\(\int \frac {A+B x+C x^2}{(-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3)^{7/2}} \, dx\) [79]

Optimal result

Integrand size = 43, antiderivative size = 609 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=-\frac {2 C}{15 c^3 \left (-64+(b+c x)^3\right )^{5/2}}-\frac {(b+c x) \left (A c^2-b^2 C+c (B c-2 b C) x\right )}{480 c^3 \left (-64+(b+c x)^3\right )^{5/2}}-\frac {(b+c x) \left (13 \left (b B c-A c^2-b^2 C\right )-11 (B c-2 b C) (b+c x)\right )}{276480 c^3 \left (-64+(b+c x)^3\right )^{3/2}}+\frac {(b+c x) \left (91 \left (b B c-A c^2-b^2 C\right )-55 (B c-2 b C) (b+c x)\right )}{53084160 c^3 \sqrt {-64+(b+c x)^3}}-\frac {11 (B c-2 b C) \sqrt {-64+(b+c x)^3}}{10616832 c^3 \left (4-4 \sqrt {3}-b-c x\right )}+\frac {11 \sqrt {2+\sqrt {3}} (B c-2 b C) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} E\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right )|-7+4 \sqrt {3}\right )}{3538944\ 3^{3/4} c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}}-\frac {\sqrt {2-\sqrt {3}} \left (220 \left (1+\sqrt {3}\right ) (B c-2 b C)+91 \left (b B c-A c^2-b^2 C\right )\right ) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{106168320 \sqrt [4]{3} c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:

-2/15*C/c^3/(-64+(c*x+b)^3)^(5/2)-1/480*(c*x+b)*(A*c^2-C*b^2+c*(B*c-2*C*b) 
*x)/c^3/(-64+(c*x+b)^3)^(5/2)-1/276480*(c*x+b)*(-13*A*c^2+13*B*b*c-13*C*b^ 
2-11*(B*c-2*C*b)*(c*x+b))/c^3/(-64+(c*x+b)^3)^(3/2)+1/53084160*(c*x+b)*(-9 
1*A*c^2+91*B*b*c-91*C*b^2-55*(B*c-2*C*b)*(c*x+b))/c^3/(-64+(c*x+b)^3)^(1/2 
)-11/10616832*(B*c-2*C*b)*(-64+(c*x+b)^3)^(1/2)/c^3/(4-4*3^(1/2)-b-c*x)+11 
/10616832*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(B*c-2*C*b)*(-c*x-b+4)*((16+4* 
c*x+4*b+(c*x+b)^2)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)*EllipticE((4+4*3^(1/2)-b-c 
*x)/(4-4*3^(1/2)-b-c*x),2*I-I*3^(1/2))/c^3/(-(-c*x-b+4)/(4-4*3^(1/2)-b-c*x 
)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)-1/318504960*(1/2*6^(1/2)-1/2*2^(1/2))*(22 
0*(1+3^(1/2))*(B*c-2*C*b)-91*A*c^2+91*B*b*c-91*C*b^2)*(-c*x-b+4)*((16+4*c* 
x+4*b+(c*x+b)^2)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)*EllipticF((4+4*3^(1/2)-b-c*x 
)/(4-4*3^(1/2)-b-c*x),2*I-I*3^(1/2))*3^(3/4)/c^3/(-(-c*x-b+4)/(4-4*3^(1/2) 
-b-c*x)^2)^(1/2)/(-64+(c*x+b)^3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 15.14 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.77 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\frac {2 \left (36 b^2 B c-91 A b c^2+19 b^3 C-19 b B c^2 x-91 A c^3 x+129 b^2 c C x-55 B c^3 x^2+110 b c^2 C x^2-\frac {192 (b+c x) \left (9 b^2 C-c^2 (13 A+11 B x)+2 b c (B+11 C x)\right )}{-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}-\frac {110592 \left (A c^2 (b+c x)+B c^2 x (b+c x)-C \left (-64+b^3+3 b^2 c x+2 b c^2 x^2\right )\right )}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^2}\right )+\sqrt {2} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \left (110 \left (3 i+\sqrt {3}\right ) (B c-2 b C) E\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )-i \left ((220+91 b) B c-91 A c^2-b (440+91 b) C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{106168320 c^3 \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}} \] Input:

