\(\int \frac {(f+g x) (a+b x+c x^2)^{3/4}}{(d+e x)^2} \, dx\) [1106]

Optimal result

Integrand size = 27, antiderivative size = 1114 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/4}}{(d+e x)^2} \, dx =\text {Too large to display} \] Output:

-1/3*(c*d*(-5*d*g+3*e*f)-e*(2*a*e*g-5*b*d*g+3*b*e*f)-3*c*e*(-d*g+e*f)*x)*( 
c*x^2+b*x+a)^(3/4)/e^2/(a*e^2-b*d*e+c*d^2)-(-d*g+e*f)*(c*x^2+b*x+a)^(7/4)/ 
(a*e^2-b*d*e+c*d^2)/(e*x+d)-1/4*(4*a*c-b^2)^(1/4)*(2*c*d*(-5*d*g+3*e*f)-e* 
(4*a*e*g-7*b*d*g+3*b*e*f))*(-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^(7/2)/(a*e^2-b*d*e+c*d^2)^(1/4)/ 
(c*x^2+b*x+a)^(1/4)+1/4*(4*a*c-b^2)^(1/4)*(2*c*d*(-5*d*g+3*e*f)-e*(4*a*e*g 
-7*b*d*g+3*b*e*f))*(-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^(7/2)/(a*e^2-b*d*e+c*d^2)^(1/4)/(c*x^2+ 
b*x+a)^(1/4)+1/2*(-4*a*c+b^2)^(1/2)*(b*e*g-5*c*d*g+3*c*e*f)*(-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^3/(c*x^2+b*x+a)^(1/4)+1/8*(4*a*c-b^2)^(1/2)*(-b 
*e+2*c*d)*(2*c*d*(-5*d*g+3*e*f)-e*(4*a*e*g-7*b*d*g+3*b*e*f))*((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^4/(a*e^2-b*d*e+c*d^2)^(1/2)/(2*c*x+b)/ 
(c*x^2+b*x+a)^(1/4)-1/8*(4*a*c-b^2)^(1/2)*(-b*e+2*c*d)*(2*c*d*(-5*d*g+3*e* 
f)-e*(4*a*e*g-7*b*d*g+3*b*e*f))*((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...
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 3.66 (sec) , antiderivative size = 1367, normalized size of antiderivative = 1.23, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {1230, 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.

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

\(\Big \downarrow \) 1230

\(\displaystyle -\frac {\int -\frac {3 b e f-5 b d g+4 a e g+2 (3 c e f-5 c d g+b e g) x}{2 (d+e x) \sqrt [4]{c x^2+b x+a}}dx}{2 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 b e f-5 b d g+4 a e g+2 (3 c e f-5 c d g+b e g) x}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {2 (b e g-5 c d g+3 c e f) \int \frac {1}{\sqrt [4]{c x^2+b x+a}}dx}{e}-\frac {(2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{e}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 1094

\(\displaystyle \frac {\frac {8 \sqrt {(b+2 c x)^2} (b e g-5 c d g+3 c e f) \int \frac {\sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{e (b+2 c x)}-\frac {(2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{e}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {\frac {8 \sqrt {(b+2 c x)^2} (b e g-5 c d g+3 c e f) \left (\frac {\sqrt {b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{e (b+2 c x)}-\frac {(2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{e}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {\frac {8 \sqrt {(b+2 c x)^2} (b e g-5 c d g+3 c e f) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{e (b+2 c x)}-\frac {(2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int \frac {1}{(d+e x) \sqrt [4]{c x^2+b x+a}}dx}{e}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 1174

\(\displaystyle \frac {\frac {8 \sqrt {(b+2 c x)^2} (b e g-5 c d g+3 c e f) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{e (b+2 c x)}-\frac {\sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int \frac {1}{(d+e x) \sqrt [4]{-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{e \sqrt [4]{a+b x+c x^2}}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 1173

\(\displaystyle \frac {\frac {8 \sqrt {(b+2 c x)^2} (b e g-5 c d g+3 c e f) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{e (b+2 c x)}-\frac {\sqrt {2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int -\frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \sqrt [4]{a+b x+c x^2}}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {8 \sqrt {(b+2 c x)^2} (b e g-5 c d g+3 c e f) \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {4 c \left (a+b x+c x^2\right )-4 a c+b^2}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{e (b+2 c x)}+\frac {\sqrt {2} \sqrt [4]{-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d (3 e f-5 d g)-e (4 a e g-7 b d g+3 b e f)) \int \frac {1}{\left (\frac {c (2 c d-b e)}{b^2-4 a c}-e \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right ) \sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}}}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{e \sqrt [4]{a+b x+c x^2}}}{4 e^2}-\frac {\left (a+b x+c x^2\right )^{3/4} (-5 d g+3 e f-2 e g x)}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 504

