Integrand size = 19, antiderivative size = 174 \[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\frac {5 a (9 b c-a d) x \left (a+b x^3\right )^{2/3}}{162 b}+\frac {(9 b c-a d) x \left (a+b x^3\right )^{5/3}}{54 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}+\frac {5 a^2 (9 b c-a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3} b^{4/3}}-\frac {5 a^2 (9 b c-a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{162 b^{4/3}} \] Output:
5/162*a*(-a*d+9*b*c)*x*(b*x^3+a)^(2/3)/b+1/54*(-a*d+9*b*c)*x*(b*x^3+a)^(5/ 3)/b+1/9*d*x*(b*x^3+a)^(8/3)/b+5/243*a^2*(-a*d+9*b*c)*arctan(1/3*(1+2*b^(1 /3)*x/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/b^(4/3)-5/162*a^2*(-a*d+9*b*c)*ln( -b^(1/3)*x+(b*x^3+a)^(1/3))/b^(4/3)
Time = 0.83 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.20 \[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\frac {3 \sqrt [3]{b} x \left (a+b x^3\right )^{2/3} \left (10 a^2 d+9 b^2 x^3 \left (3 c+2 d x^3\right )+a b \left (72 c+33 d x^3\right )\right )-10 \sqrt {3} a^2 (-9 b c+a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )+10 a^2 (-9 b c+a d) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-5 a^2 (-9 b c+a d) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{486 b^{4/3}} \] Input:
Integrate[(a + b*x^3)^(5/3)*(c + d*x^3),x]
Output:
(3*b^(1/3)*x*(a + b*x^3)^(2/3)*(10*a^2*d + 9*b^2*x^3*(3*c + 2*d*x^3) + a*b *(72*c + 33*d*x^3)) - 10*Sqrt[3]*a^2*(-9*b*c + a*d)*ArcTan[(Sqrt[3]*b^(1/3 )*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] + 10*a^2*(-9*b*c + a*d)*Log[-(b^(1 /3)*x) + (a + b*x^3)^(1/3)] - 5*a^2*(-9*b*c + a*d)*Log[b^(2/3)*x^2 + b^(1/ 3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(486*b^(4/3))
Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {913, 748, 748, 769}
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.
| \(\displaystyle \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {(9 b c-a d) \int \left (b x^3+a\right )^{5/3}dx}{9 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {(9 b c-a d) \left (\frac {5}{6} a \int \left (b x^3+a\right )^{2/3}dx+\frac {1}{6} x \left (a+b x^3\right )^{5/3}\right )}{9 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {(9 b c-a d) \left (\frac {5}{6} a \left (\frac {2}{3} a \int \frac {1}{\sqrt [3]{b x^3+a}}dx+\frac {1}{3} x \left (a+b x^3\right )^{2/3}\right )+\frac {1}{6} x \left (a+b x^3\right )^{5/3}\right )}{9 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {(9 b c-a d) \left (\frac {5}{6} a \left (\frac {2}{3} a \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )+\frac {1}{3} x \left (a+b x^3\right )^{2/3}\right )+\frac {1}{6} x \left (a+b x^3\right )^{5/3}\right )}{9 b}+\frac {d x \left (a+b x^3\right )^{8/3}}{9 b}\) |
Input:
Int[(a + b*x^3)^(5/3)*(c + d*x^3),x]
Output:
(d*x*(a + b*x^3)^(8/3))/(9*b) + ((9*b*c - a*d)*((x*(a + b*x^3)^(5/3))/6 + (5*a*((x*(a + b*x^3)^(2/3))/3 + (2*a*(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))))/3))/6))/(9*b)
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Simp[a*n*(p/(n*p + 1)) Int[(a + b*x^n)^(p - 1), x], x] /; Fre eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat or[p + 1/n], Denominator[p]])
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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Time = 1.23 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(\frac {\frac {4 a \left (b \,x^{3}+a \right )^{\frac {2}{3}} x \left (\frac {11 d \,x^{3}}{24}+c \right ) b^{\frac {4}{3}}}{9}+\frac {\left (\frac {2 d \,x^{3}}{3}+c \right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{4} b^{\frac {7}{3}}}{6}+\frac {5 a^{2} \left (d x \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {1}{3}}-\frac {\left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right ) \left (a d -9 b c \right )}{6}\right )}{81}}{b^{\frac {4}{3}}}\) | \(180\) |
