Optimal. Leaf size=104 \[ -\frac{\left (a+b x^2\right )^{2/3}}{2 x^2}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}+\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{b \log (x)}{3 \sqrt [3]{a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0663066, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 47, 55, 617, 204, 31} \[ -\frac{\left (a+b x^2\right )^{2/3}}{2 x^2}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}+\frac{b \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{b \log (x)}{3 \sqrt [3]{a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 47
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{2/3}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{2/3}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{2 x^2}+\frac{1}{3} b \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{2 x^2}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{2 x^2}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}-\frac{b \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{\sqrt [3]{a}}\\ &=-\frac{\left (a+b x^2\right )^{2/3}}{2 x^2}+\frac{b \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a}}-\frac{b \log (x)}{3 \sqrt [3]{a}}+\frac{b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{2 \sqrt [3]{a}}\\ \end{align*}
Mathematica [C] time = 0.0078281, size = 37, normalized size = 0.36 \[ \frac{3 b \left (a+b x^2\right )^{5/3} \, _2F_1\left (\frac{5}{3},2;\frac{8}{3};\frac{b x^2}{a}+1\right )}{10 a^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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 = 1.8534, size = 819, normalized size = 7.88 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} a b x^{2} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} \log \left (\frac{2 \, b x^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{\frac{2}{3}} -{\left (b x^{2} + a\right )}^{\frac{1}{3}} a - a^{\frac{4}{3}}\right )} \sqrt{-\frac{1}{a^{\frac{2}{3}}}} - 3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{\frac{2}{3}} + 3 \, a}{x^{2}}\right ) - a^{\frac{2}{3}} b x^{2} \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 ) + 2 \, a^{\frac{2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - 3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{6 \, a x^{2}}, \frac{6 \, \sqrt{\frac{1}{3}} a^{\frac{2}{3}} b x^{2} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{a^{\frac{1}{3}}}\right ) - a^{\frac{2}{3}} b x^{2} \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 ) + 2 \, a^{\frac{2}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - 3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a}{6 \, a x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 1.31117, size = 42, normalized size = 0.4 \begin{align*} - \frac{b^{\frac{2}{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 5.31429, size = 144, normalized size = 1.38 \begin{align*} \frac{1}{6} \,{\left (\frac{2 \, \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{1}{3}}} - \frac{\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{1}{3}}} + \frac{2 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{1}{3}}} - \frac{3 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{b x^{2}}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]