Integrand size = 21, antiderivative size = 463 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=-\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}+\frac {b (3 b c+2 a d) x}{12 a c (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {b \left (9 b^2 c^2-42 a b c d-2 a^2 d^2\right ) x}{12 a^2 c (b c-a d)^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )}+\frac {d \left (27 b^3 c^3-135 a b^2 c^2 d-42 a^2 b c d^2+10 a^3 d^3\right ) x \left (a+b x^3\right )^{2/3}}{36 a^2 c^2 (b c-a d)^4 \left (c+d x^3\right )}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{13/3}}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{13/3}}-\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{13/3}} \]
-1/6*d*x/c/(-a*d+b*c)/(b*x^3+a)^(4/3)/(d*x^3+c)^2+1/12*b*(2*a*d+3*b*c)*x/a /c/(-a*d+b*c)^2/(b*x^3+a)^(4/3)/(d*x^3+c)+1/12*b*(-2*a^2*d^2-42*a*b*c*d+9* b^2*c^2)*x/a^2/c/(-a*d+b*c)^3/(b*x^3+a)^(1/3)/(d*x^3+c)+1/36*d*(10*a^3*d^3 -42*a^2*b*c*d^2-135*a*b^2*c^2*d+27*b^3*c^3)*x*(b*x^3+a)^(2/3)/a^2/c^2/(-a* d+b*c)^4/(d*x^3+c)+1/54*d^2*(5*a^2*d^2-24*a*b*c*d+54*b^2*c^2)*ln(d*x^3+c)/ c^(8/3)/(-a*d+b*c)^(13/3)-1/18*d^2*(5*a^2*d^2-24*a*b*c*d+54*b^2*c^2)*ln((- a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(8/3)/(-a*d+b*c)^(13/3)+1/27*d ^2*(5*a^2*d^2-24*a*b*c*d+54*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)/(-a*d+b*c)^(13/3)*3^(1/2)
Time = 15.75 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.73 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\frac {1}{36} x \left (a+b x^3\right )^{2/3} \left (-\frac {9 b^3}{a (-b c+a d)^3 \left (a+b x^3\right )^2}+\frac {27 b^3 (b c-5 a d)}{a^2 (b c-a d)^4 \left (a+b x^3\right )}-\frac {6 d^3}{c (b c-a d)^3 \left (c+d x^3\right )^2}+\frac {2 d^3 (-21 b c+5 a d)}{c^2 (b c-a d)^4 \left (c+d x^3\right )}\right )+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{54 c^{8/3} (b c-a d)^{13/3}} \]
(x*(a + b*x^3)^(2/3)*((-9*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^3)^2) + (27*b^ 3*(b*c - 5*a*d))/(a^2*(b*c - a*d)^4*(a + b*x^3)) - (6*d^3)/(c*(b*c - a*d)^ 3*(c + d*x^3)^2) + (2*d^3*(-21*b*c + 5*a*d))/(c^2*(b*c - a*d)^4*(c + d*x^3 ))))/36 + (d^2*(54*b^2*c^2 - 24*a*b*c*d + 5*a^2*d^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)]))/(54*c^(8/3)*(b*c - a*d)^(13/3))
Time = 0.63 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {931, 1024, 27, 1024, 25, 27, 1024, 27, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 931 |
| \(\displaystyle \frac {\int \frac {-9 b d x^3+6 b c-5 a d}{\left (b x^3+a\right )^{7/3} \left (d x^3+c\right )^2}dx}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}-\frac {\int -\frac {2 \left (6 b d (3 b c+2 a d) x^3+9 b^2 c^2+10 a^2 d^2-24 a b c d\right )}{\left (b x^3+a\right )^{4/3} \left (d x^3+c\right )^2}dx}{4 a (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {6 b d (3 b c+2 a d) x^3+9 b^2 c^2+10 a^2 d^2-24 a b c d}{\left (b x^3+a\right )^{4/3} \left (d x^3+c\right )^2}dx}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}-\frac {\int -\frac {d \left (3 b \left (9 b^2 c^2-42 a b d c-2 a^2 d^2\right ) x^3+a \left (9 b^2 c^2+36 a b d c-10 a^2 d^2\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )^2}dx}{a (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {d \left (3 b \left (9 b^2 c^2-42 a b d c-2 a^2 d^2\right ) x^3+a \left (9 b^2 c^2+36 a b d c-10 a^2 d^2\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )^2}dx}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {d \int \frac {3 b \left (9 b^2 c^2-42 a b d c-2 a^2 d^2\right ) x^3+a \left (9 b^2 c^2+36 a b d c-10 a^2 d^2\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )^2}dx}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 1024 |
