\(\int \frac {A+B x+C x^2}{(-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3)^{5/2}} \, dx\) [78]

Optimal result

Integrand size = 43, antiderivative size = 549 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=-\frac {2 C}{9 c^3 \left (-64+(b+c x)^3\right )^{3/2}}-\frac {(b+c x) \left (A c^2-b^2 C+c (B c-2 b C) x\right )}{288 c^3 \left (-64+(b+c x)^3\right )^{3/2}}-\frac {(b+c x) \left (7 \left (b B c-A c^2-b^2 C\right )-5 (B c-2 b C) (b+c x)\right )}{55296 c^3 \sqrt {-64+(b+c x)^3}}+\frac {5 (B c-2 b C) \sqrt {-64+(b+c x)^3}}{55296 c^3 \left (4-4 \sqrt {3}-b-c x\right )}-\frac {5 \sqrt {2+\sqrt {3}} (B c-2 b C) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} E\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right )|-7+4 \sqrt {3}\right )}{18432\ 3^{3/4} c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}}+\frac {\sqrt {2-\sqrt {3}} \left (20 \left (1+\sqrt {3}\right ) (B c-2 b C)+7 \left (b B c-A c^2-b^2 C\right )\right ) (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{110592 \sqrt [4]{3} c^3 \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:

-2/9*C/c^3/(-64+(c*x+b)^3)^(3/2)-1/288*(c*x+b)*(A*c^2-C*b^2+c*(B*c-2*C*b)* 
x)/c^3/(-64+(c*x+b)^3)^(3/2)-1/55296*(c*x+b)*(-7*A*c^2+7*B*b*c-7*C*b^2-5*( 
B*c-2*C*b)*(c*x+b))/c^3/(-64+(c*x+b)^3)^(1/2)+5/55296*(B*c-2*C*b)*(-64+(c* 
x+b)^3)^(1/2)/c^3/(4-4*3^(1/2)-b-c*x)-5/55296*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)+1/331 
776*(1/2*6^(1/2)-1/2*2^(1/2))*(20*(1+3^(1/2))*(B*c-2*C*b)-7*A*c^2+7*B*b*c- 
7*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 = 12.25 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.73 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=\frac {2 \left (-2 b^2 B c+7 A b c^2-3 b^3 C+3 b B c^2 x+7 A c^3 x-13 b^2 c C x+5 B c^3 x^2-10 b c^2 C x^2-\frac {192 \left (A c^2 (b+c x)+B c^2 x (b+c x)-C \left (-64+b^3+3 b^2 c x+2 b c^2 x^2\right )\right )}{-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}\right )+\sqrt {2} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \left (-10 \left (3 i+\sqrt {3}\right ) (B c-2 b C) E\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right )|\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )+i \left ((20+7 b) B c-7 A c^2-b (40+7 b) C\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{110592 c^3 \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}} \] 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 
)^(5/2),x]
 

Output:

(2*(-2*b^2*B*c + 7*A*b*c^2 - 3*b^3*C + 3*b*B*c^2*x + 7*A*c^3*x - 13*b^2*c* 
C*x + 5*B*c^3*x^2 - 10*b*c^2*C*x^2 - (192*(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)) + 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)]*(-10*(3*I + Sqrt[3])* 
(B*c - 2*b*C)*EllipticE[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1 
/4))], (2*Sqrt[3])/(3*I + Sqrt[3])] + I*((20 + 7*b)*B*c - 7*A*c^2 - b*(40 
+ 7*b)*C)*EllipticF[ArcSin[Sqrt[2*I + 2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4)) 
], (2*Sqrt[3])/(3*I + Sqrt[3])]))/(110592*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 = 1.49 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {2459, 2393, 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.