Integrate[(A + B*x + C*x^2)/(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3 
)^(7/2),x]
 

Output:

(2*(36*b^2*B*c - 91*A*b*c^2 + 19*b^3*C - 19*b*B*c^2*x - 91*A*c^3*x + 129*b 
^2*c*C*x - 55*B*c^3*x^2 + 110*b*c^2*C*x^2 - (192*(b + c*x)*(9*b^2*C - c^2* 
(13*A + 11*B*x) + 2*b*c*(B + 11*C*x)))/(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^ 
2 + c^3*x^3) - (110592*(A*c^2*(b + c*x) + B*c^2*x*(b + c*x) - C*(-64 + b^3 
 + 3*b^2*c*x + 2*b*c^2*x^2)))/(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x 
^3)^2) + Sqrt[2]*Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3])]*Sqrt[16 + b^2 
 + 4*c*x + c^2*x^2 + 2*b*(2 + c*x)]*(110*(3*I + Sqrt[3])*(B*c - 2*b*C)*Ell 
ipticE[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3] 
)/(3*I + Sqrt[3])] - I*((220 + 91*b)*B*c - 91*A*c^2 - b*(440 + 91*b)*C)*El 
lipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3 
])/(3*I + Sqrt[3])]))/(106168320*c^3*Sqrt[-64 + b^3 + 3*b^2*c*x + 3*b*c^2* 
x^2 + c^3*x^3])
 

Rubi [A] (warning: unable to verify)

Time = 1.70 (sec) , antiderivative size = 733, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2459, 2393, 27, 2394, 27, 2394, 27, 2419, 760, 2418}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{\left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 2459

\(\displaystyle \int \frac {A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )+C \left (\frac {b}{c}+x\right )^2}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{7/2}}d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 2393

\(\displaystyle \frac {1}{480} \int -\frac {13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}d\left (\frac {b}{c}+x\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{960} \int \frac {13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}d\left (\frac {b}{c}+x\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 2394

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {1}{288} \int \frac {91 \left (-C b^2+B c b-A c^2\right )-55 c^2 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{2 c^2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}d\left (\frac {b}{c}+x\right )\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {\int \frac {91 \left (-C b^2+B c b-A c^2\right )-55 c (B c-2 b C) \left (\frac {b}{c}+x\right )}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}d\left (\frac {b}{c}+x\right )}{576 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 2394

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {\frac {1}{96} \int -\frac {91 \left (-C b^2+B c b-A c^2\right )+55 c (B c-2 b C) \left (\frac {b}{c}+x\right )}{2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\frac {\left (\frac {b}{c}+x\right ) \left (91 \left (-A c^2+b^2 (-C)+b B c\right )-55 c \left (\frac {b}{c}+x\right ) (B c-2 b C)\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}}{576 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {-\frac {1}{192} \int \frac {91 \left (-C b^2+B c b-A c^2\right )+55 c (B c-2 b C) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\frac {\left (\frac {b}{c}+x\right ) \left (91 \left (-A c^2+b^2 (-C)+b B c\right )-55 c \left (\frac {b}{c}+x\right ) (B c-2 b C)\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}}{576 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 2419

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {\frac {1}{192} \left (55 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\left (91 \left (-A c^2+b^2 (-C)+b B c\right )+220 \left (1+\sqrt {3}\right ) (B c-2 b C)\right ) \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )-\frac {\left (\frac {b}{c}+x\right ) \left (91 \left (-A c^2+b^2 (-C)+b B c\right )-55 c \left (\frac {b}{c}+x\right ) (B c-2 b C)\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}}{576 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {\frac {1}{192} \left (55 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )+\frac {\sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (91 \left (-A c^2+b^2 (-C)+b B c\right )+220 \left (1+\sqrt {3}\right ) (B c-2 b C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )-\frac {\left (\frac {b}{c}+x\right ) \left (91 \left (-A c^2+b^2 (-C)+b B c\right )-55 c \left (\frac {b}{c}+x\right ) (B c-2 b C)\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}}{576 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {1}{960} \left (\frac {\left (\frac {b}{c}+x\right ) \left (13 \left (A-\frac {b (B c-b C)}{c^2}\right )+11 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{288 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}-\frac {\frac {1}{192} \left (\frac {\sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (91 \left (-A c^2+b^2 (-C)+b B c\right )+220 \left (1+\sqrt {3}\right ) (B c-2 b C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}+55 (B c-2 b C) \left (\frac {2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}{c \left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )}-\frac {2 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} E\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right )|-7+4 \sqrt {3}\right )}{c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}\right )\right )-\frac {\left (\frac {b}{c}+x\right ) \left (91 \left (-A c^2+b^2 (-C)+b B c\right )-55 c \left (\frac {b}{c}+x\right ) (B c-2 b C)\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}}{576 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{480 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}}\)