| \(\displaystyle \frac {\frac {\frac {d \left (\frac {\int \frac {4 a^2 d \left (54 b^2 c^2-24 a b d c+5 a^2 d^2\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c (b c-a d)}+\frac {x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{3 c \left (c+d x^3\right ) (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {d \left (\frac {4 a^2 d \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c (b c-a d)}+\frac {x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{3 c \left (c+d x^3\right ) (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {\frac {\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}+\frac {d \left (\frac {4 a^2 d \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{3 c (b c-a d)}+\frac {x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{3 c \left (c+d x^3\right ) (b c-a d)}\right )}{a (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\) |
-1/6*(d*x)/(c*(b*c - a*d)*(a + b*x^3)^(4/3)*(c + d*x^3)^2) + ((b*(3*b*c + 2*a*d)*x)/(2*a*(b*c - a*d)*(a + b*x^3)^(4/3)*(c + d*x^3)) + ((b*(9*b^2*c^2 - 42*a*b*c*d - 2*a^2*d^2)*x)/(a*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3) ) + (d*(((27*b^3*c^3 - 135*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 10*a^3*d^3)*x*(a + b*x^3)^(2/3))/(3*c*(b*c - a*d)*(c + d*x^3)) + (4*a^2*d*(54*b^2*c^2 - 24 *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))))/(3*c*(b*c - a*d))))/(a*(b*c - a*d)))/(2*a*(b*c - a*d)))/(6*c*(b*c - a*d))
3.2.16.3.1 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[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[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && !( !IntegerQ[p] && IntegerQ[q] && LtQ[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 + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( p + 1)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b *c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
Time = 4.69 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.03
| method | result | size |
| pseudoelliptic | \(\frac {\frac {5 \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (a^{2} d^{2}-\frac {24}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) d^{2} a^{2} \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{27}+\frac {4 \left (\frac {5 a^{3} x^{3} \left (b \,x^{3}+a \right )^{2} d^{5}}{8}+\left (b \,x^{3}+a \right )^{2} \left (-\frac {21 b \,x^{3}}{8}+a \right ) c \,a^{2} d^{4}-3 b \left (\frac {45}{16} b^{3} x^{9}+4 a \,b^{2} x^{6}+2 a^{2} b \,x^{3}+a^{3}\right ) c^{2} a \,d^{3}-18 x^{3} b^{3} \left (-\frac {3}{32} b^{2} x^{6}+\frac {13}{16} a b \,x^{3}+a^{2}\right ) c^{3} d^{2}-9 b^{3} \left (-\frac {3}{8} b^{2} x^{6}+\frac {7}{16} a b \,x^{3}+a^{2}\right ) c^{4} d +\frac {9 b^{4} \left (\frac {3 b \,x^{3}}{4}+a \right ) c^{5}}{4}\right ) x c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{9}+\frac {5 \left (\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}-\frac {\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a^{2} d^{2}-\frac {24}{5} a b c d +\frac {54}{5} b^{2} c^{2}\right ) d^{2} a^{2} \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2}}{27}}{\left (d \,x^{3}+c \right )^{2} c^{3} \left (a d -b c \right )^{4} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {4}{3}} a^{2}}\) | \(475\) |
5/27/((a*d-b*c)/c)^(1/3)/(b*x^3+a)^(4/3)*((b*x^3+a)^(4/3)*(a^2*d^2-24/5*a* b*c*d+54/5*b^2*c^2)*d^2*a^2*(d*x^3+c)^2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a )^(1/3))/x)+12/5*(5/8*a^3*x^3*(b*x^3+a)^2*d^5+(b*x^3+a)^2*(-21/8*b*x^3+a)* c*a^2*d^4-3*b*(45/16*b^3*x^9+4*a*b^2*x^6+2*a^2*b*x^3+a^3)*c^2*a*d^3-18*x^3 *b^3*(-3/32*b^2*x^6+13/16*a*b*x^3+a^2)*c^3*d^2-9*b^3*(-3/8*b^2*x^6+7/16*a* b*x^3+a^2)*c^4*d+9/4*b^4*(3/4*b*x^3+a)*c^5)*x*c*((a*d-b*c)/c)^(1/3)+(arcta n(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)-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))*(a^2*d^2-24/5*a*b*c*d+54/5*b^2*c^2)*d^2*a^ 2*(b*x^3+a)^(4/3)*(d*x^3+c)^2)/(d*x^3+c)^2/c^3/(a*d-b*c)^4/a^2
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^3} \,d x \]