\(\int \frac {(a+b x^3)^{8/3}}{(c+d x^3)^2} \, dx\) [150]

Optimal result

Integrand size = 21, antiderivative size = 351 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\frac {b (2 b c-a d) x \left (a+b x^3\right )^{2/3}}{3 c d^2}-\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{3 c d \left (c+d x^3\right )}-\frac {2 b^{5/3} (3 b c-4 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} d^3}+\frac {2 (b c-a d)^{5/3} (3 b c+a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^{5/3} d^3}+\frac {(b c-a d)^{5/3} (3 b c+a d) \log \left (c+d x^3\right )}{9 c^{5/3} d^3}-\frac {(b c-a d)^{5/3} (3 b c+a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{3 c^{5/3} d^3}+\frac {b^{5/3} (3 b c-4 a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 d^3} \] Output:

1/3*b*(-a*d+2*b*c)*x*(b*x^3+a)^(2/3)/c/d^2-1/3*(-a*d+b*c)*x*(b*x^3+a)^(5/3 
)/c/d/(d*x^3+c)-2/9*b^(5/3)*(-4*a*d+3*b*c)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^ 
3+a)^(1/3))*3^(1/2))*3^(1/2)/d^3+2/9*(-a*d+b*c)^(5/3)*(a*d+3*b*c)*arctan(1 
/3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(5/ 
3)/d^3+1/9*(-a*d+b*c)^(5/3)*(a*d+3*b*c)*ln(d*x^3+c)/c^(5/3)/d^3-1/3*(-a*d+ 
b*c)^(5/3)*(a*d+3*b*c)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(5 
/3)/d^3+1/3*b^(5/3)*(-4*a*d+3*b*c)*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/d^3
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 11.02 (sec) , antiderivative size = 698, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\frac {1}{18} \left (\frac {6 x \left (a+b x^3\right )^{2/3} \left (b^2+\frac {(b c-a d)^2}{c \left (c+d x^3\right )}\right )}{d^2}-\frac {9 b^3 x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{d^2 \sqrt [3]{a+b x^3}}+\frac {12 a b^2 x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c d \sqrt [3]{a+b x^3}}+\frac {2 a^3 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{c^{5/3} \sqrt [3]{b c-a d}}-\frac {2 a b^2 \sqrt [3]{c} \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac {2 a^2 b \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{c^{2/3} d \sqrt [3]{b c-a d}}\right ) \] Input:

Integrate[(a + b*x^3)^(8/3)/(c + d*x^3)^2,x]
 

Output:

((6*x*(a + b*x^3)^(2/3)*(b^2 + (b*c - a*d)^2/(c*(c + d*x^3))))/d^2 - (9*b^ 
3*x^4*(1 + (b*x^3)/a)^(1/3)*AppellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d* 
x^3)/c)])/(d^2*(a + b*x^3)^(1/3)) + (12*a*b^2*x^4*(1 + (b*x^3)/a)^(1/3)*Ap 
pellF1[4/3, 1/3, 1, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/(c*d*(a + b*x^3)^(1/ 
3)) + (2*a^3*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + 
a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3 
)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1 
/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(c^(5/3)*(b*c - a*d)^(1/3)) 
- (2*a*b^2*c^(1/3)*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3) 
*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + 
 a*x^3)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + 
 (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(d^2*(b*c - a*d)^(1/3) 
) + (2*a^2*b*(2*Sqrt[3]*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + 
a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3 
)^(1/3)] + Log[c^(2/3) + ((b*c - a*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1 
/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)]))/(c^(2/3)*d*(b*c - a*d)^(1/3) 
))/18
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.97, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {930, 1025, 27, 1026, 769, 901}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx\)

\(\Big \downarrow \) 930

\(\displaystyle \frac {\int \frac {\left (b x^3+a\right )^{2/3} \left (3 b (2 b c-a d) x^3+a (b c+2 a d)\right )}{d x^3+c}dx}{3 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{3 c d \left (c+d x^3\right )}\)

