Integrand size = 24, antiderivative size = 336 \[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=-\frac {(6 b c-a d) x^2 \sqrt [3]{a+b x^3}}{18 b d^2}+\frac {x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\left (9 b^2 c^2-3 a b c d-a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{5/3} d^3}+\frac {c^{5/3} \sqrt [3]{b c-a d} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^3}-\frac {c^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{6 d^3}-\frac {\left (9 b^2 c^2-3 a b c d-a^2 d^2\right ) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{5/3} d^3}+\frac {c^{5/3} \sqrt [3]{b c-a d} \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^3} \]
-1/18*(-a*d+6*b*c)*x^2*(b*x^3+a)^(1/3)/b/d^2+1/6*x^5*(b*x^3+a)^(1/3)/d-1/6 *c^(5/3)*(-a*d+b*c)^(1/3)*ln(d*x^3+c)/d^3-1/18*(-a^2*d^2-3*a*b*c*d+9*b^2*c ^2)*ln(b^(1/3)*x-(b*x^3+a)^(1/3))/b^(5/3)/d^3+1/2*c^(5/3)*(-a*d+b*c)^(1/3) *ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/d^3-1/27*(-a^2*d^2-3*a*b*c *d+9*b^2*c^2)*arctan(1/3*(1+2*b^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/b^(5/3)/ d^3*3^(1/2)+1/3*c^(5/3)*(-a*d+b*c)^(1/3)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)* x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/d^3*3^(1/2)
Result contains complex when optimal does not.
Time = 6.45 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.57 \[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\frac {\frac {6 d x^2 \sqrt [3]{a+b x^3} \left (-6 b c+a d+3 b d x^3\right )}{b}-\frac {4 \sqrt {3} \left (9 b^2 c^2-3 a b c d-a^2 d^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{b^{5/3}}-18 \sqrt {-6-6 i \sqrt {3}} c^{5/3} \sqrt [3]{b c-a d} \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )+\frac {4 \left (-9 b^2 c^2+3 a b c d+a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{b^{5/3}}+18 i \left (i+\sqrt {3}\right ) c^{5/3} \sqrt [3]{b c-a d} \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+\frac {2 \left (9 b^2 c^2-3 a b c d-a^2 d^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{b^{5/3}}+9 \left (1-i \sqrt {3}\right ) c^{5/3} \sqrt [3]{b c-a d} \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{108 d^3} \]
((6*d*x^2*(a + b*x^3)^(1/3)*(-6*b*c + a*d + 3*b*d*x^3))/b - (4*Sqrt[3]*(9* b^2*c^2 - 3*a*b*c*d - a^2*d^2)*ArcTan[(Sqrt[3]*b^(1/3)*x)/(b^(1/3)*x + 2*( a + b*x^3)^(1/3))])/b^(5/3) - 18*Sqrt[-6 - (6*I)*Sqrt[3]]*c^(5/3)*(b*c - a *d)^(1/3)*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - (3 *I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] + (4*(-9*b^2*c^2 + 3*a*b*c*d + a ^2*d^2)*Log[-(b^(1/3)*x) + (a + b*x^3)^(1/3)])/b^(5/3) + (18*I)*(I + Sqrt[ 3])*c^(5/3)*(b*c - a*d)^(1/3)*Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])* c^(1/3)*(a + b*x^3)^(1/3)] + (2*(9*b^2*c^2 - 3*a*b*c*d - 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)])/b^(5/3) + 9*(1 - I*Sqrt[3])*c^(5/3)*(b*c - a*d)^(1/3)*Log[2*(b*c - a*d)^(2/3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3 ])*c^(2/3)*(a + b*x^3)^(2/3)])/(108*d^3)
Time = 0.58 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {978, 1052, 27, 1054, 2009}
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.