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \sqrt {(b+2 c x)^2} \left (\frac {\left (b^2-4 a c\right )^{3/4} \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right ) \sqrt {\frac {b^2-4 a c+4 c \left (c x^2+b x+a\right )}{\left (b^2-4 a c\right ) \left (\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right ),\frac {1}{2}\right )}{4 \sqrt {2} c^{3/4} \sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}-\frac {\sqrt {b^2-4 a c} \int \frac {1-\frac {2 \sqrt {c} \sqrt {c x^2+b x+a}}{\sqrt {b^2-4 a c}}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [4]{c x^2+b x+a}}{2 \sqrt {c}}\right )}{e (b+2 c x)}-\frac {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \sqrt [4]{-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \left (-\frac {c (2 c d-b e) \int \frac {1}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )}{b^2-4 a c}-e \int \frac {-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}}{\sqrt [4]{1-\frac {\left (b^2-4 a c\right ) \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2}{c^2}} \left (\frac {c^2 (2 c d-b e)^2}{\left (b^2-4 a c\right )^2}-e^2 \left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )^2\right )}d\left (-\frac {2 x c^2}{b^2-4 a c}-\frac {b c}{b^2-4 a c}\right )\right )}{e \sqrt [4]{c x^2+b x+a}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 1537

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {\frac {8 (3 c e f-5 c d g+b e g) \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 {\sqrt {2} (2 c d (3 e f-5 d g)-e (3 b e f-7 b d g+4 a e g)) \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}}}{4 e^2}-\frac {(3 e f-5 d g-2 e g x) \left (c x^2+b x+a\right )^{3/4}}{3 e^2 (d+e x)}\)

Input:

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

Output:

-1/3*((3*e*f - 5*d*g - 2*e*g*x)*(a + b*x + c*x^2)^(3/4))/(e^2*(d + e*x)) + 
 ((8*(3*c*e*f - 5*c*d*g + b*e*g)*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])/(Sqr 
t[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])*S 
qrt[(b^2 - 4*a*c + 4*c*(a + b*x + c*x^2))/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*S 
qrt[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)])))/(e*(b + 2*c*x)) - (Sqrt[ 
2]*(2*c*d*(3*e*f - 5*d*g) - e*(3*b*e*f - 7*b*d*g + 4*a*e*g))*(-((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...
 

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 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 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*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]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^{3/4}}{(d+e x)^2} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-1,[0,1,1,3,0,0]%%%}+%%%{1,[0,0,0,3,1,1]%%%} / %%%{1,[0,0, 
0,0,1,0]%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 24*(a + b*x + c*x**2)**(3/4)*a*e*g + 28*(a + b*x + c*x**2)**(3/4)*b*d* 
g - 24*(a + b*x + c*x**2)**(3/4)*b*e*f + 4*(a + b*x + c*x**2)**(3/4)*b*e*g 
*x - 24*(a + b*x + c*x**2)**(3/4)*c*d*g*x - 24*int((a + b*x + c*x**2)**(3/ 
4)/(a*b*d**2*e + 2*a*b*d*e**2*x + a*b*e**3*x**2 - 6*a*c*d**3 - 12*a*c*d**2 
*e*x - 6*a*c*d*e**2*x**2 + b**2*d**2*e*x + 2*b**2*d*e**2*x**2 + b**2*e**3* 
x**3 - 6*b*c*d**3*x - 11*b*c*d**2*e*x**2 - 4*b*c*d*e**2*x**3 + b*c*e**3*x* 
*4 - 6*c**2*d**3*x**2 - 12*c**2*d**2*e*x**3 - 6*c**2*d*e**2*x**4),x)*a**2* 
b*d*e**3*g - 24*int((a + b*x + c*x**2)**(3/4)/(a*b*d**2*e + 2*a*b*d*e**2*x 
 + a*b*e**3*x**2 - 6*a*c*d**3 - 12*a*c*d**2*e*x - 6*a*c*d*e**2*x**2 + b**2 
*d**2*e*x + 2*b**2*d*e**2*x**2 + b**2*e**3*x**3 - 6*b*c*d**3*x - 11*b*c*d* 
*2*e*x**2 - 4*b*c*d*e**2*x**3 + b*c*e**3*x**4 - 6*c**2*d**3*x**2 - 12*c**2 
*d**2*e*x**3 - 6*c**2*d*e**2*x**4),x)*a**2*b*e**4*g*x + 144*int((a + b*x + 
 c*x**2)**(3/4)/(a*b*d**2*e + 2*a*b*d*e**2*x + a*b*e**3*x**2 - 6*a*c*d**3 
- 12*a*c*d**2*e*x - 6*a*c*d*e**2*x**2 + b**2*d**2*e*x + 2*b**2*d*e**2*x**2 
 + b**2*e**3*x**3 - 6*b*c*d**3*x - 11*b*c*d**2*e*x**2 - 4*b*c*d*e**2*x**3 
+ b*c*e**3*x**4 - 6*c**2*d**3*x**2 - 12*c**2*d**2*e*x**3 - 6*c**2*d*e**2*x 
**4),x)*a**2*c*d**2*e**2*g + 144*int((a + b*x + c*x**2)**(3/4)/(a*b*d**2*e 
 + 2*a*b*d*e**2*x + a*b*e**3*x**2 - 6*a*c*d**3 - 12*a*c*d**2*e*x - 6*a*c*d 
*e**2*x**2 + b**2*d**2*e*x + 2*b**2*d*e**2*x**2 + b**2*e**3*x**3 - 6*b*c*d 
**3*x - 11*b*c*d**2*e*x**2 - 4*b*c*d*e**2*x**3 + b*c*e**3*x**4 - 6*c**2...