\(\displaystyle \int \frac {A+B x+C x^2}{\left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 2459

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

\(\Big \downarrow \) 2393

\(\displaystyle \frac {1}{288} \int -\frac {7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{2 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}d\left (\frac {b}{c}+x\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{576} \int \frac {7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right )}{\left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}d\left (\frac {b}{c}+x\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

\(\Big \downarrow \) 2394

\(\displaystyle \frac {1}{576} \left (\frac {\left (\frac {b}{c}+x\right ) \left (7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {1}{96} \int \frac {5 \left (B-\frac {2 b C}{c}\right ) \left (\frac {b}{c}+x\right ) c^2+7 \left (-C b^2+B c b-A c^2\right )}{2 c^2 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{576} \left (\frac {\left (\frac {b}{c}+x\right ) \left (7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {\int \frac {7 \left (-C b^2+B c b-A c^2\right )+5 c (B c-2 b C) \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{192 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

\(\Big \downarrow \) 2419

\(\displaystyle \frac {1}{576} \left (\frac {\left (\frac {b}{c}+x\right ) \left (7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {\left (7 \left (-A c^2+b^2 (-C)+b B c\right )+20 \left (1+\sqrt {3}\right ) (B c-2 b C)\right ) \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-5 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )}{192 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {1}{576} \left (\frac {\left (\frac {b}{c}+x\right ) \left (7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {-5 (B c-2 b C) \int \frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )-\frac {\sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \left (7 \left (-A c^2+b^2 (-C)+b B c\right )+20 \left (1+\sqrt {3}\right ) (B c-2 b C)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}}{192 c^2}\right )-\frac {c^3 \left (\frac {b}{c}+x\right ) \left (A-\frac {b (B c-b C)}{c^2}+\left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )+64 C}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {1}{576} \left (\frac {\left (\frac {b}{c}+x\right ) \left (7 \left (A-\frac {b (B c-b C)}{c^2}\right )+5 \left (\frac {b}{c}+x\right ) \left (B-\frac {2 b C}{c}\right )\right )}{96 \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}-\frac {-\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 (7 \left (-A c^2+b^2 (-C)+b B c\right )+20 \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}}-5 (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 )}{192 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}{288 c^3 \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}}\)

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)^(5/2 
),x]
 

Output:

-1/288*(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)^(3/2)) + (((b/c + x)*(7*(A - (b*(B* 
c - b*C))/c^2) + 5*(B - (2*b*C)/c)*(b/c + x)))/(96*Sqrt[-64 + c^3*(b/c + x 
)^3]) - (-5*(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]]*(20*(1 
 + Sqrt[3])*(B*c - 2*b*C) + 7*(b*B*c - A*c^2 - b^2*C))*(4 - c*(b/c + x))*S 
qrt[(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 + Sqrt[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*c^2))/576
 

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. 987 vs. \(2 (485 ) = 970\).

Time = 1.62 (sec) , antiderivative size = 988, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\frac {\left (-\frac {\left (B c -2 b C \right ) x^{2}}{288 c^{7}}-\frac {\left (A \,c^{2}+b B c -3 b^{2} C \right ) x}{288 c^{8}}-\frac {c^{2} b A -C \,b^{3}+64 C}{288 c^{9}}\right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right )^{2}}-\frac {2 c^{3} \left (-\frac {5 \left (B c -2 b C \right ) x^{2}}{110592 c^{4}}-\frac {\left (7 A \,c^{2}+3 b B c -13 b^{2} C \right ) x}{110592 c^{5}}-\frac {b \left (7 A \,c^{2}-2 b B c -3 b^{2} C \right )}{110592 c^{6}}\right )}{\sqrt {\left (x^{3}+\frac {3 b \,x^{2}}{c}+\frac {3 b^{2} x}{c^{2}}+\frac {b^{3}-64}{c^{3}}\right ) c^{3}}}+\frac {2 \left (\frac {7 A \,c^{2}-2 b B c -3 b^{2} C}{36864 c^{2}}-\frac {7 A \,c^{2}+3 b B c -13 b^{2} C}{55296 c^{2}}\right ) \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}-\frac {5 \left (B c -2 b C \right ) \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \left (\left (-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )+\frac {\left (-b -2+2 i \sqrt {3}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{c}\right )}{55296 c \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}\)	\(988\)
default	\(\text {Expression too large to display}\)	\(2140\)

