Integrand size = 19, antiderivative size = 1357 \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\frac {3 c^4 d^2 \sqrt [3]{a+b x^3}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {3 c^2 d^4 x^2 \sqrt [3]{a+b x^3}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {5 b c^4 d^2 \sqrt [3]{a+b x^3}}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c d^2 \left (b c^3-6 a d^3\right ) \sqrt [3]{a+b x^3}}{6 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {d^4 \left (9 b c^3-4 a d^3\right ) x^2 \sqrt [3]{a+b x^3}}{6 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {d^4 \left (3 b c^3+2 a d^3\right ) x^2 \sqrt [3]{a+b x^3}}{3 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{c^3 \left (a+b x^3\right )^{2/3}}-\frac {7 d^3 x^4 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},3,\frac {7}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{4 c^6 \left (a+b x^3\right )^{2/3}}+\frac {d^6 x^7 \left (1+\frac {b x^3}{a}\right )^{2/3} \operatorname {AppellF1}\left (\frac {7}{3},\frac {2}{3},3,\frac {10}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right )}{7 c^9 \left (a+b x^3\right )^{2/3}}+\frac {2 a d^4 \left (6 b c^3-a d^3\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b c^3 d^3+2 a^2 d^6\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {10 b^2 c^4 d \arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \arctan \left (\frac {1-\frac {2 d \sqrt [3]{a+b x^3}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}-\frac {5 b^2 c^4 d \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{8/3}}-\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {d \left (9 b^2 c^6-6 a b c^3 d^3+2 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{18 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{3 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b c^3 d^3+2 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{a+b x^3}\right )}{6 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b^2 c^4 d \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{3 \left (b c^3-a d^3\right )^{8/3}}-\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (\sqrt [3]{b c^3-a d^3}+d \sqrt [3]{a+b x^3}\right )}{6 \left (b c^3-a d^3\right )^{8/3}} \]
3/2*c^4*d^2*(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2+3/2*c^2*d^4*x^2 *(b*x^3+a)^(1/3)/(-a*d^3+b*c^3)/(d^3*x^3+c^3)^2+5/3*b*c^4*d^2*(b*x^3+a)^(1 /3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)-1/6*c*d^2*(-6*a*d^3+b*c^3)*(b*x^3+a)^(1 /3)/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+1/6*d^4*(-4*a*d^3+9*b*c^3)*x^2*(b*x^3+a )^(1/3)/c/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+1/3*d^4*(2*a*d^3+3*b*c^3)*x^2*(b* x^3+a)^(1/3)/c/(-a*d^3+b*c^3)^2/(d^3*x^3+c^3)+x*(1+b*x^3/a)^(2/3)*AppellF1 (1/3,2/3,3,4/3,-b*x^3/a,-d^3*x^3/c^3)/c^3/(b*x^3+a)^(2/3)-7/4*d^3*x^4*(1+b *x^3/a)^(2/3)*AppellF1(4/3,2/3,3,7/3,-b*x^3/a,-d^3*x^3/c^3)/c^6/(b*x^3+a)^ (2/3)+1/7*d^6*x^7*(1+b*x^3/a)^(2/3)*AppellF1(7/3,2/3,3,10/3,-b*x^3/a,-d^3* x^3/c^3)/c^9/(b*x^3+a)^(2/3)-5/9*b^2*c^4*d*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^ (8/3)+1/18*b*c*d*(-6*a*d^3+b*c^3)*ln(d^3*x^3+c^3)/(-a*d^3+b*c^3)^(8/3)-1/9 *a*d^4*(-a*d^3+6*b*c^3)*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^(8/3)-1/18*d*(2 *a^2*d^6-6*a*b*c^3*d^3+9*b^2*c^6)*ln(d^3*x^3+c^3)/c^2/(-a*d^3+b*c^3)^(8/3) +1/3*a*d^4*(-a*d^3+6*b*c^3)*ln((-a*d^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c ^2/(-a*d^3+b*c^3)^(8/3)+1/6*d*(2*a^2*d^6-6*a*b*c^3*d^3+9*b^2*c^6)*ln((-a*d ^3+b*c^3)^(1/3)*x/c-(b*x^3+a)^(1/3))/c^2/(-a*d^3+b*c^3)^(8/3)+5/3*b^2*c^4* d*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)^(1/3))/(-a*d^3+b*c^3)^(8/3)-1/6*b*c* d*(-6*a*d^3+b*c^3)*ln((-a*d^3+b*c^3)^(1/3)+d*(b*x^3+a)^(1/3))/(-a*d^3+b*c^ 3)^(8/3)+2/9*a*d^4*(-a*d^3+6*b*c^3)*arctan(1/3*(1+2*(-a*d^3+b*c^3)^(1/3)*x /c/(b*x^3+a)^(1/3))*3^(1/2))/c^2/(-a*d^3+b*c^3)^(8/3)*3^(1/2)+1/9*d*(2*...