\(\displaystyle \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx\) |
\(\Big \downarrow \) 978 |
\(\displaystyle \frac {x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\int \frac {x^4 \left ((6 b c-a d) x^3+5 a c\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{6 d}\) |
\(\Big \downarrow \) 1052 |
\(\displaystyle \frac {x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-a d)}{3 b d}-\frac {\int \frac {2 x \left (\left (9 b^2 c^2-3 a b d c-a^2 d^2\right ) x^3+a c (6 b c-a d)\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{3 b d}}{6 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-a d)}{3 b d}-\frac {2 \int \frac {x \left (\left (9 b^2 c^2-3 a b d c-a^2 d^2\right ) x^3+a c (6 b c-a d)\right )}{\left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}dx}{3 b d}}{6 d}\) |
\(\Big \downarrow \) 1054 |
\(\displaystyle \frac {x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-a d)}{3 b d}-\frac {2 \int \left (\frac {\left (9 b^2 c^2-3 a b d c-a^2 d^2\right ) x}{d \left (b x^3+a\right )^{2/3}}+\frac {9 \left (a b c^2 d-b^2 c^3\right ) x}{d \left (b x^3+a\right )^{2/3} \left (d x^3+c\right )}\right )dx}{3 b d}}{6 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^5 \sqrt [3]{a+b x^3}}{6 d}-\frac {\frac {x^2 \sqrt [3]{a+b x^3} (6 b c-a d)}{3 b d}-\frac {2 \left (-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (-a^2 d^2-3 a b c d+9 b^2 c^2\right )}{\sqrt {3} b^{2/3} d}-\frac {\left (-a^2 d^2-3 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{2 b^{2/3} d}+\frac {3 \sqrt {3} b c^{5/3} \sqrt [3]{b c-a d} \arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{d}-\frac {3 b c^{5/3} \sqrt [3]{b c-a d} \log \left (c+d x^3\right )}{2 d}+\frac {9 b c^{5/3} \sqrt [3]{b c-a d} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d}\right )}{3 b d}}{6 d}\) |
(x^5*(a + b*x^3)^(1/3))/(6*d) - (((6*b*c - a*d)*x^2*(a + b*x^3)^(1/3))/(3* b*d) - (2*(-(((9*b^2*c^2 - 3*a*b*c*d - a^2*d^2)*ArcTan[(1 + (2*b^(1/3)*x)/ (a + b*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*b^(2/3)*d)) + (3*Sqrt[3]*b*c^(5/3)*( b*c - a*d)^(1/3)*ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^ (1/3)))/Sqrt[3]])/d - (3*b*c^(5/3)*(b*c - a*d)^(1/3)*Log[c + d*x^3])/(2*d) - ((9*b^2*c^2 - 3*a*b*c*d - a^2*d^2)*Log[b^(1/3)*x - (a + b*x^3)^(1/3)])/ (2*b^(2/3)*d) + (9*b*c^(5/3)*(b*c - a*d)^(1/3)*Log[((b*c - a*d)^(1/3)*x)/c ^(1/3) - (a + b*x^3)^(1/3)])/(2*d)))/(3*b*d))/(6*d)
3.7.65.3.1 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[((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[((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]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 6.38 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.32
method | result | size |
pseudoelliptic | \(\frac {-\frac {\left (b^{\frac {11}{3}} c -b^{\frac {8}{3}} a d \right ) c \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}-\left (b^{\frac {11}{3}} c -b^{\frac {8}{3}} a d \right ) \sqrt {3}\, c \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )-\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} b \left (a^{2} d^{2}+3 a b c d -9 b^{2} c^{2}\right ) \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{18}+\left (b^{\frac {11}{3}} c -b^{\frac {8}{3}} a d \right ) c \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\frac {\left (-\sqrt {3}\, b \left (a^{2} d^{2}+3 a b c d -9 b^{2} c^{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+b \left (a^{2} d^{2}+3 a b c d -9 b^{2} c^{2}\right ) \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )+\frac {3 x^{2} \left (\left (3 d \,x^{3}-6 c \right ) b^{\frac {8}{3}}+a \,b^{\frac {5}{3}} d \right ) d \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{2}\right ) \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}}}{9}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} b^{\frac {8}{3}} d^{3}}\) | \(445\) |
