\(\int \frac {(a+b x^3)^{8/3}}{(c+d x^3)^3} \, dx\) [163]

Optimal result

Integrand size = 21, antiderivative size = 391 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (6 b c+5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^3}+\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^3}-\frac {b^{8/3} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 d^3} \] Output:

-1/6*(-a*d+b*c)*x*(b*x^3+a)^(5/3)/c/d/(d*x^3+c)^2-1/18*(-a*d+b*c)*(5*a*d+6 
*b*c)*x*(b*x^3+a)^(2/3)/c^2/d^2/(d*x^3+c)+1/3*b^(8/3)*arctan(1/3*(1+2*b^(1 
/3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/d^3-1/27*(-a*d+b*c)^(2/3)*(5*a^2*d 
^2+6*a*b*c*d+9*b^2*c^2)*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^(8/3)/d^3-1/54*(-a*d+b*c)^(2/3)*(5*a^2*d^2+6* 
a*b*c*d+9*b^2*c^2)*ln(d*x^3+c)/c^(8/3)/d^3+1/18*(-a*d+b*c)^(2/3)*(5*a^2*d^ 
2+6*a*b*c*d+9*b^2*c^2)*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(8 
/3)/d^3-1/2*b^(8/3)*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 = 11.12 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\frac {\frac {6 c^{2/3} (-b c+a d) x \left (a+b x^3\right )^{2/3} \left (3 b c \left (2 c+3 d x^3\right )+a d \left (8 c+5 d x^3\right )\right )}{d^2 \left (c+d x^3\right )^2}+\frac {27 b^3 c^{5/3} x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{d^2 \sqrt [3]{a+b x^3}}+\frac {10 a^3 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{\sqrt [3]{b c-a d}}+\frac {6 a b^2 c^2 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac {2 a^2 b c \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d \sqrt [3]{b c-a d}}}{108 c^{8/3}} \] Input:

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

Output:

((6*c^(2/3)*(-(b*c) + a*d)*x*(a + b*x^3)^(2/3)*(3*b*c*(2*c + 3*d*x^3) + a* 
d*(8*c + 5*d*x^3)))/(d^2*(c + d*x^3)^2) + (27*b^3*c^(5/3)*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)) + (10*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)]))/(b*c - a*d)^(1/ 
3) + (6*a*b^2*c^2*(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*c*(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*(b*c - a*d)^(1/3)))/(108 
*c^(8/3))
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {930, 1023, 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.

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

\(\Big \downarrow \) 930

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

\(\Big \downarrow \) 1023

\(\displaystyle \frac {-\frac {\int -\frac {2 \left (9 b^3 c^2 x^3+a \left (3 b^2 c^2+a d (b c+5 a d)\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c d}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{3 c d \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \int \frac {9 b^3 c^2 x^3+a \left (3 b^2 c^2+a d (b c+5 a d)\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c d}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{3 c d \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 1026

\(\displaystyle \frac {\frac {2 \left (\frac {9 b^3 c^2 \int \frac {1}{\sqrt [3]{b x^3+a}}dx}{d}-\frac {(b c-a d) \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}\right )}{3 c d}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{3 c d \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 769

\(\displaystyle \frac {\frac {2 \left (\frac {9 b^3 c^2 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {(b c-a d) \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{d}\right )}{3 c d}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{3 c d \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {\frac {2 \left (\frac {9 b^3 c^2 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {(b c-a d) \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{d}\right )}{3 c d}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{3 c d \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\)

Input:

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

Output:

-1/6*((b*c - a*d)*x*(a + b*x^3)^(5/3))/(c*d*(c + d*x^3)^2) + (-1/3*((b*c - 
 a*d)*(6*b*c + 5*a*d)*x*(a + b*x^3)^(2/3))/(c*d*(c + d*x^3)) + (2*(-(((b*c 
 - a*d)*(9*b^2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*(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) + 
(9*b^3*c^2*(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))/(3*c*d 
))/(6*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 1023