\(\Big \downarrow \) 1025

\(\displaystyle \frac {\frac {\int -\frac {6 \left (b^2 c (3 b c-4 a d) x^3+a \left (b^2 c^2-a b d c-a^2 d^2\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 d}+\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d)}{d}}{3 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{3 c d \left (c+d x^3\right )}\)

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1026

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

\(\Big \downarrow \) 769

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

\(\Big \downarrow \) 901

\(\displaystyle \frac {\frac {b x \left (a+b x^3\right )^{2/3} (2 b c-a d)}{d}-\frac {2 \left (\frac {b^2 c (3 b c-4 a d) \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {(b c-a d)^2 (a d+3 b c) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{d}\right )}{d}}{3 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{3 c d \left (c+d x^3\right )}\)

Input:

Int[(a + b*x^3)^(8/3)/(c + d*x^3)^2,x]
 

Output:

-1/3*((b*c - a*d)*x*(a + b*x^3)^(5/3))/(c*d*(c + d*x^3)) + ((b*(2*b*c - a* 
d)*x*(a + b*x^3)^(2/3))/d - (2*(-(((b*c - a*d)^2*(3*b*c + a*d)*(ArcTan[(1 
+ (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c 
^(2/3)*(b*c - a*d)^(1/3)) + Log[c + d*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - 
 Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - 
a*d)^(1/3))))/d) + (b^2*c*(3*b*c - 4*a*d)*(ArcTan[(1 + (2*b^(1/3)*x)/(a + 
b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^3)^ 
(1/3)]/(2*b^(1/3))))/d))/d)/(3*c*d)
 

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 769

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]* 
(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^ 
3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 930

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(a*d - c*b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*b*n*(p + 
1))), x] - Simp[1/(a*b*n*(p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q 
- 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*( 
p + q) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 
 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]
 

rule 1025

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( 
f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( 
b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1))   Int[(a + b*x 
^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e 
- a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ 
{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
 

rule 1026

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)* 
(x_)^(n_)), x_Symbol] :> Simp[f/d   Int[(a + b*x^n)^p, x], x] + Simp[(d*e - 
 c*f)/d   Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, f, 
 p, n}, x]
 

Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.36

method	result	size
pseudoelliptic	\(\frac {\frac {4 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{2} \left (d \,x^{3}+c \right ) \left (a \,b^{\frac {5}{3}} d -\frac {3 b^{\frac {8}{3}} c}{4}\right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{9}+\frac {2 \left (d \,x^{3}+c \right ) \left (a d +3 b c \right ) \left (a d -b c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{9}-\frac {8 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{2} \left (d \,x^{3}+c \right ) \left (a \,b^{\frac {5}{3}} d -\frac {3 b^{\frac {8}{3}} c}{4}\right ) \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{b^{\frac {1}{3}}}+x \right )}{3 x}\right )}{9}-\frac {8 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} c^{2} \left (d \,x^{3}+c \right ) \left (a \,b^{\frac {5}{3}} d -\frac {3 b^{\frac {8}{3}} c}{4}\right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{9}+\frac {c d \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (2 b^{2} c^{2}-2 d b \left (-\frac {b \,x^{3}}{2}+a \right ) c +a^{2} d^{2}\right ) x \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{3}+\frac {2 \left (a d -b c \right )^{2} \left (a d +3 b c \right ) \left (d \,x^{3}+c \right ) \left (\arctan \left (\frac {\sqrt {3}\, \left (-\frac {2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}+x \right )}{3 x}\right ) \sqrt {3}-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )}{9}}{d^{3} c^{2} \left (d \,x^{3}+c \right ) \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}\)	\(477\)

Input:

int((b*x^3+a)^(8/3)/(d*x^3+c)^2,x,method=_RETURNVERBOSE)
 

Output:

