Optimal. Leaf size=138 \[ \frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{8/3}}-\frac{5 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{8/3}}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b \sqrt [3]{a+b x^2}}{12 a^2 x^2}-\frac{\sqrt [3]{a+b x^2}}{4 a x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0883603, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 51, 57, 617, 204, 31} \[ \frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{8/3}}-\frac{5 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{6 \sqrt{3} a^{8/3}}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b \sqrt [3]{a+b x^2}}{12 a^2 x^2}-\frac{\sqrt [3]{a+b x^2}}{4 a x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+b x^2\right )^{2/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 a x^4}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{2/3}} \, dx,x,x^2\right )}{12 a}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 a x^4}+\frac{5 b \sqrt [3]{a+b x^2}}{12 a^2 x^2}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^2\right )}{18 a^2}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 a x^4}+\frac{5 b \sqrt [3]{a+b x^2}}{12 a^2 x^2}-\frac{5 b^2 \log (x)}{18 a^{8/3}}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{8/3}}-\frac{\left (5 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^2}\right )}{12 a^{7/3}}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 a x^4}+\frac{5 b \sqrt [3]{a+b x^2}}{12 a^2 x^2}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{8/3}}+\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{6 a^{8/3}}\\ &=-\frac{\sqrt [3]{a+b x^2}}{4 a x^4}+\frac{5 b \sqrt [3]{a+b x^2}}{12 a^2 x^2}-\frac{5 b^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 \sqrt{3} a^{8/3}}-\frac{5 b^2 \log (x)}{18 a^{8/3}}+\frac{5 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{8/3}}\\ \end{align*}
Mathematica [C] time = 0.0070714, size = 39, normalized size = 0.28 \[ -\frac{3 b^2 \sqrt [3]{a+b x^2} \, _2F_1\left (\frac{1}{3},3;\frac{4}{3};\frac{b x^2}{a}+1\right )}{2 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.77968, size = 474, normalized size = 3.43 \begin{align*} -\frac{10 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}} a b^{2} x^{4} \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^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \, a^{2}}\right ) + 5 \,{\left (a^{2}\right )}^{\frac{2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{2} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 10 \,{\left (a^{2}\right )}^{\frac{2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 3 \,{\left (5 \, a^{2} b x^{2} - 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{36 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.73589, size = 41, normalized size = 0.3 \begin{align*} - \frac{\Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac{2}{3}} x^{\frac{16}{3}} \Gamma \left (\frac{11}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 4.09054, size = 171, normalized size = 1.24 \begin{align*} -\frac{1}{36} \, b^{2}{\left (\frac{10 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{8}{3}}} + \frac{5 \, \log \left ({\left (b x^{2} + a\right )}^{\frac{2}{3}} +{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{8}{3}}} - \frac{10 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{8}{3}}} - \frac{3 \,{\left (5 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} - 8 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a\right )}}{a^{2} b^{2} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]