Integrand size = 21, antiderivative size = 219 \[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=\frac {\left (27 b^2 c^2-9 a b c d+2 a^2 d^2\right ) x \left (a+b x^3\right )^{2/3}}{81 b^2}+\frac {2 d (3 b c-a d) x \left (a+b x^3\right )^{5/3}}{27 b^2}+\frac {d x \left (a+b x^3\right )^{5/3} \left (c+d x^3\right )}{9 b}+\frac {2 a \left (27 b^2 c^2-9 a b c d+2 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3} b^{7/3}}-\frac {a \left (27 b^2 c^2-9 a b c d+2 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{81 b^{7/3}} \]
1/81*(2*a^2*d^2-9*a*b*c*d+27*b^2*c^2)*x*(b*x^3+a)^(2/3)/b^2+2/27*d*(-a*d+3 *b*c)*x*(b*x^3+a)^(5/3)/b^2+1/9*d*x*(b*x^3+a)^(5/3)*(d*x^3+c)/b-1/81*a*(2* a^2*d^2-9*a*b*c*d+27*b^2*c^2)*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(7/3)+2/243 *a*(2*a^2*d^2-9*a*b*c*d+27*b^2*c^2)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1 /3))*3^(1/2))/b^(7/3)*3^(1/2)
Time = 1.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.17 \[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=\frac {3 \sqrt [3]{b} x \left (a+b x^3\right )^{2/3} \left (-4 a^2 d^2+3 a b d \left (6 c+d x^3\right )+9 b^2 \left (3 c^2+3 c d x^3+d^2 x^6\right )\right )+2 \sqrt {3} a \left (27 b^2 c^2-9 a b c d+2 a^2 d^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 a \left (27 b^2 c^2-9 a b c d+2 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+a \left (27 b^2 c^2-9 a b c d+2 a^2 d^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{243 b^{7/3}} \]
(3*b^(1/3)*x*(a + b*x^3)^(2/3)*(-4*a^2*d^2 + 3*a*b*d*(6*c + d*x^3) + 9*b^2 *(3*c^2 + 3*c*d*x^3 + d^2*x^6)) + 2*Sqrt[3]*a*(27*b^2*c^2 - 9*a*b*c*d + 2* a^2*d^2)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/3))] - 2 *a*(27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/ 3)] + a*(27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*Log[b^(2/3)*x^2 + b^(1/3)*x*( a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)])/(243*b^(7/3))
Time = 0.29 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {933, 913, 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.
| \(\displaystyle \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx\) |
| \(\Big \downarrow \) 933 |
| \(\displaystyle \frac {\int \left (b x^3+a\right )^{2/3} \left (4 d (3 b c-a d) x^3+c (9 b c-a d)\right )dx}{9 b}+\frac {d x \left (a+b x^3\right )^{5/3} \left (c+d x^3\right )}{9 b}\) |
| \(\Big \downarrow \) 913 |
| \(\displaystyle \frac {\frac {\left (2 a^2 d^2-9 a b c d+27 b^2 c^2\right ) \int \left (b x^3+a\right )^{2/3}dx}{3 b}+\frac {2 d x \left (a+b x^3\right )^{5/3} (3 b c-a d)}{3 b}}{9 b}+\frac {d x \left (a+b x^3\right )^{5/3} \left (c+d x^3\right )}{9 b}\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {\frac {\left (2 a^2 d^2-9 a b c d+27 b^2 c^2\right ) \left (\frac {2}{3} a \int \frac {1}{\sqrt [3]{b x^3+a}}dx+\frac {1}{3} x \left (a+b x^3\right )^{2/3}\right )}{3 b}+\frac {2 d x \left (a+b x^3\right )^{5/3} (3 b c-a d)}{3 b}}{9 b}+\frac {d x \left (a+b x^3\right )^{5/3} \left (c+d x^3\right )}{9 b}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {\frac {\left (2 a^2 d^2-9 a b c d+27 b^2 c^2\right ) \left (\frac {2}{3} a \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )+\frac {1}{3} x \left (a+b x^3\right )^{2/3}\right )}{3 b}+\frac {2 d x \left (a+b x^3\right )^{5/3} (3 b c-a d)}{3 b}}{9 b}+\frac {d x \left (a+b x^3\right )^{5/3} \left (c+d x^3\right )}{9 b}\) |
(d*x*(a + b*x^3)^(5/3)*(c + d*x^3))/(9*b) + ((2*d*(3*b*c - a*d)*x*(a + b*x ^3)^(5/3))/(3*b) + ((27*b^2*c^2 - 9*a*b*c*d + 2*a^2*d^2)*((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]]/(Sqr t[3]*b^(1/3)) - Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)]/(2*b^(1/3))))/3))/(3 *b))/(9*b)