2/9/((a*d-b*c)/c)^(1/3)*(2*((a*d-b*c)/c)^(1/3)*c^2*(d*x^3+c)*(a*b^(5/3)*d- 
3/4*b^(8/3)*c)*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/ 
x^2)+(d*x^3+c)*(a*d+3*b*c)*(a*d-b*c)^2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a) 
^(1/3))/x)-4*((a*d-b*c)/c)^(1/3)*c^2*(d*x^3+c)*(a*b^(5/3)*d-3/4*b^(8/3)*c) 
*3^(1/2)*arctan(1/3*3^(1/2)*(2*(b*x^3+a)^(1/3)/b^(1/3)+x)/x)-4*((a*d-b*c)/ 
c)^(1/3)*c^2*(d*x^3+c)*(a*b^(5/3)*d-3/4*b^(8/3)*c)*ln((-b^(1/3)*x+(b*x^3+a 
)^(1/3))/x)+3/2*c*d*(b*x^3+a)^(2/3)*(2*b^2*c^2-2*d*b*(-1/2*b*x^3+a)*c+a^2* 
d^2)*x*((a*d-b*c)/c)^(1/3)+(a*d-b*c)^2*(a*d+3*b*c)*(d*x^3+c)*(arctan(1/3*3 
^(1/2)*(-2/((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x)/x)*3^(1/2)-1/2*ln((((a*d 
-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/ 
x^2)))/d^3/c^2/(d*x^3+c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (291) = 582\).

Time = 3.52 (sec) , antiderivative size = 819, normalized size of antiderivative = 2.33 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:

integrate((b*x^3+a)^(8/3)/(d*x^3+c)^2,x, algorithm="fricas")
 

Output:

1/9*(2*sqrt(3)*(3*b^2*c^3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b 
*c*d^2 - a^2*d^3)*x^3)*((b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*arctan( 
-1/3*(sqrt(3)*(b*c - a*d)*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*c*((b^2*c^2 - 2* 
a*b*c*d + a^2*d^2)/c^2)^(1/3))/((b*c - a*d)*x)) + 2*sqrt(3)*(3*b^2*c^3 - 4 
*a*b*c^2*d + (3*b^2*c^2*d - 4*a*b*c*d^2)*x^3)*(-b^2)^(1/3)*arctan(-1/3*(sq 
rt(3)*b*x - 2*sqrt(3)*(b*x^3 + a)^(1/3)*(-b^2)^(1/3))/(b*x)) - 2*(3*b^2*c^ 
3 - 2*a*b*c^2*d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*x^3)*( 
(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/c^2)^(1/3)*log((c*x*((b^2*c^2 - 2*a*b*c*d 
+ a^2*d^2)/c^2)^(2/3) - (b*x^3 + a)^(1/3)*(b*c - a*d))/x) - 2*(3*b^2*c^3 - 
 4*a*b*c^2*d + (3*b^2*c^2*d - 4*a*b*c*d^2)*x^3)*(-b^2)^(1/3)*log(-((-b^2)^ 
(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (3*b^2*c^3 - 4*a*b*c^2*d + (3*b^2*c^2* 
d - 4*a*b*c*d^2)*x^3)*(-b^2)^(1/3)*log(-((-b^2)^(1/3)*b*x^2 - (b*x^3 + a)^ 
(1/3)*(-b^2)^(2/3)*x - (b*x^3 + a)^(2/3)*b)/x^2) + (3*b^2*c^3 - 2*a*b*c^2* 
d - a^2*c*d^2 + (3*b^2*c^2*d - 2*a*b*c*d^2 - a^2*d^3)*x^3)*((b^2*c^2 - 2*a 
*b*c*d + a^2*d^2)/c^2)^(1/3)*log(-((b*c - a*d)*x^2*((b^2*c^2 - 2*a*b*c*d + 
 a^2*d^2)/c^2)^(1/3) + (b*x^3 + a)^(1/3)*c*x*((b^2*c^2 - 2*a*b*c*d + a^2*d 
^2)/c^2)^(2/3) + (b*x^3 + a)^(2/3)*(b*c - a*d))/x^2) + 3*(b^2*c*d^2*x^4 + 
(2*b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*x)*(b*x^3 + a)^(2/3))/(c*d^4*x^3 + c 
^2*d^3)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\text {Timed out} \] Input:

integrate((b*x**3+a)**(8/3)/(d*x**3+c)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{{\left (d x^{3} + c\right )}^{2}} \,d x } \] Input:

integrate((b*x^3+a)^(8/3)/(d*x^3+c)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^(8/3)/(d*x^3 + c)^2, x)
 