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)^(5/2),x,method=_R 
ETURNVERBOSE)
 

Output:

(-1/288*(B*c-2*C*b)/c^7*x^2-1/288*(A*c^2+B*b*c-3*C*b^2)/c^8*x-1/288*(A*b*c 
^2-C*b^3+64*C)/c^9)*(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*(-5/110592*(B*c-2*C*b)/c^4*x^2-1/1 
10592/c^5*(7*A*c^2+3*B*b*c-13*C*b^2)*x-1/110592*b/c^6*(7*A*c^2-2*B*b*c-3*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*(1/36864/c^2 
*(7*A*c^2-2*B*b*c-3*C*b^2)-1/55296/c^2*(7*A*c^2+3*B*b*c-13*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))-5/55296/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-4)/c-(-b-2+2*I*3^(1 
/2))/c))^(1/2))+(-b-2+2*I*3^(1/2))/c*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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 923 vs. \(2 (439) = 878\).

Time = 0.11 (sec) , antiderivative size = 923, normalized size of antiderivative = 1.68 \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx =\text {Too large to display} \] 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)^(5/2),x, al 
gorithm="fricas")
 

Output:

1/55296*(7*(C*b^8 + (C*b^2*c^6 - B*b*c^7 + A*c^8)*x^6 - 128*C*b^5 + 6*(C*b 
^3*c^5 - B*b^2*c^6 + A*b*c^7)*x^5 + 15*(C*b^4*c^4 - B*b^3*c^5 + A*b^2*c^6) 
*x^4 + 4*((5*A*b^3 - 32*A)*c^5 - (5*B*b^4 - 32*B*b)*c^4 + (5*C*b^5 - 32*C* 
b^2)*c^3)*x^3 + 4096*C*b^2 + (A*b^6 - 128*A*b^3 + 4096*A)*c^2 + 3*((5*A*b^ 
4 - 128*A*b)*c^4 - (5*B*b^5 - 128*B*b^2)*c^3 + (5*C*b^6 - 128*C*b^3)*c^2)* 
x^2 - (B*b^7 - 128*B*b^4 + 4096*B*b)*c + 6*((A*b^5 - 64*A*b^2)*c^3 - (B*b^ 
6 - 64*B*b^3)*c^2 + (C*b^7 - 64*C*b^4)*c)*x)*sqrt(c^3)*weierstrassPInverse 
(0, 256/c^3, (c*x + b)/c) - 5*((2*C*b*c^7 - B*c^8)*x^6 + 6*(2*C*b^2*c^6 - 
B*b*c^7)*x^5 + 15*(2*C*b^3*c^5 - B*b^2*c^6)*x^4 - 4*((5*B*b^3 - 32*B)*c^5 
- 2*(5*C*b^4 - 32*C*b)*c^4)*x^3 - (B*b^6 - 128*B*b^3 + 4096*B)*c^2 - 3*((5 
*B*b^4 - 128*B*b)*c^4 - 2*(5*C*b^5 - 128*C*b^2)*c^3)*x^2 + 2*(C*b^7 - 128* 
C*b^4 + 4096*C*b)*c - 6*((B*b^5 - 64*B*b^2)*c^3 - 2*(C*b^6 - 64*C*b^3)*c^2 
)*x)*sqrt(c^3)*weierstrassZeta(0, 256/c^3, weierstrassPInverse(0, 256/c^3, 
 (c*x + b)/c)) - sqrt(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)*(5*(2* 
C*b*c^7 - B*c^8)*x^5 - (7*A*b^4 - 640*A*b)*c^4 + (43*C*b^2*c^6 - 18*B*b*c^ 
7 - 7*A*c^8)*x^4 + 2*(B*b^5 - 64*B*b^2)*c^3 + 2*(36*C*b^3*c^5 - 11*B*b^2*c 
^6 - 14*A*b*c^7)*x^3 + 3*(C*b^6 - 128*C*b^3 + 4096*C)*c^2 - 2*(21*A*b^2*c^ 
6 + 4*(B*b^3 - 64*B)*c^5 - (29*C*b^4 - 512*C*b)*c^4)*x^2 - (4*(7*A*b^3 - 1 
60*A)*c^5 - 3*(B*b^4 + 128*B*b)*c^4 - 22*(C*b^5 - 64*C*b^2)*c^3)*x))/(c^11 
*x^6 + 6*b*c^10*x^5 + 15*b^2*c^9*x^4 + 4*(5*b^3 - 32)*c^8*x^3 + 3*(5*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=\text {Timed out} \] 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)**(5/ 
2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {5}{2}}} \,d x } \] 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)^(5/2),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) 
^(5/2), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {5}{2}}} \,d x } \] 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)^(5/2),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) 
^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=\int \frac {C\,x^2+B\,x+A}{{\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{5/2}} \,d x \] 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)^(5/2) 
,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)^(5/2) 
, x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2}} \, dx=\text {too large to display} \] 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)^(5/2),x)
 