3.1.71.3.1 Defintions of rubi rules used
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]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), x] + Simp[1/(b*(n*(p + q) + 1)) Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d , 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[ a, b, c, d, n, p, q, x]
Time = 4.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.93
| method | result | size |
| pseudoelliptic | \(-\frac {4 \left (-\frac {27 x \left (b \,x^{3}+a \right )^{\frac {2}{3}} d a \left (\frac {d \,x^{3}}{6}+c \right ) b^{\frac {4}{3}}}{2}-\frac {81 x \left (\frac {1}{3} d^{2} x^{6}+c d \,x^{3}+c^{2}\right ) \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {7}{3}}}{4}+\left (3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,d^{2} x \,b^{\frac {1}{3}}+\left (a^{2} d^{2}-\frac {9}{2} a b c d +\frac {27}{2} b^{2} c^{2}\right ) \left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right )\right ) a \right )}{243 b^{\frac {7}{3}}}\) | \(203\) |
-4/243*(-27/2*x*(b*x^3+a)^(2/3)*d*a*(1/6*d*x^3+c)*b^(4/3)-81/4*x*(1/3*d^2* x^6+c*d*x^3+c^2)*(b*x^3+a)^(2/3)*b^(7/3)+(3*(b*x^3+a)^(2/3)*a*d^2*x*b^(1/3 )+(a^2*d^2-9/2*a*b*c*d+27/2*b^2*c^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^(1/3)*x+(b*x^3+a)^(1/3))/x)-1/2*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)/b^(7/3)
Time = 0.33 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.89 \[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (27 \, a b^{3} c^{2} - 9 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) - 2 \, {\left (27 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + {\left (27 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (9 \, b^{3} d^{2} x^{7} + 3 \, {\left (9 \, b^{3} c d + a b^{2} d^{2}\right )} x^{4} + {\left (27 \, b^{3} c^{2} + 18 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{243 \, b^{3}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (27 \, a b^{3} c^{2} - 9 \, a^{2} b^{2} c d + 2 \, a^{3} b d^{2}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) + 2 \, {\left (27 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - {\left (27 \, a b^{2} c^{2} - 9 \, a^{2} b c d + 2 \, a^{3} d^{2}\right )} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (9 \, b^{3} d^{2} x^{7} + 3 \, {\left (9 \, b^{3} c d + a b^{2} d^{2}\right )} x^{4} + {\left (27 \, b^{3} c^{2} + 18 \, a b^{2} c d - 4 \, a^{2} b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{243 \, b^{3}}\right ] \]
[1/243*(3*sqrt(1/3)*(27*a*b^3*c^2 - 9*a^2*b^2*c*d + 2*a^3*b*d^2)*sqrt((-b) ^(1/3)/b)*log(3*b*x^3 - 3*(b*x^3 + a)^(1/3)*(-b)^(2/3)*x^2 - 3*sqrt(1/3)*( (-b)^(1/3)*b*x^3 - (b*x^3 + a)^(1/3)*b*x^2 + 2*(b*x^3 + a)^(2/3)*(-b)^(2/3 )*x)*sqrt((-b)^(1/3)/b) + 2*a) - 2*(27*a*b^2*c^2 - 9*a^2*b*c*d + 2*a^3*d^2 )*(-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) + (27*a*b^2*c^2 - 9 *a^2*b*c*d + 2*a^3*d^2)*(-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*(9*b^3*d^2*x^7 + 3*(9*b^3*c*d + a*b^2*d^2)*x^4 + (27*b^3*c^2 + 18*a*b^2*c*d - 4*a^2*b*d^2)*x)*(b*x^3 + a )^(2/3))/b^3, -1/243*(6*sqrt(1/3)*(27*a*b^3*c^2 - 9*a^2*b^2*c*d + 2*a^3*b* d^2)*sqrt(-(-b)^(1/3)/b)*arctan(-sqrt(1/3)*((-b)^(1/3)*x - 2*(b*x^3 + a)^( 1/3))*sqrt(-(-b)^(1/3)/b)/x) + 2*(27*a*b^2*c^2 - 9*a^2*b*c*d + 2*a^3*d^2)* (-b)^(2/3)*log(((-b)^(1/3)*x + (b*x^3 + a)^(1/3))/x) - (27*a*b^2*c^2 - 9*a ^2*b*c*d + 2*a^3*d^2)*(-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*(9*b^3*d^2*x^7 + 3*(9*b^3*c*d + a*b^2*d^2)*x^4 + (27*b^3*c^2 + 18*a*b^2*c*d - 4*a^2*b*d^2)*x)*(b*x^3 + a)^ (2/3))/b^3]
Result contains complex when optimal does not.