Giac [F]

\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{{\left (d x^{3} + c\right )}^{2}} \,d x } \] Input:

integrate((b*x^3+a)^(8/3)/(d*x^3+c)^2,x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^(8/3)/(d*x^3 + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{8/3}}{{\left (d\,x^3+c\right )}^2} \,d x \] Input:

int((a + b*x^3)^(8/3)/(c + d*x^3)^2,x)
 

Output:

int((a + b*x^3)^(8/3)/(c + d*x^3)^2, x)
                                                                                    
                                                                                    
 

Reduce [F]

\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^2} \, dx=\text {too large to display} \] Input:

int((b*x^3+a)^(8/3)/(d*x^3+c)^2,x)
 

Output:

( - 9*(a + b*x**3)**(2/3)*a**2*b*d*x + 4*(a + b*x**3)**(2/3)*a*b**2*c*x + 
2*(a + b*x**3)**(2/3)*a*b**2*d*x**4 - 3*(a + b*x**3)**(2/3)*b**3*c*x**4 + 
12*int((a + b*x**3)**(2/3)/(2*a**2*c**2*d + 4*a**2*c*d**2*x**3 + 2*a**2*d* 
*3*x**6 - 3*a*b*c**3 - 4*a*b*c**2*d*x**3 + a*b*c*d**2*x**6 + 2*a*b*d**3*x* 
*9 - 3*b**2*c**3*x**3 - 6*b**2*c**2*d*x**6 - 3*b**2*c*d**2*x**9),x)*a**5*c 
*d**3 + 12*int((a + b*x**3)**(2/3)/(2*a**2*c**2*d + 4*a**2*c*d**2*x**3 + 2 
*a**2*d**3*x**6 - 3*a*b*c**3 - 4*a*b*c**2*d*x**3 + a*b*c*d**2*x**6 + 2*a*b 
*d**3*x**9 - 3*b**2*c**3*x**3 - 6*b**2*c**2*d*x**6 - 3*b**2*c*d**2*x**9),x 
)*a**5*d**4*x**3 - 18*int((a + b*x**3)**(2/3)/(2*a**2*c**2*d + 4*a**2*c*d* 
*2*x**3 + 2*a**2*d**3*x**6 - 3*a*b*c**3 - 4*a*b*c**2*d*x**3 + a*b*c*d**2*x 
**6 + 2*a*b*d**3*x**9 - 3*b**2*c**3*x**3 - 6*b**2*c**2*d*x**6 - 3*b**2*c*d 
**2*x**9),x)*a**4*b*c**2*d**2 - 18*int((a + b*x**3)**(2/3)/(2*a**2*c**2*d 
+ 4*a**2*c*d**2*x**3 + 2*a**2*d**3*x**6 - 3*a*b*c**3 - 4*a*b*c**2*d*x**3 + 
 a*b*c*d**2*x**6 + 2*a*b*d**3*x**9 - 3*b**2*c**3*x**3 - 6*b**2*c**2*d*x**6 
 - 3*b**2*c*d**2*x**9),x)*a**4*b*c*d**3*x**3 - 8*int((a + b*x**3)**(2/3)/( 
2*a**2*c**2*d + 4*a**2*c*d**2*x**3 + 2*a**2*d**3*x**6 - 3*a*b*c**3 - 4*a*b 
*c**2*d*x**3 + a*b*c*d**2*x**6 + 2*a*b*d**3*x**9 - 3*b**2*c**3*x**3 - 6*b* 
*2*c**2*d*x**6 - 3*b**2*c*d**2*x**9),x)*a**3*b**2*c**3*d - 8*int((a + b*x* 
*3)**(2/3)/(2*a**2*c**2*d + 4*a**2*c*d**2*x**3 + 2*a**2*d**3*x**6 - 3*a*b* 
c**3 - 4*a*b*c**2*d*x**3 + a*b*c*d**2*x**6 + 2*a*b*d**3*x**9 - 3*b**2*c...