Optimal. Leaf size=132 \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b^2 \log (x)}{9 a^{2/3}}-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4} \]
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Rubi [A] time = 0.087383, antiderivative size = 132, 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, 47, 57, 617, 204, 31} \[ \frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}-\frac{b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b^2 \log (x)}{9 a^{2/3}}-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 47
Rule 57
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{4/3}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^{4/3}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4}+\frac{1}{3} b \operatorname{Subst}\left (\int \frac{\sqrt [3]{a+b x}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4}+\frac{1}{9} b^2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4}-\frac{b^2 \log (x)}{9 a^{2/3}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{6 \sqrt [3]{a}}\\ &=-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4}-\frac{b^2 \log (x)}{9 a^{2/3}}+\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{3 a^{2/3}}\\ &=-\frac{b \sqrt [3]{a+b x^2}}{3 x^2}-\frac{\left (a+b x^2\right )^{4/3}}{4 x^4}-\frac{b^2 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt{3} a^{2/3}}-\frac{b^2 \log (x)}{9 a^{2/3}}+\frac{b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.009028, size = 39, normalized size = 0.3 \[ -\frac{3 b^2 \left (a+b x^2\right )^{7/3} \, _2F_1\left (\frac{7}{3},3;\frac{10}{3};\frac{b x^2}{a}+1\right )}{14 a^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{5}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.74868, size = 471, normalized size = 3.57 \begin{align*} -\frac{4 \, \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 ) + 2 \,{\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 ) - 4 \,{\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 (7 \, a^{2} b x^{2} + 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{36 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.84654, size = 42, normalized size = 0.32 \begin{align*} - \frac{b^{\frac{4}{3}} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac{4}{3}} \Gamma \left (\frac{5}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.57951, size = 167, normalized size = 1.27 \begin{align*} -\frac{1}{36} \, b^{2}{\left (\frac{4 \, \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{2}{3}}} + \frac{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 )}{a^{\frac{2}{3}}} - \frac{4 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}}} + \frac{3 \,{\left (7 \,{\left (b x^{2} + a\right )}^{\frac{4}{3}} - 4 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} a\right )}}{b^{2} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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