Input:

Int[(A + B*x + C*x^2)/(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(7/2 
),x]
 

Output:

-1/480*(64*C + c^3*(b/c + x)*(A - (b*(B*c - b*C))/c^2 + (B - (2*b*C)/c)*(b 
/c + x)))/(c^3*(-64 + c^3*(b/c + x)^3)^(5/2)) + (((b/c + x)*(13*(A - (b*(B 
*c - b*C))/c^2) + 11*(B - (2*b*C)/c)*(b/c + x)))/(288*(-64 + c^3*(b/c + x) 
^3)^(3/2)) - (-1/96*((b/c + x)*(91*(b*B*c - A*c^2 - b^2*C) - 55*c*(B*c - 2 
*b*C)*(b/c + x)))/Sqrt[-64 + c^3*(b/c + x)^3] + (55*(B*c - 2*b*C)*((2*Sqrt 
[-64 + c^3*(b/c + x)^3])/(c*(4*(1 - Sqrt[3]) - c*(b/c + x))) - (2*3^(1/4)* 
Sqrt[2 + Sqrt[3]]*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x) + c^2*(b/c + 
x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x))^2]*EllipticE[ArcSin[(4*(1 + Sqrt[3]) 
 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqrt[3]])/(c*Sqrt 
[-((4 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]*Sqrt[-64 + c^3*(b 
/c + x)^3])) + (Sqrt[2 - Sqrt[3]]*(220*(1 + Sqrt[3])*(B*c - 2*b*C) + 91*(b 
*B*c - A*c^2 - b^2*C))*(4 - c*(b/c + x))*Sqrt[(16 + 4*c*(b/c + x) + c^2*(b 
/c + x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x))^2]*EllipticF[ArcSin[(4*(1 + Sqr 
t[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))], -7 + 4*Sqrt[3]])/(3 
^(1/4)*c*Sqrt[-((4 - c*(b/c + x))/(4*(1 - Sqrt[3]) - c*(b/c + x))^2)]*Sqrt 
[-64 + c^3*(b/c + x)^3]))/192)/(576*c^2))/960
 

Defintions of rubi rules used

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 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 2393

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, 
 x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q 
, x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1))   In 
t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( 
p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n 
, 0] && LtQ[p, -1]
 

rule 2394

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b 
*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1))   Int[ExpandToSum[n 
*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x 
] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S 
imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S 
qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 
3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && 
 EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

rule 2419

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(c*r - (1 + Sqrt[3])*d*s)/r 
  Int[1/Sqrt[a + b*x^3], x], x] + Simp[d/r   Int[((1 + Sqrt[3])*s + r*x)/Sq 
rt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && NeQ[b*c^3 - 
2*(5 + 3*Sqrt[3])*a*d^3, 0]
 

rule 2459

Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 
]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x 
 -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial 
Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - 
> x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ 
[Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] &&  !(MonomialQ[Qx, x] 
&& IGtQ[p, 0])
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1116 vs. \(2 (539 ) = 1078\).

Time = 1.78 (sec) , antiderivative size = 1117, normalized size of antiderivative = 1.83

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1117\)
default	\(\text {Expression too large to display}\)	\(2396\)

Input:

int((C*x^2+B*x+A)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x,method=_R 
ETURNVERBOSE)
 

Output:

