]{a+b x+c x^2}}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1173

\(\displaystyle \frac {\frac {8 c \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}+\frac {\sqrt {2} e^2 \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{\sqrt [4]{a+b x+c x^2}}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {8 c \sqrt {(b+2 c x)^2} (2 c d-b e) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}-\frac {\sqrt {2} e^2 \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{\sqrt [4]{a+b x+c x^2}}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {4 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt [4]{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (-\frac {c (2 c d-b e) \int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (-\frac {1}{2} e \int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 c^2 e \int \frac {\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{c^2 \left (\frac {e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}{b^2-4 a c}+\frac {4 c \left (c d^2-b e d+a e^2\right )}{\left (b^2-4 a c\right )^2}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \int \frac {\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}{\frac {e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^8}{b^2-4 a c}+\frac {4 c \left (c d^2-b e d+a e^2\right )}{\left (b^2-4 a c\right )^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{3/2} \int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}-\frac {\left (4 a c-b^2\right )^{3/2} \int \frac {1}{\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{3/2} \int \frac {1}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^4}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}-\frac {\left (4 a c-b^2\right )^{5/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}-\frac {\left (4 a c-b^2\right )^{5/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \int \frac {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (-\left (\left (1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}\right ) e^2\right )+e^2-\frac {(2 c d-b e)^2}{b^2-4 a c}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}-\frac {\left (4 a c-b^2\right )^{5/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\sqrt {4 a c-b^2} \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}-\frac {\sqrt {4 a c-b^2} \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\sqrt {4 a c-b^2} \sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} e+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}\right )}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}-\frac {\left (4 a c-b^2\right )^{5/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\sqrt {4 a c-b^2} \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}-\frac {\sqrt {4 a c-b^2} \int \frac {1}{\sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\sqrt {4 a c-b^2} \sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} e+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}\right )}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt [4]{c x^2+b x+a} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 1537

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}-\frac {\left (4 a c-b^2\right )^{5/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\sqrt {4 a c-b^2} \int \frac {1}{\sqrt {1-\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}} \sqrt {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}+1} \left (2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}-\sqrt {4 a c-b^2} e \sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}-\frac {\sqrt {4 a c-b^2} \int \frac {1}{\sqrt {1-\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}} \sqrt {\sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}+1} \left (\sqrt {4 a c-b^2} \sqrt {1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} e+2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}\right )}d\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{2 e}\right )}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt [4]{c x^2+b x+a} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\frac {\sqrt {2} \left (b^2-4 a c\right ) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (\frac {2 e \left (\frac {\left (4 a c-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}-\frac {\left (4 a c-b^2\right )^{5/4} \arctan \left (\frac {\sqrt [4]{4 a c-b^2} \sqrt {e} \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{2 \sqrt {2} \sqrt [4]{c} e^{3/2} \sqrt [4]{c d^2-b e d+a e^2}}\right )}{b^2-4 a c}-\frac {2 c (2 c d-b e) \sqrt {\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {\sqrt {4 a c-b^2} \operatorname {EllipticPi}\left (-\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\frac {2 x c^2}{b^2-4 a c}+\frac {b c}{b^2-4 a c}\right ),-1\right )}{4 \sqrt {c} e \sqrt {c d^2-b e d+a e^2}}-\frac {\sqrt {4 a c-b^2} \operatorname {EllipticPi}\left (\frac {\sqrt {4 a c-b^2} e}{2 \sqrt {c} \sqrt {c d^2-b e d+a e^2}},\arcsin \left (\frac {2 x c^2}{b^2-4 a c}+\frac {b c}{b^2-4 a c}\right ),-1\right )}{4 \sqrt {c} e \sqrt {c d^2-b e d+a e^2}}\right )}{\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}\right ) e^2}{\sqrt [4]{c x^2+b x+a}}+\frac {8 c (2 c d-b e) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \left (\frac {\sqrt [4]{b^2-4 a c} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac {1}{2}\right )}{\sqrt {2} \sqrt [4]{c} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt [4]{c x^2+b x+a} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )}\right )}{2 \sqrt {c}}\right )}{b+2 c x}}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {4 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt [4]{c x^2+b x+a}}\)

Input:

Int[1/((d + e*x)*(a + b*x + c*x^2)^(5/4)),x]
 

Output:

(-4*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - 
 b*d*e + a*e^2)*(a + b*x + c*x^2)^(1/4)) + ((8*c*(2*c*d - b*e)*Sqrt[(b + 2 
*c*x)^2]*(-1/2*(Sqrt[b^2 - 4*a*c]*(-(((a + b*x + c*x^2)^(1/4)*Sqrt[b^2 - 4 
*a*c + 4*c*(a + b*x + c*x^2)])/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x 
 + c*x^2])/Sqrt[b^2 - 4*a*c]))) + ((b^2 - 4*a*c)^(1/4)*(1 + (2*Sqrt[c]*Sqr 
t[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + 
c*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4 
*a*c])^2)]*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b 
^2 - 4*a*c)^(1/4)], 1/2])/(Sqrt[2]*c^(1/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x 
 + c*x^2)])))/Sqrt[c] + ((b^2 - 4*a*c)^(3/4)*(1 + (2*Sqrt[c]*Sqrt[a + b*x 
+ c*x^2])/Sqrt[b^2 - 4*a*c])*Sqrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/(( 
b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)] 
*EllipticF[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c 
)^(1/4)], 1/2])/(4*Sqrt[2]*c^(3/4)*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2 
)])))/(b + 2*c*x) + (Sqrt[2]*(b^2 - 4*a*c)*e^2*(-((c*(a + b*x + c*x^2))/(b 
^2 - 4*a*c)))^(1/4)*((2*e*(-1/2*((-b^2 + 4*a*c)^(5/4)*ArcTan[((-b^2 + 4*a* 
c)^(1/4)*Sqrt[e]*(1 - ((b^2 - 4*a*c)*(-((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/( 
b^2 - 4*a*c))^2)/c^2)^(1/4))/(Sqrt[2]*c^(1/4)*(c*d^2 - b*d*e + a*e^2)^(1/4 
))])/(Sqrt[2]*c^(1/4)*e^(3/2)*(c*d^2 - b*d*e + a*e^2)^(1/4)) + ((-b^2 + 4* 
a*c)^(5/4)*ArcTanh[((-b^2 + 4*a*c)^(1/4)*Sqrt[e]*(1 - ((b^2 - 4*a*c)*(-...
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 310

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[2*(Sqrt[(-b)*(x^2/a)]/x)   Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* 
x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 504

Int[((a_) + (b_.)*(x_)^2)^(p_)/((c_) + (d_.)*(x_)), x_Symbol] :> Simp[c   I 
nt[(a + b*x^2)^p/(c^2 - d^2*x^2), x], x] - Simp[d   Int[x*((a + b*x^2)^p/(c 
^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, p}, x]
 

rule 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 0]
 

rule 1094

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[4*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(4*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^4], x], x, (a + b*x + c*x^2)^(1/4)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[4*p]
 

rule 1165

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) 
*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 
2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d 
+ e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p 
+ 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + 
 b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] 
 && IntQuadraticQ[a, b, c, d, e, m, p, x]
 

rule 1173

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol 
] :> Simp[1/(-4*(c/(b^2 - 4*a*c)))^p   Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c) 
, x]^p/Simp[2*c*d - b*e + e*x, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, 
 d, e, p}, x] && GtQ[4*a - b^2/c, 0] && IntegerQ[4*p]
 

rule 1174

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)/((d_.) + (e_.)*(x_)), x_Symbol 
] :> Simp[(a + b*x + c*x^2)^p/((-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c)))^p 
Int[((-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) - c^2*(x^2/(b^2 - 4*a*c 
)))^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] &&  !GtQ[4*a - b^2/ 
c, 0] && IntegerQ[4*p]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

rule 1537

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[(-a)*c, 2]}, Simp[Sqrt[-c]   Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqr 
t[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] & 
& GtQ[a, 0] && LtQ[c, 0]
 

Maple [F]

\[\int \frac {1}{\left (e x +d \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{4}}}d x\]

Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^(5/4),x)
 

Output:

int(1/(e*x+d)/(c*x^2+b*x+a)^(5/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(5/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac {5}{4}}}\, dx \] Input:

integrate(1/(e*x+d)/(c*x**2+b*x+a)**(5/4),x)
 

Output:

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(5/4)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(5/4),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)), x)
 

Giac [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{4}} {\left (e x + d\right )}} \,d x } \] Input:

integrate(1/(e*x+d)/(c*x^2+b*x+a)^(5/4),x, algorithm="giac")
 

Output:

integrate(1/((c*x^2 + b*x + a)^(5/4)*(e*x + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/4}} \,d x \] Input:

int(1/((d + e*x)*(a + b*x + c*x^2)^(5/4)),x)
 

Output:

int(1/((d + e*x)*(a + b*x + c*x^2)^(5/4)), x)
 

Reduce [F]

\[ \int \frac {1}{(d+e x) \left (a+b x+c x^2\right )^{5/4}} \, dx=\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a d +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} a e x +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b d x +\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} b e \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} c d \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{4}} c e \,x^{3}}d x \] Input:

int(1/(e*x+d)/(c*x^2+b*x+a)^(5/4),x)
 

Output:

int(1/((a + b*x + c*x**2)**(1/4)*a*d + (a + b*x + c*x**2)**(1/4)*a*e*x + ( 
a + b*x + c*x**2)**(1/4)*b*d*x + (a + b*x + c*x**2)**(1/4)*b*e*x**2 + (a + 
 b*x + c*x**2)**(1/4)*c*d*x**2 + (a + b*x + c*x**2)**(1/4)*c*e*x**3),x)