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^q/(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 - 1)*Simp[c*(b*e*n*(p + 1) + b*e - a*f) + d*(b*e*n*( 
p + 1) + (b*e - a*f)*(n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
 n}, x] && LtQ[p, -1] && GtQ[q, 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 = 1.98 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(\frac {-\frac {5 \left (a d -b c \right ) \left (d \,x^{3}+c \right )^{2} \left (a^{2} d^{2}+\frac {6}{5} a b c d +\frac {9}{5} b^{2} c^{2}\right ) \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 )}{54}+\frac {b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \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 )}{6}+\frac {5 \left (a d -b c \right ) \left (d \,x^{3}+c \right )^{2} \left (a^{2} d^{2}+\frac {6}{5} a b c d +\frac {9}{5} b^{2} c^{2}\right ) \sqrt {3}\, \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 )}{27}+\frac {5 \left (a d -b c \right ) \left (d \,x^{3}+c \right )^{2} \left (a^{2} d^{2}+\frac {6}{5} a b c d +\frac {9}{5} b^{2} c^{2}\right ) \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 )}{27}+\frac {4 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (-\frac {3 b^{\frac {8}{3}} \sqrt {3}\, c^{2} \left (d \,x^{3}+c \right )^{2} \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 )}{4}-\frac {3 b^{\frac {8}{3}} c^{2} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{4}+\left (a d -b c \right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (\frac {3 b \,c^{2}}{4}+d \left (\frac {9 b \,x^{3}}{8}+a \right ) c +\frac {5 a \,d^{2} x^{3}}{8}\right ) d x \right ) c}{9}}{\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} d^{3} \left (d \,x^{3}+c \right )^{2} c^{3}}\)	\(498\)

Input:

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

Output:

5/27/((a*d-b*c)/c)^(1/3)*(-1/2*(a*d-b*c)*(d*x^3+c)^2*(a^2*d^2+6/5*a*b*c*d+ 
9/5*b^2*c^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)+9/10*b^(8/3)*c^3*((a*d-b*c)/c)^(1/3)*(d*x^3+c)^ 
2*ln((b^(2/3)*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)+(a*d-b*c 
)*(d*x^3+c)^2*(a^2*d^2+6/5*a*b*c*d+9/5*b^2*c^2)*3^(1/2)*arctan(1/3*3^(1/2) 
*(-2/((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)+x)/x)+(a*d-b*c)*(d*x^3+c)^2*(a^2* 
d^2+6/5*a*b*c*d+9/5*b^2*c^2)*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x) 
+12/5*((a*d-b*c)/c)^(1/3)*(-3/4*b^(8/3)*3^(1/2)*c^2*(d*x^3+c)^2*arctan(1/3 
*3^(1/2)*(2*(b*x^3+a)^(1/3)/b^(1/3)+x)/x)-3/4*b^(8/3)*c^2*(d*x^3+c)^2*ln(( 
-b^(1/3)*x+(b*x^3+a)^(1/3))/x)+(a*d-b*c)*(b*x^3+a)^(2/3)*(3/4*b*c^2+d*(9/8 
*b*x^3+a)*c+5/8*a*d^2*x^3)*d*x)*c)/d^3/(d*x^3+c)^2/c^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (334) = 668\).

