\(\int \frac {(a+b x+c x^2)^{3/4}}{d+e x} \, dx\) [732]

Optimal result

Integrand size = 22, antiderivative size = 855 \[ \int \frac {\left (a+b x+c x^2\right )^{3/4}}{d+e x} \, dx=\frac {2 \left (a+b x+c x^2\right )^{3/4}}{3 e}+\frac {\sqrt [4]{-b^2+4 a c} \left (c d^2-b d e+a e^2\right )^{3/4} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \arctan \left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt [4]{-b^2+4 a c} \left (c d^2-b d e+a e^2\right )^{3/4} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \text {arctanh}\left (\frac {\sqrt [4]{-b^2+4 a c} \sqrt {e} \sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}}{\sqrt {2} \sqrt [4]{c} \sqrt [4]{c d^2-b d e+a e^2}}\right )}{\sqrt [4]{c} e^{5/2} \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt {b^2-4 a c} (2 c d-b e) \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\left .\frac {1}{2} \arcsin \left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )\right |2\right )}{\sqrt {2} c e^2 \sqrt [4]{a+b x+c x^2}}-\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {c d^2-b d e+a e^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} e^3 (b+2 c x) \sqrt [4]{a+b x+c x^2}}+\frac {\sqrt {-b^2+4 a c} (2 c d-b e) \sqrt {c d^2-b d e+a e^2} \sqrt {\frac {(b+2 c x)^2}{b^2-4 a c}} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {\sqrt {-b^2+4 a c} e}{2 \sqrt {c} \sqrt {c d^2-b d e+a e^2}},\arcsin \left (\sqrt [4]{1-\frac {(b+2 c x)^2}{b^2-4 a c}}\right ),-1\right )}{\sqrt {2} \sqrt {c} e^3 (b+2 c x) \sqrt [4]{a+b x+c x^2}} \] Output:

2/3*(c*x^2+b*x+a)^(3/4)/e+(4*a*c-b^2)^(1/4)*(a*e^2-b*d*e+c*d^2)^(3/4)*(-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)/e^(5/2)/(c*x^2+b*x+a)^(1/4)-(4*a*c-b^2)^(1/4)*(a*e^2-b*d*e+c*d^2) 
^(3/4)*(-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)/e^(5/2)/(c*x^2+b*x+a)^(1/4)-1/2*(-4*a*c+b^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))*2^(1/2)/c/e^2/(c*x^2+b*x+a)^(1/4)- 
1/2*(4*a*c-b^2)^(1/2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^(1/2)*((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)/e^3/(2*c*x+b)/(c*x^2+b*x+a)^(1/4)+1/2*(4 
*a*c-b^2)^(1/2)*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^(1/2)*((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)/e^3/(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 = 11.93 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.21 \[ \int \frac {\left (a+b x+c x^2\right )^{3/4}}{d+e x} \, dx=\frac {4 \sqrt {2} (a+x (b+c x))^{3/4} \operatorname {AppellF1}\left (-\frac {3}{2},-\frac {3}{4},-\frac {3}{4},-\frac {1}{2},\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )}{3 e \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{3/4} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{3/4}} \] Input:

Integrate[(a + b*x + c*x^2)^(3/4)/(d + e*x),x]
 

Output:

(4*Sqrt[2]*(a + x*(b + c*x))^(3/4)*AppellF1[-3/2, -3/4, -3/4, -1/2, (2*c*d 
 - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4 
*a*c]*e)/(2*c*d + 2*c*e*x)])/(3*e*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c* 
(d + e*x)))^(3/4)*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^(3/4 
))
 

Rubi [A] (warning: unable to verify)

Time = 1.99 (sec) , antiderivative size = 1323, normalized size of antiderivative = 1.55, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {1162, 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.

\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/4}}{d+e x} \, dx\)

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1094

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

\(\Big \downarrow \) 834

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1174

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

\(\Big \downarrow \) 1173

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 504

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

\(\Big \downarrow \) 310

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 1537

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {2 \left (c x^2+b x+a\right )^{3/4}}{3 e}-\frac {\frac {4 (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 )}{e (b+2 c x)}-\frac {2 \sqrt {2} \left (c d^2-b e d+a e^2\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 \sqrt [4]{c x^2+b x+a}}}{2 e}\)

Input:

Int[(a + b*x + c*x^2)^(3/4)/(d + e*x),x]
 

Output:

(2*(a + b*x + c*x^2)^(3/4))/(3*e) - ((4*(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]*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) 
]*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)]*Elliptic 
F[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)])))/(e* 
(b + 2*c*x)) - (2*Sqrt[2]*(c*d^2 - b*d*e + a*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)*(- 
((b*c)/(b^2 - 4*a*c)) - (2*c^2*x)/(b^2 - 4*a*c))^2)/c^2)^(1/4))/(Sqrt[2...
 

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 1162

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 0] && 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]

\[\int \frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{4}}}{e x +d}d x\]

Input:

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

Output:

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

Fricas [F(-1)]

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

integrate((c*x^2+b*x+a)^(3/4)/(e*x+d),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((c*x^2+b*x+a)^(3/4)/(e*x+d),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + b*x + a)^(3/4)/(e*x + d), x)
 

Giac [F]

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

integrate((c*x^2+b*x+a)^(3/4)/(e*x+d),x, algorithm="giac")
 

Output:

integrate((c*x^2 + b*x + a)^(3/4)/(e*x + d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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