Time = 5.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.60 \[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=\frac {a^{\frac {2}{3}} c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {2 a^{\frac {2}{3}} c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {2}{3}} d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]
a**(2/3)*c**2*x*gamma(1/3)*hyper((-2/3, 1/3), (4/3,), b*x**3*exp_polar(I*p i)/a)/(3*gamma(4/3)) + 2*a**(2/3)*c*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)*d**2*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. 552 vs. \(2 (188) = 376\).
Time = 0.28 (sec) , antiderivative size = 552, normalized size of antiderivative = 2.52 \[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=-\frac {1}{9} \, {\left (\frac {2 \, \sqrt {3} a \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 {a \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 {2 \, a \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 (b x^{3} + a\right )}^{\frac {2}{3}} a}{{\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x^{2}}\right )} c^{2} + \frac {1}{27} \, {\left (\frac {2 \, \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 {4}{3}}} - \frac {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 {4}{3}}} + \frac {2 \, a^{2} \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 {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{3} - \frac {2 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b}{x^{6}}}\right )} c d - \frac {1}{243} \, {\left (\frac {4 \, \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 {7}{3}}} - \frac {2 \, 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 {7}{3}}} + \frac {4 \, a^{3} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (\frac {2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3} b^{2}}{x^{2}} + \frac {11 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{3} b}{x^{5}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}\right )}}{b^{5} - \frac {3 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}}}\right )} d^{2} \]
-1/9*(2*sqrt(3)*a*arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^( 1/3))/b^(1/3) - a*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) + 2*a*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(1/3) + 3*( b*x^3 + a)^(2/3)*a/((b - (b*x^3 + a)/x^3)*x^2))*c^2 + 1/27*(2*sqrt(3)*a^2* arctan(1/3*sqrt(3)*(b^(1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(4/3) - 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^(4/ 3) + 2*a^2*log(-b^(1/3) + (b*x^3 + a)^(1/3)/x)/b^(4/3) + 3*((b*x^3 + a)^(2 /3)*a^2*b/x^2 + 2*(b*x^3 + a)^(5/3)*a^2/x^5)/(b^3 - 2*(b*x^3 + a)*b^2/x^3 + (b*x^3 + a)^2*b/x^6))*c*d - 1/243*(4*sqrt(3)*a^3*arctan(1/3*sqrt(3)*(b^( 1/3) + 2*(b*x^3 + a)^(1/3)/x)/b^(1/3))/b^(7/3) - 2*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^(7/3) + 4*a^3*log(-b^(1/ 3) + (b*x^3 + a)^(1/3)/x)/b^(7/3) + 3*(2*(b*x^3 + a)^(2/3)*a^3*b^2/x^2 + 1 1*(b*x^3 + a)^(5/3)*a^3*b/x^5 - 4*(b*x^3 + a)^(8/3)*a^3/x^8)/(b^5 - 3*(b*x ^3 + a)*b^4/x^3 + 3*(b*x^3 + a)^2*b^3/x^6 - (b*x^3 + a)^3*b^2/x^9))*d^2
\[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}^{2} \,d x } \]
Timed out. \[ \int \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^2 \, dx=\int {\left (b\,x^3+a\right )}^{2/3}\,{\left (d\,x^3+c\right )}^2 \,d x \]