Output:

( - 2*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) + 18*int(sq 
rt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)/(b**9 + 9*b**8*c*x 
+ 36*b**7*c**2*x**2 + 84*b**6*c**3*x**3 - 192*b**6 + 126*b**5*c**4*x**4 - 
1152*b**5*c*x + 126*b**4*c**5*x**5 - 2880*b**4*c**2*x**2 + 84*b**3*c**6*x* 
*6 - 3840*b**3*c**3*x**3 + 12288*b**3 + 36*b**2*c**7*x**7 - 2880*b**2*c**4 
*x**4 + 36864*b**2*c*x + 9*b*c**8*x**8 - 1152*b*c**5*x**5 + 36864*b*c**2*x 
**2 + c**9*x**9 - 192*c**6*x**6 + 12288*c**3*x**3 - 262144),x)*a*b**6*c**2 
 + 108*int(sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)/(b**9 
+ 9*b**8*c*x + 36*b**7*c**2*x**2 + 84*b**6*c**3*x**3 - 192*b**6 + 126*b**5 
*c**4*x**4 - 1152*b**5*c*x + 126*b**4*c**5*x**5 - 2880*b**4*c**2*x**2 + 84 
*b**3*c**6*x**6 - 3840*b**3*c**3*x**3 + 12288*b**3 + 36*b**2*c**7*x**7 - 2 
880*b**2*c**4*x**4 + 36864*b**2*c*x + 9*b*c**8*x**8 - 1152*b*c**5*x**5 + 3 
6864*b*c**2*x**2 + c**9*x**9 - 192*c**6*x**6 + 12288*c**3*x**3 - 262144),x 
)*a*b**5*c**3*x + 270*int(sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x* 
*3 - 64)/(b**9 + 9*b**8*c*x + 36*b**7*c**2*x**2 + 84*b**6*c**3*x**3 - 192* 
b**6 + 126*b**5*c**4*x**4 - 1152*b**5*c*x + 126*b**4*c**5*x**5 - 2880*b**4 
*c**2*x**2 + 84*b**3*c**6*x**6 - 3840*b**3*c**3*x**3 + 12288*b**3 + 36*b** 
2*c**7*x**7 - 2880*b**2*c**4*x**4 + 36864*b**2*c*x + 9*b*c**8*x**8 - 1152* 
b*c**5*x**5 + 36864*b*c**2*x**2 + c**9*x**9 - 192*c**6*x**6 + 12288*c**3*x 
**3 - 262144),x)*a*b**4*c**4*x**2 + 360*int(sqrt(b**3 + 3*b**2*c*x + 3*...