1/3*(-1/2*(b^(11/3)*c-b^(8/3)*a*d)*c*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)-(b^(11/3)*c-b^(8/3)*a*d )*3^(1/2)*c*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)-1/18*((a*d-b*c)/c)^(2/3)*b*(a^2*d^2+3*a*b*c*d-9*b^2* 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)+(b^(1 1/3)*c-b^(8/3)*a*d)*c*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)+1/9*(- 3^(1/2)*b*(a^2*d^2+3*a*b*c*d-9*b^2*c^2)*arctan(1/3*3^(1/2)*(b^(1/3)*x+2*(b *x^3+a)^(1/3))/b^(1/3)/x)+b*(a^2*d^2+3*a*b*c*d-9*b^2*c^2)*ln((-b^(1/3)*x+( b*x^3+a)^(1/3))/x)+3/2*x^2*((3*d*x^3-6*c)*b^(8/3)+a*b^(5/3)*d)*d*(b*x^3+a) ^(1/3))*((a*d-b*c)/c)^(2/3))/((a*d-b*c)/c)^(2/3)/b^(8/3)/d^3
Time = 1.15 (sec) , antiderivative size = 494, normalized size of antiderivative = 1.47 \[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\frac {18 \, \sqrt {3} {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} b^{3} c \arctan \left (-\frac {\sqrt {3} {\left (b c^{2} - a c d\right )} x + 2 \, \sqrt {3} {\left (b c^{3} - a c^{2} d\right )}^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{3 \, {\left (b c^{2} - a c d\right )} x}\right ) + 18 \, {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} b^{3} c \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} c - {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} x}{x}\right ) - 9 \, {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} b^{3} c \log \left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} c^{2} + {\left (b c^{3} - a c^{2} d\right )}^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} c x + {\left (b c^{3} - a c^{2} d\right )}^{\frac {2}{3}} x^{2}}{x^{2}}\right ) + 2 \, \sqrt {3} {\left (9 \, b^{3} c^{2} - 3 \, a b^{2} c d - a^{2} b d^{2}\right )} {\left (b^{2}\right )}^{\frac {1}{6}} \arctan \left (\frac {{\left (\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}}\right )} {\left (b^{2}\right )}^{\frac {1}{6}}}{3 \, b^{2} x}\right ) - 2 \, {\left (9 \, b^{2} c^{2} - 3 \, a b c d - a^{2} d^{2}\right )} {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2}\right )}^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + {\left (9 \, b^{2} c^{2} - 3 \, a b c d - a^{2} d^{2}\right )} {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (b^{2}\right )}^{\frac {1}{3}} b x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) + 3 \, {\left (3 \, b^{3} d^{2} x^{5} - {\left (6 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, b^{3} d^{3}} \]
1/54*(18*sqrt(3)*(b*c^3 - a*c^2*d)^(1/3)*b^3*c*arctan(-1/3*(sqrt(3)*(b*c^2 - a*c*d)*x + 2*sqrt(3)*(b*c^3 - a*c^2*d)^(2/3)*(b*x^3 + a)^(1/3))/((b*c^2 - a*c*d)*x)) + 18*(b*c^3 - a*c^2*d)^(1/3)*b^3*c*log(((b*x^3 + a)^(1/3)*c - (b*c^3 - a*c^2*d)^(1/3)*x)/x) - 9*(b*c^3 - a*c^2*d)^(1/3)*b^3*c*log(((b* x^3 + a)^(2/3)*c^2 + (b*c^3 - a*c^2*d)^(1/3)*(b*x^3 + a)^(1/3)*c*x + (b*c^ 3 - a*c^2*d)^(2/3)*x^2)/x^2) + 2*sqrt(3)*(9*b^3*c^2 - 3*a*b^2*c*d - a^2*b* d^2)*(b^2)^(1/6)*arctan(1/3*(sqrt(3)*(b^2)^(1/3)*b*x + 2*sqrt(3)*(b*x^3 + a)^(1/3)*(b^2)^(2/3))*(b^2)^(1/6)/(b^2*x)) - 2*(9*b^2*c^2 - 3*a*b*c*d - a^ 2*d^2)*(b^2)^(2/3)*log(-((b^2)^(2/3)*x - (b*x^3 + a)^(1/3)*b)/x) + (9*b^2* c^2 - 3*a*b*c*d - a^2*d^2)*(b^2)^(2/3)*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) + 3*(3*b^3*d^2*x^5 - (6* b^3*c*d - a*b^2*d^2)*x^2)*(b*x^3 + a)^(1/3))/(b^3*d^3)
\[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int \frac {x^{7} \sqrt [3]{a + b x^{3}}}{c + d x^{3}}\, dx \]
\[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{7}}{d x^{3} + c} \,d x } \]
\[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{7}}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {x^7 \sqrt [3]{a+b x^3}}{c+d x^3} \, dx=\int \frac {x^7\,{\left (b\,x^3+a\right )}^{1/3}}{d\,x^3+c} \,d x \]