Integrand size = 21, antiderivative size = 391 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {(b c-a d) x \left (a+b x^3\right )^{5/3}}{6 c d \left (c+d x^3\right )^2}-\frac {(b c-a d) (6 b c+5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 d^2 \left (c+d x^3\right )}+\frac {b^{8/3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} d^3}-\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} d^3}+\frac {(b c-a d)^{2/3} \left (9 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} d^3}-\frac {b^{8/3} \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{2 d^3} \]
-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)-1/54*(-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+1/3*b^(8/3)*arct an(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/d^3*3^(1/2)-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))/c^(8/3)/d^3*3^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 11.00 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.66 \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\frac {\frac {6 c^{2/3} (-b c+a d) x \left (a+b x^3\right )^{2/3} \left (3 b c \left (2 c+3 d x^3\right )+a d \left (8 c+5 d x^3\right )\right )}{d^2 \left (c+d x^3\right )^2}+\frac {27 b^3 c^{5/3} x^4 \sqrt [3]{1+\frac {b x^3}{a}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{3},1,\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{d^2 \sqrt [3]{a+b x^3}}+\frac {10 a^3 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{\sqrt [3]{b c-a d}}+\frac {6 a b^2 c^2 \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d^2 \sqrt [3]{b c-a d}}+\frac {2 a^2 b c \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{d \sqrt [3]{b c-a d}}}{108 c^{8/3}} \]
((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))
Time = 0.50 (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 |
\(\displaystyle \frac {\frac {2 \left (\frac {9 b^3 c^2 \left (\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{2 \sqrt [3]{b}}\right )}{d}-\frac {(b c-a d) \left (5 a^2 d^2+6 a b c d+9 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{d}\right )}{3 c d}-\frac {x \left (a+b x^3\right )^{2/3} (b c-a d) (5 a d+6 b c)}{3 c d \left (c+d x^3\right )}}{6 c d}-\frac {x \left (a+b x^3\right )^{5/3} (b c-a d)}{6 c d \left (c+d x^3\right )^2}\) |
-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)
3.2.11.3.1 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[((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]
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]
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]
Time = 4.80 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.29
method | result | size |
pseudoelliptic | \(\frac {\frac {9 b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\left (5 a^{2} d^{2}+6 a b c d +9 b^{2} c^{2}\right ) \left (d \,x^{3}+c \right )^{2} \left (a d -b c \right ) \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-9 \sqrt {3}\, b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )-9 b^{\frac {8}{3}} c^{3} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\frac {\left (3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} x d \left (5 a \,d^{2} x^{3}+9 b c d \,x^{3}+8 a c d +6 b \,c^{2}\right ) c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}-\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )\right ) \left (5 a^{2} d^{2}+6 a b c d +9 b^{2} c^{2}\right ) \left (d \,x^{3}+c \right )^{2}\right ) \left (a d -b c \right )}{2}}{27 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )^{2} d^{3} c^{3}}\) | \(505\) |
1/27*(9/2*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)+(5*a^2*d^2+6*a*b*c*d+9*b^2*c^2 )*(d*x^3+c)^2*(a*d-b*c)*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)-9*3^ (1/2)*b^(8/3)*c^3*((a*d-b*c)/c)^(1/3)*(d*x^3+c)^2*arctan(1/3*3^(1/2)*(b^(1 /3)*x+2*(b*x^3+a)^(1/3))/b^(1/3)/x)-9*b^(8/3)*c^3*((a*d-b*c)/c)^(1/3)*(d*x ^3+c)^2*ln((-b^(1/3)*x+(b*x^3+a)^(1/3))/x)+1/2*(3*(b*x^3+a)^(2/3)*x*d*(5*a *d^2*x^3+9*b*c*d*x^3+8*a*c*d+6*b*c^2)*c*((a*d-b*c)/c)^(1/3)-(-2*arctan(1/3 *3^(1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/x)* 3^(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))*(5*a^2*d^2+6*a*b*c*d+9*b^2*c^2)*(d*x^3+c)^2)*(a*d-b *c))/((a*d-b*c)/c)^(1/3)/(d*x^3+c)^2/d^3/c^3
Leaf count of result is larger than twice the leaf count of optimal. 954 vs. \(2 (334) = 668\).
Time = 1.95 (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=-\frac {2 \, \sqrt {3} {\left ({\left (9 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 9 \, b^{2} c^{4} + 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (9 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} {\left (b c - a d\right )} x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}}}{3 \, {\left (b c - a d\right )} x}\right ) + 18 \, \sqrt {3} {\left (b^{2} c^{2} d^{2} x^{6} + 2 \, b^{2} c^{3} d x^{3} + b^{2} c^{4}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \arctan \left (-\frac {\sqrt {3} b x - 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}}}{3 \, b x}\right ) - 2 \, {\left ({\left (9 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 9 \, b^{2} c^{4} + 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (9 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (\frac {c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}}{x}\right ) - 18 \, {\left (b^{2} c^{2} d^{2} x^{6} + 2 \, b^{2} c^{3} d x^{3} + b^{2} c^{4}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + 9 \, {\left (b^{2} c^{2} d^{2} x^{6} + 2 \, b^{2} c^{3} d x^{3} + b^{2} c^{4}\right )} \left (-b^{2}\right )^{\frac {1}{3}} \log \left (-\frac {\left (-b^{2}\right )^{\frac {1}{3}} b x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) + {\left ({\left (9 \, b^{2} c^{2} d^{2} + 6 \, a b c d^{3} + 5 \, a^{2} d^{4}\right )} x^{6} + 9 \, b^{2} c^{4} + 6 \, a b c^{3} d + 5 \, a^{2} c^{2} d^{2} + 2 \, {\left (9 \, b^{2} c^{3} d + 6 \, a b c^{2} d^{2} + 5 \, a^{2} c d^{3}\right )} x^{3}\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (b c - a d\right )} x^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{c^{2}}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c - a d\right )}}{x^{2}}\right ) + 3 \, {\left ({\left (9 \, b^{2} c^{2} d^{2} - 4 \, a b c d^{3} - 5 \, a^{2} d^{4}\right )} x^{4} + 2 \, {\left (3 \, b^{2} c^{3} d + a b c^{2} d^{2} - 4 \, a^{2} c d^{3}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, {\left (c^{2} d^{5} x^{6} + 2 \, c^{3} d^{4} x^{3} + c^{4} d^{3}\right )}} \]
-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)...
Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {8}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{8/3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{8/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]