(-1/480*(B*c-2*C*b)/c^10*x^2-1/480*(A*c^2+B*b*c-3*C*b^2)/c^11*x-1/480*(A*b 
*c^2-C*b^3+64*C)/c^12)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)/(x^3+3 
*b/c*x^2+3*b^2/c^2*x+(b^3-64)/c^3)^3+(11/276480*(B*c-2*C*b)/c^7*x^2+1/2764 
80/c^8*(13*A*c^2+9*B*b*c-31*C*b^2)*x+1/276480*b/c^9*(13*A*c^2-2*B*b*c-9*C* 
b^2))*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)/(x^3+3*b/c*x^2+3*b^2/c^ 
2*x+(b^3-64)/c^3)^2-2*c^3*(11/21233664*(B*c-2*C*b)/c^4*x^2+1/106168320/c^5 
*(91*A*c^2+19*B*b*c-129*C*b^2)*x+1/106168320*b/c^6*(91*A*c^2-36*B*b*c-19*C 
*b^2))/((x^3+3*b/c*x^2+3*b^2/c^2*x+(b^3-64)/c^3)*c^3)^(1/2)+2*(-1/35389440 
/c^2*(91*A*c^2-36*B*b*c-19*C*b^2)+1/53084160/c^2*(91*A*c^2+19*B*b*c-129*C* 
b^2))*((-b-2-2*I*3^(1/2))/c+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b 
-4)/c))^(1/2)*((x-(-b-2+2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^( 
1/2)*((x-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2-2*I*3^(1/2))/c))^(1/2)/(c^3 
*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)*EllipticF(((x+(b-4)/c)/((-b-2-2*I 
*3^(1/2))/c+(b-4)/c))^(1/2),((-(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b 
-2+2*I*3^(1/2))/c))^(1/2))+11/10616832/c*(B*c-2*C*b)*((-b-2-2*I*3^(1/2))/c 
+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2)*((x-(-b-2+2*I 
*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2)*((x-(-b-2-2*I*3^(1/2)) 
/c)/(-(b-4)/c-(-b-2-2*I*3^(1/2))/c))^(1/2)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+ 
b^3-64)^(1/2)*((-(b-4)/c-(-b-2+2*I*3^(1/2))/c)*EllipticE(((x+(b-4)/c)/((-b 
-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2),((-(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(-(b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (494) = 988\).

Time = 0.19 (sec) , antiderivative size = 1608, normalized size of antiderivative = 2.64 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x, al 
gorithm="fricas")
 

Output:

-1/53084160*(91*(C*b^11 + (C*b^2*c^9 - B*b*c^10 + A*c^11)*x^9 - 192*C*b^8 
+ 9*(C*b^3*c^8 - B*b^2*c^9 + A*b*c^10)*x^8 + 36*(C*b^4*c^7 - B*b^3*c^8 + A 
*b^2*c^9)*x^7 + 12*((7*A*b^3 - 16*A)*c^8 - (7*B*b^4 - 16*B*b)*c^7 + (7*C*b 
^5 - 16*C*b^2)*c^6)*x^6 + 12288*C*b^5 + 18*((7*A*b^4 - 64*A*b)*c^7 - (7*B* 
b^5 - 64*B*b^2)*c^6 + (7*C*b^6 - 64*C*b^3)*c^5)*x^5 + 18*((7*A*b^5 - 160*A 
*b^2)*c^6 - (7*B*b^6 - 160*B*b^3)*c^5 + (7*C*b^7 - 160*C*b^4)*c^4)*x^4 + 1 
2*((7*A*b^6 - 320*A*b^3 + 1024*A)*c^5 - (7*B*b^7 - 320*B*b^4 + 1024*B*b)*c 
^4 + (7*C*b^8 - 320*C*b^5 + 1024*C*b^2)*c^3)*x^3 - 262144*C*b^2 + (A*b^9 - 
 192*A*b^6 + 12288*A*b^3 - 262144*A)*c^2 + 36*((A*b^7 - 80*A*b^4 + 1024*A* 
b)*c^4 - (B*b^8 - 80*B*b^5 + 1024*B*b^2)*c^3 + (C*b^9 - 80*C*b^6 + 1024*C* 
b^3)*c^2)*x^2 - (B*b^10 - 192*B*b^7 + 12288*B*b^4 - 262144*B*b)*c + 9*((A* 
b^8 - 128*A*b^5 + 4096*A*b^2)*c^3 - (B*b^9 - 128*B*b^6 + 4096*B*b^3)*c^2 + 
 (C*b^10 - 128*C*b^7 + 4096*C*b^4)*c)*x)*sqrt(c^3)*weierstrassPInverse(0, 
256/c^3, (c*x + b)/c) - 55*((2*C*b*c^10 - B*c^11)*x^9 + 9*(2*C*b^2*c^9 - B 
*b*c^10)*x^8 + 36*(2*C*b^3*c^8 - B*b^2*c^9)*x^7 - 12*((7*B*b^3 - 16*B)*c^8 
 - 2*(7*C*b^4 - 16*C*b)*c^7)*x^6 - 18*((7*B*b^4 - 64*B*b)*c^7 - 2*(7*C*b^5 
 - 64*C*b^2)*c^6)*x^5 - 18*((7*B*b^5 - 160*B*b^2)*c^6 - 2*(7*C*b^6 - 160*C 
*b^3)*c^5)*x^4 - 12*((7*B*b^6 - 320*B*b^3 + 1024*B)*c^5 - 2*(7*C*b^7 - 320 
*C*b^4 + 1024*C*b)*c^4)*x^3 - (B*b^9 - 192*B*b^6 + 12288*B*b^3 - 262144*B) 
*c^2 - 36*((B*b^7 - 80*B*b^4 + 1024*B*b)*c^4 - 2*(C*b^8 - 80*C*b^5 + 10...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate((C*x**2+B*x+A)/(c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**(7/ 
2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x, al 
gorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64) 
^(7/2), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x, al 
gorithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)/(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64) 
^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\int \frac {C\,x^2+B\,x+A}{{\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{7/2}} \,d x \] Input:

int((A + B*x + C*x^2)/(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(7/2) 
,x)
 

Output:

int((A + B*x + C*x^2)/(b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(7/2) 
, x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{7/2}} \, dx=\text {too large to display} \] Input:

int((C*x^2+B*x+A)/(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(7/2),x)
 

Output:

( - 2*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) + 30*int(sq 
rt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)/(b**12 + 12*b**11*c 
*x + 66*b**10*c**2*x**2 + 220*b**9*c**3*x**3 - 256*b**9 + 495*b**8*c**4*x* 
*4 - 2304*b**8*c*x + 792*b**7*c**5*x**5 - 9216*b**7*c**2*x**2 + 924*b**6*c 
**6*x**6 - 21504*b**6*c**3*x**3 + 24576*b**6 + 792*b**5*c**7*x**7 - 32256* 
b**5*c**4*x**4 + 147456*b**5*c*x + 495*b**4*c**8*x**8 - 32256*b**4*c**5*x* 
*5 + 368640*b**4*c**2*x**2 + 220*b**3*c**9*x**9 - 21504*b**3*c**6*x**6 + 4 
91520*b**3*c**3*x**3 - 1048576*b**3 + 66*b**2*c**10*x**10 - 9216*b**2*c**7 
*x**7 + 368640*b**2*c**4*x**4 - 3145728*b**2*c*x + 12*b*c**11*x**11 - 2304 
*b*c**8*x**8 + 147456*b*c**5*x**5 - 3145728*b*c**2*x**2 + c**12*x**12 - 25 
6*c**9*x**9 + 24576*c**6*x**6 - 1048576*c**3*x**3 + 16777216),x)*a*b**9*c* 
*2 + 270*int(sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)/(b** 
12 + 12*b**11*c*x + 66*b**10*c**2*x**2 + 220*b**9*c**3*x**3 - 256*b**9 + 4 
95*b**8*c**4*x**4 - 2304*b**8*c*x + 792*b**7*c**5*x**5 - 9216*b**7*c**2*x* 
*2 + 924*b**6*c**6*x**6 - 21504*b**6*c**3*x**3 + 24576*b**6 + 792*b**5*c** 
7*x**7 - 32256*b**5*c**4*x**4 + 147456*b**5*c*x + 495*b**4*c**8*x**8 - 322 
56*b**4*c**5*x**5 + 368640*b**4*c**2*x**2 + 220*b**3*c**9*x**9 - 21504*b** 
3*c**6*x**6 + 491520*b**3*c**3*x**3 - 1048576*b**3 + 66*b**2*c**10*x**10 - 
 9216*b**2*c**7*x**7 + 368640*b**2*c**4*x**4 - 3145728*b**2*c*x + 12*b*c** 
11*x**11 - 2304*b*c**8*x**8 + 147456*b*c**5*x**5 - 3145728*b*c**2*x**2 ...