\[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx \]
Time = 2.04 (sec) , antiderivative size = 1357, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2581, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^3\right )^{2/3} (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 2581 |
| \(\displaystyle \int \left (-\frac {7 c^3 d^3 x^3}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}+\frac {d^6 x^6}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}-\frac {3 c d^5 x^5}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}+\frac {c^6}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}-\frac {3 c^5 d x}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}+\frac {6 c^4 d^2 x^2}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}+\frac {6 c^2 d^4 x^4}{\left (a+b x^3\right )^{2/3} \left (c^3+d^3 x^3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^6 \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {7}{3},\frac {2}{3},3,\frac {10}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^7}{7 c^9 \left (b x^3+a\right )^{2/3}}-\frac {7 d^3 \left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {4}{3},\frac {2}{3},3,\frac {7}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x^4}{4 c^6 \left (b x^3+a\right )^{2/3}}+\frac {d^4 \left (3 b c^3+2 a d^3\right ) \sqrt [3]{b x^3+a} x^2}{3 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {d^4 \left (9 b c^3-4 a d^3\right ) \sqrt [3]{b x^3+a} x^2}{6 c \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^2 d^4 \sqrt [3]{b x^3+a} x^2}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}+\frac {\left (\frac {b x^3}{a}+1\right )^{2/3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {2}{3},3,\frac {4}{3},-\frac {b x^3}{a},-\frac {d^3 x^3}{c^3}\right ) x}{c^3 \left (b x^3+a\right )^{2/3}}+\frac {2 a d^4 \left (6 b c^3-a d^3\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \arctan \left (\frac {\frac {2 \sqrt [3]{b c^3-a d^3} x}{c \sqrt [3]{b x^3+a}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {10 b^2 c^4 d \arctan \left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \arctan \left (\frac {1-\frac {2 d \sqrt [3]{b x^3+a}}{\sqrt [3]{b c^3-a d^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \left (b c^3-a d^3\right )^{8/3}}-\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (c^3+d^3 x^3\right )}{9 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \log \left (c^3+d^3 x^3\right )}{18 c^2 \left (b c^3-a d^3\right )^{8/3}}-\frac {5 b^2 c^4 d \log \left (c^3+d^3 x^3\right )}{9 \left (b c^3-a d^3\right )^{8/3}}+\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (c^3+d^3 x^3\right )}{18 \left (b c^3-a d^3\right )^{8/3}}+\frac {a d^4 \left (6 b c^3-a d^3\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{3 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {d \left (9 b^2 c^6-6 a b d^3 c^3+2 a^2 d^6\right ) \log \left (\frac {\sqrt [3]{b c^3-a d^3} x}{c}-\sqrt [3]{b