Integrand size = 28, antiderivative size = 264 \[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {4 a x \left (a+b x^3\right )^{2/3}}{9 b^2 d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}-\frac {14 a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3} d}+\frac {2^{2/3} a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} b^{7/3} d}+\frac {a^2 \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} b^{7/3} d}-\frac {a^2 \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} b^{7/3} d}+\frac {7 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{9 b^{7/3} d} \]
-4/9*a*x*(b*x^3+a)^(2/3)/b^2/d-1/6*x^4*(b*x^3+a)^(2/3)/b/d+1/6*a^2*ln(-b*d *x^3+a*d)*2^(2/3)/b^(7/3)/d-1/2*a^2*ln(2^(1/3)*b^(1/3)*x-(b*x^3+a)^(1/3))* 2^(2/3)/b^(7/3)/d+7/9*a^2*ln(-b^(1/3)*x+(b*x^3+a)^(1/3))/b^(7/3)/d-14/27*a ^2*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(7/3)/d*3^(1/2)+1 /3*2^(2/3)*a^2*arctan(1/3*(1+2*2^(1/3)*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2)) /b^(7/3)/d*3^(1/2)
Time = 1.38 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.23 \[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=-\frac {24 a \sqrt [3]{b} x \left (a+b x^3\right )^{2/3}+9 b^{4/3} x^4 \left (a+b x^3\right )^{2/3}+28 \sqrt {3} a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-18\ 2^{2/3} \sqrt {3} a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )-28 a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+18\ 2^{2/3} a^2 \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )+14 a^2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-9\ 2^{2/3} a^2 \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{54 b^{7/3} d} \]
-1/54*(24*a*b^(1/3)*x*(a + b*x^3)^(2/3) + 9*b^(4/3)*x^4*(a + b*x^3)^(2/3) + 28*Sqrt[3]*a^2*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*(a + b*x^3)^(1/ 3))] - 18*2^(2/3)*Sqrt[3]*a^2*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2^(2 /3)*(a + b*x^3)^(1/3))] - 28*a^2*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)] + 1 8*2^(2/3)*a^2*Log[-2*b^(1/3)*x + 2^(2/3)*(a + b*x^3)^(1/3)] + 14*a^2*Log[b ^(2/3)*x^2 + b^(1/3)*x*(a + b*x^3)^(1/3) + (a + b*x^3)^(2/3)] - 9*2^(2/3)* a^2*Log[2*b^(2/3)*x^2 + 2^(2/3)*b^(1/3)*x*(a + b*x^3)^(1/3) + 2^(1/3)*(a + b*x^3)^(2/3)])/(b^(7/3)*d)
Time = 0.40 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {978, 27, 1052, 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.
| \(\displaystyle \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx\) |
| \(\Big \downarrow \) 978 |
| \(\displaystyle \frac {\int \frac {4 a x^3 \left (2 b x^3+a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{6 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a \int \frac {x^3 \left (2 b x^3+a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{3 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
| \(\Big \downarrow \) 1052 |
| \(\displaystyle \frac {2 a \left (\frac {\int \frac {a b \left (7 b x^3+2 a\right )}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{3 b^2}-\frac {2 x \left (a+b x^3\right )^{2/3}}{3 b}\right )}{3 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a \left (\frac {a \int \frac {7 b x^3+2 a}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx}{3 b}-\frac {2 x \left (a+b x^3\right )^{2/3}}{3 b}\right )}{3 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
| \(\Big \downarrow \) 1026 |
| \(\displaystyle \frac {2 a \left (\frac {a \left (9 a \int \frac {1}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx-7 \int \frac {1}{\sqrt [3]{b x^3+a}}dx\right )}{3 b}-\frac {2 x \left (a+b x^3\right )^{2/3}}{3 b}\right )}{3 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
| \(\Big \downarrow \) 769 |
| \(\displaystyle \frac {2 a \left (\frac {a \left (9 a \int \frac {1}{\left (a-b x^3\right ) \sqrt [3]{b x^3+a}}dx-7 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )\right )}{3 b}-\frac {2 x \left (a+b x^3\right )^{2/3}}{3 b}\right )}{3 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {2 a \left (\frac {a \left (9 a \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3} a \sqrt [3]{b}}+\frac {\log \left (a-b x^3\right )}{6 \sqrt [3]{2} a \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{2} a \sqrt [3]{b}}\right )-7 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )\right )}{3 b}-\frac {2 x \left (a+b x^3\right )^{2/3}}{3 b}\right )}{3 b d}-\frac {x^4 \left (a+b x^3\right )^{2/3}}{6 b d}\) |
-1/6*(x^4*(a + b*x^3)^(2/3))/(b*d) + (2*a*((-2*x*(a + b*x^3)^(2/3))/(3*b) + (a*(9*a*(ArcTan[(1 + (2*2^(1/3)*b^(1/3)*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/( 2^(1/3)*Sqrt[3]*a*b^(1/3)) + Log[a - b*x^3]/(6*2^(1/3)*a*b^(1/3)) - Log[2^ (1/3)*b^(1/3)*x - (a + b*x^3)^(1/3)]/(2*2^(1/3)*a*b^(1/3))) - 7*(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*b)))/(3*b*d)
3.6.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[e^(n - 1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)* ((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Simp[e^n/(b*(m + n*(p + q) + 1)) Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c , d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
