\(\int \frac {A+B x+C x^2}{(a d+(b d+a e) x+(c d+b e) x^2+c e x^3)^{3/2}} \, dx\) [131]

Optimal result

Integrand size = 45, antiderivative size = 913 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}} \, dx=-\frac {2 (d+e x) \left (A \left (b c d-b^2 e+2 a c e\right )-a (2 B c d-b C d-b B e+2 a C e)+\left (b^2 C d+2 c (A c d-a C d+a B e)-b (B c d+A c e+a C e)\right ) x\right ) \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}}-\frac {2 e \left (b^2 \left (2 C d^2-e (B d-2 A e)\right )-b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )+2 \left (A c \left (c d^2-3 a e^2\right )+a \left (a C e^2-c d (3 C d-4 B e)\right )\right )\right ) (d+e x) \left (a+b x+c x^2\right )^2}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}}+\frac {\sqrt {2} \left (b^2 \left (2 C d^2-e (B d-2 A e)\right )-b \left (2 (A c+a C) d e+B \left (c d^2+a e^2\right )\right )+2 \left (A c \left (c d^2-3 a e^2\right )+a \left (a C e^2-c d (3 C d-4 B e)\right )\right )\right ) (d+e x)^2 \left (a+b x+c x^2\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}}-\frac {2 \sqrt {2} \left (b^2 C d+2 c (A c d-a C d+a B e)-b (B c d+A c e+a C e)\right ) (d+e x) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a+b x+c x^2\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}} \] Output:

-2*(e*x+d)*(A*(2*a*c*e-b^2*e+b*c*d)-a*(-B*b*e+2*B*c*d+2*C*a*e-C*b*d)+(b^2* 
C*d+2*c*(A*c*d+B*a*e-C*a*d)-b*(A*c*e+B*c*d+C*a*e))*x)*(c*x^2+b*x+a)/(-4*a* 
c+b^2)/(a*e^2-b*d*e+c*d^2)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(3/2)-2 
*e*(b^2*(2*C*d^2-e*(-2*A*e+B*d))-b*(2*(A*c+C*a)*d*e+B*(a*e^2+c*d^2))+2*A*c 
*(-3*a*e^2+c*d^2)+2*a*(a*C*e^2-c*d*(-4*B*e+3*C*d)))*(e*x+d)*(c*x^2+b*x+a)^ 
2/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^ 
3)^(3/2)+2^(1/2)*(b^2*(2*C*d^2-e*(-2*A*e+B*d))-b*(2*(A*c+C*a)*d*e+B*(a*e^2 
+c*d^2))+2*A*c*(-3*a*e^2+c*d^2)+2*a*(a*C*e^2-c*d*(-4*B*e+3*C*d)))*(e*x+d)^ 
2*(c*x^2+b*x+a)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1+(2* 
c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-( 
b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/(-4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^2/ 
(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(a*d+(a*e+b*d)*x+(b*e+c 
*d)*x^2+c*e*x^3)^(3/2)-2*2^(1/2)*(b^2*C*d+2*c*(A*c*d+B*a*e-C*a*d)-b*(A*c*e 
+B*c*d+C*a*e))*(e*x+d)*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)* 
(c*x^2+b*x+a)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*c* 
x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+ 
(-4*a*c+b^2)^(1/2))*e))^(1/2))/c/(-4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)/(a 
*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(3/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 36.31 (sec) , antiderivative size = 10146, normalized size of antiderivative = 11.11 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2526

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2490

\(\displaystyle -\frac {\int \frac {\frac {3 c e (b C d-3 A c e+a C e)-(c d+b e) (2 c C d-3 B c e+2 b C e)}{3 c e}+(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}\right )^{3/2}}d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2}{3 c e \left (c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}\right )^{3/2}}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

\(\Big \downarrow \) 7292

\(\displaystyle -\frac {\int \left (\frac {27 \sqrt {3} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{c e \left (\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}+\frac {81 \sqrt {3} (2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )}{\left (\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}\right )^{3/2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}-\frac {2 C}{3 c e \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3}}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 25

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

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2422\) vs. \(2(867)=1734\).

Time = 2.59 (sec) , antiderivative size = 2423, normalized size of antiderivative = 2.65

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2423\)
default	\(\text {Expression too large to display}\)	\(5320\)

Input:

int((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(3/2),x,method=_ 
RETURNVERBOSE)
 

Output:

-2*c*e*((6*A*a*c*e^2-2*A*b^2*e^2+2*A*b*c*d*e-2*A*c^2*d^2+B*a*b*e^2-8*B*a*c 
*d*e+B*b^2*d*e+B*b*c*d^2-2*C*a^2*e^2+2*C*a*b*d*e+6*C*a*c*d^2-2*C*b^2*d^2)/ 
(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+2 
*a*b^2*c*d^2*e^2-8*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^2 
*c^2*d^4)*x^2+(7*A*a*b*c*e^3-2*A*a*c^2*d*e^2-2*A*b^3*e^3+A*b^2*c*d*e^2+A*b 
*c^2*d^2*e-2*A*c^3*d^3-2*B*a^2*c*e^3+B*a*b^2*e^3-5*B*a*b*c*d*e^2-2*B*a*c^2 
*d^2*e+B*b^3*d*e^2+B*b*c^2*d^3-C*a^2*b*e^3+2*C*a^2*c*d*e^2+5*C*a*b*c*d^2*e 
+2*C*a*c^2*d^3-C*b^3*d^2*e-C*b^2*c*d^3)/c/e/(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2 
*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+2*a*b^2*c*d^2*e^2-8*a*b*c^2*d^3 
*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^2*c^2*d^4)*x+(4*A*a^2*c*e^3-A*a 
*b^2*e^3+3*A*a*b*c*d*e^2-4*A*a*c^2*d^2*e-A*b^3*d*e^2+2*A*b^2*c*d^2*e-A*b*c 
^2*d^3-6*B*a^2*c*d*e^2+2*B*a*b^2*d*e^2-2*B*a*b*c*d^2*e+2*B*a*c^2*d^3-C*a^2 
*b*d*e^2+8*C*a^2*c*d^2*e-C*a*b^2*d^2*e-C*a*b*c*d^3)/c/e/(4*a^3*c*e^4-a^2*b 
^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+2*a*b^2*c*d^2*e^2-8 
*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^2*c^2*d^4))/((x^3+( 
b*e+c*d)/c/e*x^2+(a*e+b*d)/c/e*x+a*d/c/e)*c*e)^(1/2)+2*(-(15*A*a*b*c*e^3-1 
2*A*a*c^2*d*e^2-4*A*b^3*e^3+3*A*b^2*c*d*e^2+3*A*b*c^2*d^2*e-4*A*c^3*d^3-6* 
B*a^2*c*e^3+2*B*a*b^2*e^3-8*B*a*b*c*d*e^2+2*B*a*c^2*d^2*e+2*B*b^3*d*e^2-2* 
B*b^2*c*d^2*e+2*B*b*c^2*d^3-C*a^2*b*e^3+8*C*a^2*c*d*e^2-2*C*a*b^2*d*e^2+7* 
C*a*b*c*d^2*e-C*b^3*d^2*e-C*b^2*c*d^3)/(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2763 vs. \(2 (874) = 1748\).

Time = 0.20 (sec) , antiderivative size = 2763, normalized size of antiderivative = 3.03 \[ \int \frac {A+B x+C x^2}{\left (a d+(b d+a e) x+(c d+b e) x^2+c e x^3\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(3/2),x, a 
lgorithm="fricas")
 

Output:

2/3*(((C*a*b^2*c - 2*A*a*c^3 - (6*C*a^2 - B*a*b)*c^2)*d^4 + (C*a*b^3 + (10 
*B*a^2 + 3*A*a*b)*c^2 - (C*a^2*b + 4*B*a*b^2)*c)*d^3*e - (4*C*a^2*b^2 - B* 
a*b^3 + 18*A*a^2*c^2 - (10*C*a^3 - B*a^2*b + 3*A*a*b^2)*c)*d^2*e^2 + (C*a^ 
3*b + B*a^2*b^2 - 2*A*a*b^3 - 3*(2*B*a^3 - 3*A*a^2*b)*c)*d*e^3 + ((C*b^2*c 
^2 - 2*A*c^4 - (6*C*a - B*b)*c^3)*d^3*e + (C*b^3*c + (10*B*a + 3*A*b)*c^3 
- (C*a*b + 4*B*b^2)*c^2)*d^2*e^2 - (18*A*a*c^3 - (10*C*a^2 - B*a*b + 3*A*b 
^2)*c^2 + (4*C*a*b^2 - B*b^3)*c)*d*e^3 - (3*(2*B*a^2 - 3*A*a*b)*c^2 - (C*a 
^2*b + B*a*b^2 - 2*A*b^3)*c)*e^4)*x^3 + ((C*b^2*c^2 - 2*A*c^4 - (6*C*a - B 
*b)*c^3)*d^4 + (2*C*b^3*c + (10*B*a + A*b)*c^3 - (7*C*a*b + 3*B*b^2)*c^2)* 
d^3*e + (C*b^4 - 18*A*a*c^3 + (10*C*a^2 + 9*B*a*b + 6*A*b^2)*c^2 - (5*C*a* 
b^2 + 3*B*b^3)*c)*d^2*e^2 - (4*C*a*b^3 - B*b^4 + 3*(2*B*a^2 + 3*A*a*b)*c^2 
 - (11*C*a^2*b + A*b^3)*c)*d*e^3 + (C*a^2*b^2 + B*a*b^3 - 2*A*b^4 - 3*(2*B 
*a^2*b - 3*A*a*b^2)*c)*e^4)*x^2 + ((C*b^3*c - 2*A*b*c^3 - (6*C*a*b - B*b^2 
)*c^2)*d^4 + (C*b^4 - 4*B*b^3*c - 2*A*a*c^3 - (6*C*a^2 - 11*B*a*b - 3*A*b^ 
2)*c^2)*d^3*e - (3*C*a*b^3 - B*b^4 - 5*(2*B*a^2 - 3*A*a*b)*c^2 - (9*C*a^2* 
b - 5*B*a*b^2 + 3*A*b^3)*c)*d^2*e^2 - (3*C*a^2*b^2 - 2*B*a*b^3 + 2*A*b^4 + 
 18*A*a^2*c^2 - (10*C*a^3 - 7*B*a^2*b + 12*A*a*b^2)*c)*d*e^3 + (C*a^3*b + 
B*a^2*b^2 - 2*A*a*b^3 - 3*(2*B*a^3 - 3*A*a^2*b)*c)*e^4)*x)*sqrt(c*e)*weier 
strassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/2 
7*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a...
 

Sympy [F]

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

integrate((C*x**2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3)**(3/2), 
x)
 

Output:

Integral((A + B*x + C*x**2)/((d + e*x)*(a + b*x + c*x**2))**(3/2), x)
 

Maxima [F]

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

integrate((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(3/2),x, a 
lgorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((C*x^2+B*x+A)/(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(3/2),x, a 
lgorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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