Optimal. Leaf size=159 \[ \frac{7 b^2}{3 a^3 \sqrt [3]{a+b x^2}}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{10/3}}+\frac{7 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}-\frac{7 b^2 \log (x)}{9 a^{10/3}}+\frac{7 b}{12 a^2 x^2 \sqrt [3]{a+b x^2}}-\frac{1}{4 a x^4 \sqrt [3]{a+b x^2}} \]
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Rubi [A] time = 0.105579, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {266, 51, 55, 617, 204, 31} \[ \frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{10/3}}+\frac{7 b^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{3 \sqrt{3} a^{10/3}}-\frac{7 b^2 \log (x)}{9 a^{10/3}}+\frac{7 b \left (a+b x^2\right )^{2/3}}{3 a^3 x^2}-\frac{7 \left (a+b x^2\right )^{2/3}}{4 a^2 x^4}+\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+b x^2\right )^{4/3}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}}-\frac{7 \left (a+b x^2\right )^{2/3}}{4 a^2 x^4}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{3 a^2}\\ &=\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}}-\frac{7 \left (a+b x^2\right )^{2/3}}{4 a^2 x^4}+\frac{7 b \left (a+b x^2\right )^{2/3}}{3 a^3 x^2}+\frac{\left (7 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{9 a^3}\\ &=\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}}-\frac{7 \left (a+b x^2\right )^{2/3}}{4 a^2 x^4}+\frac{7 b \left (a+b x^2\right )^{2/3}}{3 a^3 x^2}-\frac{7 b^2 \log (x)}{9 a^{10/3}}-\frac{\left (7 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{6 a^{10/3}}+\frac{\left (7 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 )}{6 a^3}\\ &=\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}}-\frac{7 \left (a+b x^2\right )^{2/3}}{4 a^2 x^4}+\frac{7 b \left (a+b x^2\right )^{2/3}}{3 a^3 x^2}-\frac{7 b^2 \log (x)}{9 a^{10/3}}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{10/3}}-\frac{\left (7 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 )}{3 a^{10/3}}\\ &=\frac{3}{2 a x^4 \sqrt [3]{a+b x^2}}-\frac{7 \left (a+b x^2\right )^{2/3}}{4 a^2 x^4}+\frac{7 b \left (a+b x^2\right )^{2/3}}{3 a^3 x^2}+\frac{7 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^{10/3}}-\frac{7 b^2 \log (x)}{9 a^{10/3}}+\frac{7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{6 a^{10/3}}\\ \end{align*}
Mathematica [C] time = 0.0074407, size = 39, normalized size = 0.25 \[ \frac{3 b^2 \, _2F_1\left (-\frac{1}{3},3;\frac{2}{3};\frac{b x^2}{a}+1\right )}{2 a^3 \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, 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 = 1.8441, size = 1107, normalized size = 6.96 \begin{align*} \left [\frac{42 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \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 ) - 14 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} a^{\frac{2}{3}} \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 ) + 28 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} a^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) + 3 \,{\left (28 \, a b^{2} x^{4} + 7 \, a^{2} b x^{2} - 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{36 \,{\left (a^{4} b x^{6} + a^{5} x^{4}\right )}}, -\frac{14 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} a^{\frac{2}{3}} \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 ) - 28 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} a^{\frac{2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}}\right ) - \frac{84 \, \sqrt{\frac{1}{3}}{\left (a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \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{1}{3}}} - 3 \,{\left (28 \, a b^{2} x^{4} + 7 \, a^{2} b x^{2} - 3 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{2}{3}}}{36 \,{\left (a^{4} b x^{6} + a^{5} x^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.99542, size = 41, normalized size = 0.26 \begin{align*} - \frac{\Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{4}{3}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac{4}{3}} x^{\frac{20}{3}} \Gamma \left (\frac{13}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.47891, size = 190, normalized size = 1.19 \begin{align*} \frac{1}{36} \, b^{2}{\left (\frac{28 \, \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{10}{3}}} - \frac{14 \, \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{10}{3}}} + \frac{28 \, \log \left ({\left |{\left (b x^{2} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{10}{3}}} + \frac{54}{{\left (b x^{2} + a\right )}^{\frac{1}{3}} a^{3}} + \frac{3 \,{\left (10 \,{\left (b x^{2} + a\right )}^{\frac{5}{3}} - 13 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a\right )}}{a^{3} b^{2} x^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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