Time = 2.34 (sec) , antiderivative size = 954, normalized size of antiderivative = 2.44 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/54*(2*sqrt(3)*((9*b^2*c^2*d^2 + 6*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 9*b^2*c^ 
4 + 6*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(9*b^2*c^3*d + 6*a*b*c^2*d^2 + 5*a^2*c 
*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)) + 18*sqrt(3)*(b^2*c^2*d^2*x^6 + 2*b^2* 
c^3*d*x^3 + b^2*c^4)*(-b^2)^(1/3)*arctan(-1/3*(sqrt(3)*b*x - 2*sqrt(3)*(b* 
x^3 + a)^(1/3)*(-b^2)^(1/3))/(b*x)) - 2*((9*b^2*c^2*d^2 + 6*a*b*c*d^3 + 5* 
a^2*d^4)*x^6 + 9*b^2*c^4 + 6*a*b*c^3*d + 5*a^2*c^2*d^2 + 2*(9*b^2*c^3*d + 
6*a*b*c^2*d^2 + 5*a^2*c*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) - 18*(b^2*c^2*d^2*x^6 + 2*b^2*c^3*d*x^3 + b^2*c^4)*(-b^ 
2)^(1/3)*log(-((-b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + 9*(b^2*c^2*d^2*x 
^6 + 2*b^2*c^3*d*x^3 + b^2*c^4)*(-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) + ((9*b^2*c^2*d 
^2 + 6*a*b*c*d^3 + 5*a^2*d^4)*x^6 + 9*b^2*c^4 + 6*a*b*c^3*d + 5*a^2*c^2*d^ 
2 + 2*(9*b^2*c^3*d + 6*a*b*c^2*d^2 + 5*a^2*c*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*((9*b^2*c^2*d^2 - 4*a 
*b*c*d^3 - 5*a^2*d^4)*x^4 + 2*(3*b^2*c^3*d + a*b*c^2*d^2 - 4*a^2*c*d^3)...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(a + b*x**3)**(2/3)*a**3*d*x - 6*(a + b*x**3)**(2/3)*a**2*b*c*x - 3*(a 
+ b*x**3)**(2/3)*a**2*b*d*x**4 - 2*(a + b*x**3)**(2/3)*a*b**2*c*x**4 + 2*i 
nt(((a + b*x**3)**(2/3)*x**6)/(c**3 + 3*c**2*d*x**3 + 3*c*d**2*x**6 + d**3 
*x**9),x)*a*b**2*c**3*d + 4*int(((a + b*x**3)**(2/3)*x**6)/(c**3 + 3*c**2* 
d*x**3 + 3*c*d**2*x**6 + d**3*x**9),x)*a*b**2*c**2*d**2*x**3 + 2*int(((a + 
 b*x**3)**(2/3)*x**6)/(c**3 + 3*c**2*d*x**3 + 3*c*d**2*x**6 + d**3*x**9),x 
)*a*b**2*c*d**3*x**6 - 6*int(((a + b*x**3)**(2/3)*x**6)/(c**3 + 3*c**2*d*x 
**3 + 3*c*d**2*x**6 + d**3*x**9),x)*b**3*c**4 - 12*int(((a + b*x**3)**(2/3 
)*x**6)/(c**3 + 3*c**2*d*x**3 + 3*c*d**2*x**6 + d**3*x**9),x)*b**3*c**3*d* 
x**3 - 6*int(((a + b*x**3)**(2/3)*x**6)/(c**3 + 3*c**2*d*x**3 + 3*c*d**2*x 
**6 + d**3*x**9),x)*b**3*c**2*d**2*x**6 + 10*int(((a + b*x**3)**(2/3)*x**3 
)/(a**2*c**3*d + 3*a**2*c**2*d**2*x**3 + 3*a**2*c*d**3*x**6 + a**2*d**4*x* 
*9 - 3*a*b*c**4 - 8*a*b*c**3*d*x**3 - 6*a*b*c**2*d**2*x**6 + a*b*d**4*x**1 
2 - 3*b**2*c**4*x**3 - 9*b**2*c**3*d*x**6 - 9*b**2*c**2*d**2*x**9 - 3*b**2 
*c*d**3*x**12),x)*a**5*c**2*d**3 + 20*int(((a + b*x**3)**(2/3)*x**3)/(a**2 
*c**3*d + 3*a**2*c**2*d**2*x**3 + 3*a**2*c*d**3*x**6 + a**2*d**4*x**9 - 3* 
a*b*c**4 - 8*a*b*c**3*d*x**3 - 6*a*b*c**2*d**2*x**6 + a*b*d**4*x**12 - 3*b 
**2*c**4*x**3 - 9*b**2*c**3*d*x**6 - 9*b**2*c**2*d**2*x**9 - 3*b**2*c*d**3 
*x**12),x)*a**5*c*d**4*x**3 + 10*int(((a + b*x**3)**(2/3)*x**3)/(a**2*c**3 
*d + 3*a**2*c**2*d**2*x**3 + 3*a**2*c*d**3*x**6 + a**2*d**4*x**9 - 3*a*...