x^3+a}\right )}{6 c^2 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b^2 c^4 d \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{3 \left (b c^3-a d^3\right )^{8/3}}-\frac {b c d \left (b c^3-6 a d^3\right ) \log \left (\sqrt [3]{b x^3+a} d+\sqrt [3]{b c^3-a d^3}\right )}{6 \left (b c^3-a d^3\right )^{8/3}}+\frac {5 b c^4 d^2 \sqrt [3]{b x^3+a}}{3 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}-\frac {c d^2 \left (b c^3-6 a d^3\right ) \sqrt [3]{b x^3+a}}{6 \left (b c^3-a d^3\right )^2 \left (c^3+d^3 x^3\right )}+\frac {3 c^4 d^2 \sqrt [3]{b x^3+a}}{2 \left (b c^3-a d^3\right ) \left (c^3+d^3 x^3\right )^2}\) |
(3*c^4*d^2*(a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (3*c ^2*d^4*x^2*(a + b*x^3)^(1/3))/(2*(b*c^3 - a*d^3)*(c^3 + d^3*x^3)^2) + (5*b *c^4*d^2*(a + b*x^3)^(1/3))/(3*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) - (c*d^2 *(b*c^3 - 6*a*d^3)*(a + b*x^3)^(1/3))/(6*(b*c^3 - a*d^3)^2*(c^3 + d^3*x^3) ) + (d^4*(9*b*c^3 - 4*a*d^3)*x^2*(a + b*x^3)^(1/3))/(6*c*(b*c^3 - a*d^3)^2 *(c^3 + d^3*x^3)) + (d^4*(3*b*c^3 + 2*a*d^3)*x^2*(a + b*x^3)^(1/3))/(3*c*( b*c^3 - a*d^3)^2*(c^3 + d^3*x^3)) + (x*(1 + (b*x^3)/a)^(2/3)*AppellF1[1/3, 2/3, 3, 4/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(c^3*(a + b*x^3)^(2/3)) - ( 7*d^3*x^4*(1 + (b*x^3)/a)^(2/3)*AppellF1[4/3, 2/3, 3, 7/3, -((b*x^3)/a), - ((d^3*x^3)/c^3)])/(4*c^6*(a + b*x^3)^(2/3)) + (d^6*x^7*(1 + (b*x^3)/a)^(2/ 3)*AppellF1[7/3, 2/3, 3, 10/3, -((b*x^3)/a), -((d^3*x^3)/c^3)])/(7*c^9*(a + b*x^3)^(2/3)) + (2*a*d^4*(6*b*c^3 - a*d^3)*ArcTan[(1 + (2*(b*c^3 - a*d^3 )^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c^2*(b*c^3 - a*d^3) ^(8/3)) + (d*(9*b^2*c^6 - 6*a*b*c^3*d^3 + 2*a^2*d^6)*ArcTan[(1 + (2*(b*c^3 - a*d^3)^(1/3)*x)/(c*(a + b*x^3)^(1/3)))/Sqrt[3]])/(3*Sqrt[3]*c^2*(b*c^3 - a*d^3)^(8/3)) - (10*b^2*c^4*d*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/Sqrt[3]])/(3*Sqrt[3]*(b*c^3 - a*d^3)^(8/3)) + (b*c*d*(b*c ^3 - 6*a*d^3)*ArcTan[(1 - (2*d*(a + b*x^3)^(1/3))/(b*c^3 - a*d^3)^(1/3))/S qrt[3]])/(3*Sqrt[3]*(b*c^3 - a*d^3)^(8/3)) - (5*b^2*c^4*d*Log[c^3 + d^3*x^ 3])/(9*(b*c^3 - a*d^3)^(8/3)) + (b*c*d*(b*c^3 - 6*a*d^3)*Log[c^3 + d^3*...
3.1.42.3.1 Defintions of rubi rules used
Int[(Px_.)*((c_) + (d_.)*(x_))^(q_)*((a_) + (b_.)*(x_)^3)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c^3 + d^3*x^3)^q*(a + b*x^3)^p, Px/(c^2 - c*d*x + d ^2*x^2)^q, x], x] /; FreeQ[{a, b, c, d, p}, x] && PolyQ[Px, x] && ILtQ[q, 0 ] && RationalQ[p] && EqQ[Denominator[p], 3]
\[\int \frac {1}{\left (d x +c \right )^{3} \left (b \,x^{3}+a \right )^{\frac {2}{3}}}d x\]
Timed out. \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x\right )^{3}}\, dx \]
\[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3}} \,d x } \]
\[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{(c+d x)^3 \left (a+b x^3\right )^{2/3}} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{2/3}\,{\left (c+d\,x\right )}^3} \,d x \]