Input:
int((b*x^3+a)^(5/3)*(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
5/81*(36/5*a*(b*x^3+a)^(2/3)*x*(11/24*d*x^3+c)*b^(4/3)+27/10*(2/3*d*x^3+c) *(b*x^3+a)^(2/3)*x^4*b^(7/3)+a^2*(d*x*(b*x^3+a)^(2/3)*b^(1/3)-1/6*(-2*3^(1 /2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)+ln((b^(2/3 )*x^2+b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-2*ln((-b^(1/3)*x+(b* x^3+a)^(1/3))/x))*(a*d-9*b*c)))/b^(4/3)
Time = 0.10 (sec) , antiderivative size = 482, normalized size of antiderivative = 2.77 \[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\left [-\frac {15 \, \sqrt {\frac {1}{3}} {\left (9 \, a^{2} b^{2} c - a^{3} b d\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 10 \, {\left (9 \, a^{2} b c - a^{3} d\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (9 \, a^{2} b c - a^{3} d\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (18 \, b^{3} d x^{7} + 3 \, {\left (9 \, b^{3} c + 11 \, a b^{2} d\right )} x^{4} + 2 \, {\left (36 \, a b^{2} c + 5 \, a^{2} b d\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{486 \, b^{2}}, -\frac {10 \, {\left (9 \, a^{2} b c - a^{3} d\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (9 \, a^{2} b c - a^{3} d\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {30 \, \sqrt {\frac {1}{3}} {\left (9 \, a^{2} b^{2} c - a^{3} b d\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right )}{b^{\frac {1}{3}}} - 3 \, {\left (18 \, b^{3} d x^{7} + 3 \, {\left (9 \, b^{3} c + 11 \, a b^{2} d\right )} x^{4} + 2 \, {\left (36 \, a b^{2} c + 5 \, a^{2} b d\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{486 \, b^{2}}\right ] \] Input:
integrate((b*x^3+a)^(5/3)*(d*x^3+c),x, algorithm="fricas")
Output:
[-1/486*(15*sqrt(1/3)*(9*a^2*b^2*c - a^3*b*d)*sqrt(-1/b^(2/3))*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*b^(2/3)*x^2 - 3*sqrt(1/3)*(b^(4/3)*x^3 + (b*x^3 + a )^(1/3)*b*x^2 - 2*(b*x^3 + a)^(2/3)*b^(2/3)*x)*sqrt(-1/b^(2/3)) + 2*a) + 1 0*(9*a^2*b*c - a^3*d)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - 5* (9*a^2*b*c - a^3*d)*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x^3 + a)^(2/3))/x^2) - 3*(18*b^3*d*x^7 + 3*(9*b^3*c + 11*a*b^2*d)*x^ 4 + 2*(36*a*b^2*c + 5*a^2*b*d)*x)*(b*x^3 + a)^(2/3))/b^2, -1/486*(10*(9*a^ 2*b*c - a^3*d)*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) - 5*(9*a^2* b*c - a^3*d)*b^(2/3)*log((b^(2/3)*x^2 + (b*x^3 + a)^(1/3)*b^(1/3)*x + (b*x ^3 + a)^(2/3))/x^2) + 30*sqrt(1/3)*(9*a^2*b^2*c - a^3*b*d)*arctan(sqrt(1/3 )*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x))/b^(1/3) - 3*(18*b^3*d*x^7 + 3*(9*b^3*c + 11*a*b^2*d)*x^4 + 2*(36*a*b^2*c + 5*a^2*b*d)*x)*(b*x^3 + a )^(2/3))/b^2]
Result contains complex when optimal does not.
Time = 6.70 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98 \[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\frac {a^{\frac {5}{3}} c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {a^{\frac {5}{3}} d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {2}{3}} b c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {2}{3}} b d x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \] Input:
integrate((b*x**3+a)**(5/3)*(d*x**3+c),x)
Output:
a**(5/3)*c*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*pi)/ a)/(3*gamma(4/3)) + a**(5/3)*d*x**4*gamma(4/3)*hyper((-2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + a**(2/3)*b*c*x**4*gamma(4/3)*hy per((-2/3, 4/3), (7/3,), b*x**3*exp_polar(I*pi)/a)/(3*gamma(7/3)) + a**(2/ 3)*b*d*x**7*gamma(7/3)*hyper((-2/3, 7/3), (10/3,), b*x**3*exp_polar(I*pi)/ a)/(3*gamma(10/3))
Leaf count of result is larger than twice the leaf count of optimal. 406 vs. \(2 (143) = 286\).
