Optimal. Leaf size=116 \[ -\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}+2 b \sqrt [3]{a+b x^2}+\sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )-\frac {2 \sqrt [3]{a} b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {2}{3} \sqrt [3]{a} b \log (x) \]
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Rubi [A] time = 0.08, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {266, 47, 50, 57, 617, 204, 31} \[ -\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}+2 b \sqrt [3]{a+b x^2}+\sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )-\frac {2 \sqrt [3]{a} b \tan ^{-1}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3}}-\frac {2}{3} \sqrt [3]{a} b \log (x) \]
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 50
Rule 57
Rule 204
Rule 266
Rule 617
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{4/3}}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^{4/3}}{x^2} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}+\frac {1}{3} (2 b) \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{x} \, dx,x,x^2\right )\\ &=2 b \sqrt [3]{a+b x^2}-\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}+\frac {1}{3} (2 a b) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{2/3}} \, dx,x,x^2\right )\\ &=2 b \sqrt [3]{a+b x^2}-\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}-\frac {2}{3} \sqrt [3]{a} b \log (x)-\left (\sqrt [3]{a} b\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )-\left (a^{2/3} b\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 )\\ &=2 b \sqrt [3]{a+b x^2}-\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}-\frac {2}{3} \sqrt [3]{a} b \log (x)+\sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )+\left (2 \sqrt [3]{a} b\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 )\\ &=2 b \sqrt [3]{a+b x^2}-\frac {\left (a+b x^2\right )^{4/3}}{2 x^2}-\frac {2 \sqrt [3]{a} b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \sqrt [3]{a} b \log (x)+\sqrt [3]{a} b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 37, normalized size = 0.32 \[ \frac {3 b \left (a+b x^2\right )^{7/3} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};\frac {b x^2}{a}+1\right )}{14 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 129, normalized size = 1.11 \[ -\frac {4 \, \sqrt {3} a^{\frac {1}{3}} b x^{2} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 2 \, a^{\frac {1}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b x^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) - 3 \, {\left (3 \, b x^{2} - a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.42, size = 131, normalized size = 1.13 \[ -\frac {4 \, \sqrt {3} a^{\frac {1}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) + 2 \, a^{\frac {1}{3}} b^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 4 \, a^{\frac {1}{3}} b^{2} \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right ) - 9 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b^{2} + \frac {3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} a b}{x^{2}}}{6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{2}+a \right )^{\frac {4}{3}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.90, size = 116, normalized size = 1.00 \[ -\frac {2}{3} \, \sqrt {3} a^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{3 \, a^{\frac {1}{3}}}\right ) - \frac {1}{3} \, a^{\frac {1}{3}} b \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} + {\left (b x^{2} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) + \frac {2}{3} \, a^{\frac {1}{3}} b \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + \frac {3}{2} \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} b - \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}} a}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.42, size = 141, normalized size = 1.22 \[ \frac {3\,b\,{\left (b\,x^2+a\right )}^{1/3}}{2}-\frac {a\,{\left (b\,x^2+a\right )}^{1/3}}{2\,x^2}+\frac {2\,a^{1/3}\,b\,\ln \left (6\,a^{4/3}\,b-6\,a\,b\,{\left (b\,x^2+a\right )}^{1/3}\right )}{3}+\frac {a^{1/3}\,b\,\ln \left (6\,a\,b\,{\left (b\,x^2+a\right )}^{1/3}-3\,a^{4/3}\,b\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{3}-\frac {a^{1/3}\,b\,\ln \left (3\,a^{4/3}\,b\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )+6\,a\,b\,{\left (b\,x^2+a\right )}^{1/3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.41, size = 46, normalized size = 0.40 \[ - \frac {b^{\frac {4}{3}} x^{\frac {2}{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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