Optimal. Leaf size=70 \[ \frac {3 b^2 x^2}{2 a^4}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2}-\frac {b^4}{2 a^5 \left (b+a x^2\right )}-\frac {2 b^3 \log \left (b+a x^2\right )}{a^5} \]
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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45}
\begin {gather*} -\frac {b^4}{2 a^5 \left (a x^2+b\right )}-\frac {2 b^3 \log \left (a x^2+b\right )}{a^5}+\frac {3 b^2 x^2}{2 a^4}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+\frac {b}{x^2}\right )^2} \, dx &=\int \frac {x^9}{\left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {3 b^2}{a^4}-\frac {2 b x}{a^3}+\frac {x^2}{a^2}+\frac {b^4}{a^4 (b+a x)^2}-\frac {4 b^3}{a^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {3 b^2 x^2}{2 a^4}-\frac {b x^4}{2 a^3}+\frac {x^6}{6 a^2}-\frac {b^4}{2 a^5 \left (b+a x^2\right )}-\frac {2 b^3 \log \left (b+a x^2\right )}{a^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 60, normalized size = 0.86 \begin {gather*} \frac {9 a b^2 x^2-3 a^2 b x^4+a^3 x^6-\frac {3 b^4}{b+a x^2}-12 b^3 \log \left (b+a x^2\right )}{6 a^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 66, normalized size = 0.94
method | result | size |
risch | \(\frac {3 b^{2} x^{2}}{2 a^{4}}-\frac {b \,x^{4}}{2 a^{3}}+\frac {x^{6}}{6 a^{2}}-\frac {b^{4}}{2 a^{5} \left (a \,x^{2}+b \right )}-\frac {2 b^{3} \ln \left (a \,x^{2}+b \right )}{a^{5}}\) | \(63\) |
norman | \(\frac {-\frac {2 b^{4}}{a^{5}}+\frac {b^{2} x^{4}}{a^{3}}+\frac {x^{8}}{6 a}-\frac {b \,x^{6}}{3 a^{2}}}{a \,x^{2}+b}-\frac {2 b^{3} \ln \left (a \,x^{2}+b \right )}{a^{5}}\) | \(64\) |
default | \(\frac {\frac {1}{6} a^{2} x^{6}-\frac {1}{2} a b \,x^{4}+\frac {3}{2} b^{2} x^{2}}{a^{4}}-\frac {b^{3} \left (\frac {b}{a \left (a \,x^{2}+b \right )}+\frac {4 \ln \left (a \,x^{2}+b \right )}{a}\right )}{2 a^{4}}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 65, normalized size = 0.93 \begin {gather*} -\frac {b^{4}}{2 \, {\left (a^{6} x^{2} + a^{5} b\right )}} - \frac {2 \, b^{3} \log \left (a x^{2} + b\right )}{a^{5}} + \frac {a^{2} x^{6} - 3 \, a b x^{4} + 9 \, b^{2} x^{2}}{6 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 81, normalized size = 1.16 \begin {gather*} \frac {a^{4} x^{8} - 2 \, a^{3} b x^{6} + 6 \, a^{2} b^{2} x^{4} + 9 \, a b^{3} x^{2} - 3 \, b^{4} - 12 \, {\left (a b^{3} x^{2} + b^{4}\right )} \log \left (a x^{2} + b\right )}{6 \, {\left (a^{6} x^{2} + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 66, normalized size = 0.94 \begin {gather*} - \frac {b^{4}}{2 a^{6} x^{2} + 2 a^{5} b} + \frac {x^{6}}{6 a^{2}} - \frac {b x^{4}}{2 a^{3}} + \frac {3 b^{2} x^{2}}{2 a^{4}} - \frac {2 b^{3} \log {\left (a x^{2} + b \right )}}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.73, size = 80, normalized size = 1.14 \begin {gather*} -\frac {2 \, b^{3} \log \left ({\left | a x^{2} + b \right |}\right )}{a^{5}} + \frac {a^{4} x^{6} - 3 \, a^{3} b x^{4} + 9 \, a^{2} b^{2} x^{2}}{6 \, a^{6}} + \frac {4 \, a b^{3} x^{2} + 3 \, b^{4}}{2 \, {\left (a x^{2} + b\right )} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 68, normalized size = 0.97 \begin {gather*} \frac {x^6}{6\,a^2}-\frac {b^4}{2\,a\,\left (a^5\,x^2+b\,a^4\right )}-\frac {b\,x^4}{2\,a^3}-\frac {2\,b^3\,\ln \left (a\,x^2+b\right )}{a^5}+\frac {3\,b^2\,x^2}{2\,a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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