Optimal. Leaf size=104 \[ \frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3}{2 a \sqrt [3]{a+b x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {266, 51, 55, 617, 204, 31} \[ \frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{2 a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3}{2 a \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 266
Rule 617
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2\right )^{4/3}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{4/3}} \, dx,x,x^2\right )\\ &=\frac {3}{2 a \sqrt [3]{a+b x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {3}{2 a \sqrt [3]{a+b x^2}}-\frac {\log (x)}{2 a^{4/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{4 a}\\ &=\frac {3}{2 a \sqrt [3]{a+b x^2}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{2 a^{4/3}}\\ &=\frac {3}{2 a \sqrt [3]{a+b x^2}}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{2 a^{4/3}}-\frac {\log (x)}{2 a^{4/3}}+\frac {3 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{4 a^{4/3}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.35 \[ \frac {3 \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};\frac {b x^2}{a}+1\right )}{2 a \sqrt [3]{a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.01, size = 327, normalized size = 3.14 \[ \left [\frac {\sqrt {3} {\left (a b x^{2} + a^{2}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{2} + \sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{2}}\right ) - {\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + 2 \, {\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{4 \, {\left (a^{2} b x^{2} + a^{3}\right )}}, -\frac {{\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 2 \, {\left (b x^{2} + a\right )} a^{\frac {2}{3}} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - \frac {2 \, \sqrt {3} {\left (a b x^{2} + a^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{a^{\frac {1}{3}}} - 6 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} a}{4 \, {\left (a^{2} b x^{2} + a^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 101, normalized size = 0.97 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{2 \, a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{4 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{2 \, a^{\frac {4}{3}}} + \frac {3}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {4}{3}} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.02, size = 100, normalized size = 0.96 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right )}{2 \, a^{\frac {4}{3}}} - \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right )}{4 \, a^{\frac {4}{3}}} + \frac {\log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{2 \, a^{\frac {4}{3}}} + \frac {3}{2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.59, size = 123, normalized size = 1.18 \[ \frac {\ln \left (18\,a\,{\left (b\,x^2+a\right )}^{1/3}-18\,a^{4/3}\right )}{2\,a^{4/3}}+\frac {3}{2\,a\,{\left (b\,x^2+a\right )}^{1/3}}+\frac {\ln \left (18\,a\,{\left (b\,x^2+a\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,a^{4/3}}-\frac {\ln \left (18\,a\,{\left (b\,x^2+a\right )}^{1/3}-\frac {9\,a^{4/3}\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,a^{4/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.13, size = 41, normalized size = 0.39 \[ - \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {4}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 b^{\frac {4}{3}} x^{\frac {8}{3}} \Gamma \left (\frac {7}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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