Optimal. Leaf size=283 \[ \frac{b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac{11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}-\frac{11 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} d}+\frac{\sqrt [3]{2} b^2 \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} d}-\frac{11 b^2 \log (x)}{18 a^{8/3} d}-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.264317, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {446, 103, 149, 156, 57, 617, 204, 31} \[ \frac{b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac{11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}-\frac{11 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{9 \sqrt{3} a^{8/3} d}+\frac{\sqrt [3]{2} b^2 \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{8/3} d}-\frac{11 b^2 \log (x)}{18 a^{8/3} d}-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 446
Rule 103
Rule 149
Rule 156
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^3}}{x^7 \left (a d-b d x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x^3 (a d-b d x)} \, dx,x,x^3\right )\\ &=-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x} \left (-\frac{4}{3} a b d-\frac{2}{3} b^2 d x\right )}{x^2 (a d-b d x)} \, dx,x,x^3\right )}{6 a^2 d}\\ &=-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{22}{9} a^2 b^2 d^2-\frac{14}{9} a b^3 d^2 x}{x (a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{6 a^3 d^2}\\ &=-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}+\frac{\left (2 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{3 a^2}+\frac{\left (11 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^3\right )}{27 a^2 d}\\ &=-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac{11 b^2 \log (x)}{18 a^{8/3} d}+\frac{b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}-\frac{\left (11 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}-\frac{\left (11 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{18 a^{7/3} d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}\\ &=-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac{11 b^2 \log (x)}{18 a^{8/3} d}+\frac{b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac{11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}+\frac{\left (11 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{9 a^{8/3} d}-\frac{\left (\sqrt [3]{2} b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{a^{8/3} d}\\ &=-\frac{2 b \sqrt [3]{a+b x^3}}{9 a^2 d x^3}-\frac{\left (a+b x^3\right )^{4/3}}{6 a^2 d x^6}-\frac{11 b^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{9 \sqrt{3} a^{8/3} d}+\frac{\sqrt [3]{2} b^2 \tan ^{-1}\left (\frac{1+\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} a^{8/3} d}-\frac{11 b^2 \log (x)}{18 a^{8/3} d}+\frac{b^2 \log \left (a-b x^3\right )}{3\ 2^{2/3} a^{8/3} d}+\frac{11 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{18 a^{8/3} d}-\frac{b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{8/3} d}\\ \end{align*}
Mathematica [A] time = 0.128286, size = 314, normalized size = 1.11 \[ -\frac{11 b^2 x^6 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-9 \sqrt [3]{2} b^2 x^6 \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+21 a^{2/3} b x^3 \sqrt [3]{a+b x^3}+9 a^{5/3} \sqrt [3]{a+b x^3}-22 b^2 x^6 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+18 \sqrt [3]{2} b^2 x^6 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )+22 \sqrt{3} b^2 x^6 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )-18 \sqrt [3]{2} \sqrt{3} b^2 x^6 \tan ^{-1}\left (\frac{\frac{2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{54 a^{8/3} d x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{7} \left ( -bd{x}^{3}+ad \right ) }\sqrt [3]{b{x}^{3}+a}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{{\left (b d x^{3} - a d\right )} x^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.38086, size = 952, normalized size = 3.36 \begin{align*} -\frac{18 \, \sqrt{3} 2^{\frac{1}{3}} a^{2} b^{2} x^{6} \left (-\frac{1}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{1}{3} \, \sqrt{3} 2^{\frac{2}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} a \left (-\frac{1}{a^{2}}\right )^{\frac{2}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 9 \cdot 2^{\frac{1}{3}} a^{2} b^{2} x^{6} \left (-\frac{1}{a^{2}}\right )^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} a^{2} \left (-\frac{1}{a^{2}}\right )^{\frac{2}{3}} - 2^{\frac{1}{3}}{\left (b x^{3} + a\right )}^{\frac{1}{3}} a \left (-\frac{1}{a^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{2}{3}}\right ) - 18 \cdot 2^{\frac{1}{3}} a^{2} b^{2} x^{6} \left (-\frac{1}{a^{2}}\right )^{\frac{1}{3}} \log \left (2^{\frac{1}{3}} a \left (-\frac{1}{a^{2}}\right )^{\frac{1}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}\right ) + 22 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}} a b^{2} x^{6} \arctan \left (\frac{{\left (a^{2}\right )}^{\frac{1}{6}}{\left (\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} a + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \, a^{2}}\right ) + 11 \,{\left (a^{2}\right )}^{\frac{2}{3}} b^{2} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 22 \,{\left (a^{2}\right )}^{\frac{2}{3}} b^{2} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) + 3 \,{\left (7 \, a^{2} b x^{3} + 3 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{54 \, a^{4} d x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]