Time = 0.11 (sec) , antiderivative size = 406, normalized size of antiderivative = 2.33 \[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=-\frac {1}{54} \, {\left (\frac {10 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {5 \, a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {1}{3}}} + \frac {10 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {1}{3}}} + \frac {3 \, {\left (\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} - \frac {8 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{2} - \frac {2 \, {\left (b x^{3} + a\right )} b}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2}}{x^{6}}}\right )} c + \frac {1}{486} \, {\left (\frac {10 \, \sqrt {3} a^{3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} - \frac {5 \, a^{3} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {10 \, a^{3} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}} + \frac {3 \, {\left (\frac {5 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3} b^{2}}{x^{2}} - \frac {13 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{3} b}{x^{5}} - \frac {10 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}\right )}}{b^{4} - \frac {3 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3} b}{x^{9}}}\right )} d \] Input:
integrate((b*x^3+a)^(5/3)*(d*x^3+c),x, algorithm="maxima")
Output:
-1/54*(10*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x) /b^(1/3))/b^(1/3) - 5*a^2*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x ^3 + a)^(2/3)/x^2)/b^(1/3) + 10*a^2*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^ (1/3) + 3*(5*(b*x^3 + a)^(2/3)*a^2*b/x^2 - 8*(b*x^3 + a)^(5/3)*a^2/x^5)/(b ^2 - 2*(b*x^3 + a)*b/x^3 + (b*x^3 + a)^2/x^6))*c + 1/486*(10*sqrt(3)*a^3*a rctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(4/3) - 5*a ^3*log(b^(2/3) + (b*x^3 + a)^(1/3)*b^(1/3)/x + (b*x^3 + a)^(2/3)/x^2)/b^(4 /3) + 10*a^3*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(4/3) + 3*(5*(b*x^3 + a )^(2/3)*a^3*b^2/x^2 - 13*(b*x^3 + a)^(5/3)*a^3*b/x^5 - 10*(b*x^3 + a)^(8/3 )*a^3/x^8)/(b^4 - 3*(b*x^3 + a)*b^3/x^3 + 3*(b*x^3 + a)^2*b^2/x^6 - (b*x^3 + a)^3*b/x^9))*d
\[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {5}{3}} {\left (d x^{3} + c\right )} \,d x } \] Input:
integrate((b*x^3+a)^(5/3)*(d*x^3+c),x, algorithm="giac")
Output:
integrate((b*x^3 + a)^(5/3)*(d*x^3 + c), x)
Timed out. \[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\int {\left (b\,x^3+a\right )}^{5/3}\,\left (d\,x^3+c\right ) \,d x \] Input:
int((a + b*x^3)^(5/3)*(c + d*x^3),x)
Output:
int((a + b*x^3)^(5/3)*(c + d*x^3), x)
\[ \int \left (a+b x^3\right )^{5/3} \left (c+d x^3\right ) \, dx=\frac {10 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a^{2} d x +72 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b c x +33 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a b d \,x^{4}+27 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} c \,x^{4}+18 \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{2} d \,x^{7}-10 \left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) a^{3} d +90 \left (\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) a^{2} b c}{162 b} \] Input:
int((b*x^3+a)^(5/3)*(d*x^3+c),x)
Output:
(10*(a + b*x**3)**(2/3)*a**2*d*x + 72*(a + b*x**3)**(2/3)*a*b*c*x + 33*(a + b*x**3)**(2/3)*a*b*d*x**4 + 27*(a + b*x**3)**(2/3)*b**2*c*x**4 + 18*(a + b*x**3)**(2/3)*b**2*d*x**7 - 10*int((a + b*x**3)**(2/3)/(a + b*x**3),x)*a **3*d + 90*int((a + b*x**3)**(2/3)/(a + b*x**3),x)*a**2*b*c)/(162*b)