Optimal. Leaf size=135 \[ -\frac {\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac {b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac {b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 135, 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 = {272, 43, 44, 57,
631, 210, 31} \begin {gather*} -\frac {b^2 \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^2}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac {\left (a+b x^2\right )^{2/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 43
Rule 44
Rule 57
Rule 210
Rule 272
Rule 631
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{2/3}}{x^5} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x^3} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{2/3}}{4 x^4}+\frac {1}{6} b \text {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{a+b x}} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac {b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^2\right )}{18 a}\\ &=-\frac {\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac {b \left (a+b x^2\right )^{2/3}}{6 a x^2}+\frac {b^2 \log (x)}{18 a^{4/3}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}-\frac {b^2 \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^2}\right )}{12 a}\\ &=-\frac {\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac {b \left (a+b x^2\right )^{2/3}}{6 a x^2}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}\right )}{6 a^{4/3}}\\ &=-\frac {\left (a+b x^2\right )^{2/3}}{4 x^4}-\frac {b \left (a+b x^2\right )^{2/3}}{6 a x^2}-\frac {b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{6 \sqrt {3} a^{4/3}}+\frac {b^2 \log (x)}{18 a^{4/3}}-\frac {b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{12 a^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 158, normalized size = 1.17 \begin {gather*} \frac {\left (-3 a-2 b x^2\right ) \left (a+b x^2\right )^{2/3}}{12 a x^4}-\frac {b^2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+b x^2}}{\sqrt {3} \sqrt [3]{a}}\right )}{6 \sqrt {3} a^{4/3}}-\frac {b^2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^2}\right )}{18 a^{4/3}}+\frac {b^2 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )}{36 a^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{\frac {2}{3}}}{x^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 155, normalized size = 1.15 \begin {gather*} -\frac {\sqrt {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 )}{18 \, a^{\frac {4}{3}}} + \frac {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 )}{36 \, a^{\frac {4}{3}}} - \frac {b^{2} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right )}{18 \, a^{\frac {4}{3}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} b^{2} + {\left (b x^{2} + a\right )}^{\frac {2}{3}} a b^{2}}{12 \, {\left ({\left (b x^{2} + a\right )}^{2} a - 2 \, {\left (b x^{2} + a\right )} a^{2} + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.97, size = 380, normalized size = 2.81 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b^{2} x^{4} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} a + \left (-a\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a\right )^{\frac {1}{3}}}{a}} - 3 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + 3 \, a}{x^{2}}\right ) + \left (-a\right )^{\frac {2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) - 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) - 3 \, {\left (2 \, a b x^{2} + 3 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{36 \, a^{2} x^{4}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b^{2} x^{4} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {1}{3}} - \left (-a\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a\right )^{\frac {1}{3}}}{a}}\right ) - \left (-a\right )^{\frac {2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac {2}{3}} - {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + \left (-a\right )^{\frac {2}{3}}\right ) + 2 \, \left (-a\right )^{\frac {2}{3}} b^{2} x^{4} \log \left ({\left (b x^{2} + a\right )}^{\frac {1}{3}} + \left (-a\right )^{\frac {1}{3}}\right ) + 3 \, {\left (2 \, a b x^{2} + 3 \, a^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}}}{36 \, a^{2} x^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.10, size = 42, normalized size = 0.31 \begin {gather*} - \frac {b^{\frac {2}{3}} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 x^{\frac {8}{3}} \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.90, size = 141, normalized size = 1.04 \begin {gather*} -\frac {\frac {2 \, \sqrt {3} b^{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 {4}{3}}} - \frac {b^{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 )}{a^{\frac {4}{3}}} + \frac {2 \, b^{3} \log \left ({\left | {\left (b x^{2} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}} \right |}\right )}{a^{\frac {4}{3}}} + \frac {3 \, {\left (2 \, {\left (b x^{2} + a\right )}^{\frac {5}{3}} b^{3} + {\left (b x^{2} + a\right )}^{\frac {2}{3}} a b^{3}\right )}}{a b^{2} x^{4}}}{36 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.16, size = 212, normalized size = 1.57 \begin {gather*} \frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left ({\left (b\,x^2+a\right )}^{1/3}-{\left (-1\right )}^{2/3}\,a^{1/3}\right )}{18\,a^{4/3}}-\frac {\frac {b^2\,{\left (b\,x^2+a\right )}^{2/3}}{6}+\frac {b^2\,{\left (b\,x^2+a\right )}^{5/3}}{3\,a}}{2\,{\left (b\,x^2+a\right )}^2-4\,a\,\left (b\,x^2+a\right )+2\,a^2}+\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left (\frac {b^4\,{\left (b\,x^2+a\right )}^{1/3}}{36\,a^2}-\frac {{\left (-1\right )}^{2/3}\,b^4\,{\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{36\,a^{5/3}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,a^{4/3}}-\frac {{\left (-1\right )}^{1/3}\,b^2\,\ln \left (\frac {b^4\,{\left (b\,x^2+a\right )}^{1/3}}{36\,a^2}-\frac {{\left (-1\right )}^{2/3}\,b^4\,{\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}^2}{36\,a^{5/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{18\,a^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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