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]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*g^(n - 1)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q + 1) + 1))), x] - Simp[g^n/(b*d*(m + n*(p + q + 1) + 1)) Int[(g*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m - n + 1) + (a*f*d*(m + n*q + 1) + b*( f*c*(m + n*p + 1) - e*d*(m + n*(p + q + 1) + 1)))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && GtQ[m, n - 1]
Time = 4.96 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(\frac {-9 x^{4} \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {4}{3}}-18 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (2^{\frac {2}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )}{3 b^{\frac {1}{3}} x}\right ) a^{2}-18 \,2^{\frac {2}{3}} \ln \left (\frac {-2^{\frac {1}{3}} b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a^{2}+9 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} b^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a^{2}-24 a x \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {1}{3}}+28 a^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+28 a^{2} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-14 a^{2} \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{54 d \,b^{\frac {7}{3}}}\) | \(271\) |
1/54*(-9*x^4*(b*x^3+a)^(2/3)*b^(4/3)-18*3^(1/2)*2^(2/3)*arctan(1/3*3^(1/2) *(2^(2/3)*(b*x^3+a)^(1/3)+b^(1/3)*x)/b^(1/3)/x)*a^2-18*2^(2/3)*ln((-2^(1/3 )*b^(1/3)*x+(b*x^3+a)^(1/3))/x)*a^2+9*2^(2/3)*ln((2^(2/3)*b^(2/3)*x^2+2^(1 /3)*b^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)*a^2-24*a*x*(b*x^3+a)^( 2/3)*b^(1/3)+28*a^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)+28*a^2*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)-14*a^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))/d/b^(7/3)
Time = 0.34 (sec) , antiderivative size = 701, normalized size of antiderivative = 2.66 \[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\left [-\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{b}\right )^{\frac {1}{3}}}{3 \, x}\right ) - 42 \, \sqrt {\frac {1}{3}} a^{2} b \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 ) - 18 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 9 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 4^{\frac {1}{3}} b x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 28 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 14 \, a^{2} 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 (3 \, b^{2} x^{4} + 8 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3} d}, -\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{b}\right )^{\frac {1}{3}}}{3 \, x}\right ) - 18 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {4^{\frac {2}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 9 \cdot 4^{\frac {1}{3}} a^{2} b \left (-\frac {1}{b}\right )^{\frac {1}{3}} \log \left (-\frac {2 \cdot 4^{\frac {1}{3}} b x^{2} \left (-\frac {1}{b}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x \left (-\frac {1}{b}\right )^{\frac {2}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 84 \, \sqrt {\frac {1}{3}} a^{2} b^{\frac {2}{3}} \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 ) - 28 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 14 \, a^{2} 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 (3 \, b^{2} x^{4} + 8 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3} d}\right ] \]
[-1/54*(18*4^(1/3)*sqrt(3)*a^2*b*(-1/b)^(1/3)*arctan(-1/3*(sqrt(3)*x - 4^( 1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/b)^(1/3))/x) - 42*sqrt(1/3)*a^2*b*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) - 18*4^(1/3)*a^2*b*(-1/b)^(1/3)*log(-(4^(2/3)*b*x*(-1/ b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 9*4^(1/3)*a^2*b*(-1/b)^(1/3)*log(-(2* 4^(1/3)*b*x^2*(-1/b)^(1/3) - 4^(2/3)*(b*x^3 + a)^(1/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(2/3))/x^2) - 28*a^2*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^( 1/3))/x) + 14*a^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*(3*b^2*x^4 + 8*a*b*x)*(b*x^3 + a)^(2/3))/(b^3 *d), -1/54*(18*4^(1/3)*sqrt(3)*a^2*b*(-1/b)^(1/3)*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(b*x^3 + a)^(1/3)*(-1/b)^(1/3))/x) - 18*4^(1/3)*a^2*b*(-1 /b)^(1/3)*log(-(4^(2/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(1/3))/x) + 9*4^( 1/3)*a^2*b*(-1/b)^(1/3)*log(-(2*4^(1/3)*b*x^2*(-1/b)^(1/3) - 4^(2/3)*(b*x^ 3 + a)^(1/3)*b*x*(-1/b)^(2/3) - 2*(b*x^3 + a)^(2/3))/x^2) - 84*sqrt(1/3)*a ^2*b^(2/3)*arctan(sqrt(1/3)*(b^(1/3)*x + 2*(b*x^3 + a)^(1/3))/(b^(1/3)*x)) - 28*a^2*b^(2/3)*log(-(b^(1/3)*x - (b*x^3 + a)^(1/3))/x) + 14*a^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*(3*b^2*x^4 + 8*a*b*x)*(b*x^3 + a)^(2/3))/(b^3*d)]
\[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=- \frac {\int \frac {x^{6} \left (a + b x^{3}\right )^{\frac {2}{3}}}{- a + b x^{3}}\, dx}{d} \]
\[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{6}}{b d x^{3} - a d} \,d x } \]
\[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int { -\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{6}}{b d x^{3} - a d} \,d x } \]
Timed out. \[ \int \frac {x^6 \left (a+b x^3\right )^{2/3}}{a d-b d x^3} \, dx=\int \frac {x^6\,{\left (b\,x^3+a\right )}^{2/3}}{a\,d-b\,d\,x^3} \,d x \]