Optimal. Leaf size=65 \[ \frac {x^2}{2 a^3}+\frac {b^3}{4 a^4 \left (b+a x^2\right )^2}-\frac {3 b^2}{2 a^4 \left (b+a x^2\right )}-\frac {3 b \log \left (b+a x^2\right )}{2 a^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {269, 272, 45}
\begin {gather*} \frac {b^3}{4 a^4 \left (a x^2+b\right )^2}-\frac {3 b^2}{2 a^4 \left (a x^2+b\right )}-\frac {3 b \log \left (a x^2+b\right )}{2 a^4}+\frac {x^2}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {x}{\left (a+\frac {b}{x^2}\right )^3} \, dx &=\int \frac {x^7}{\left (b+a x^2\right )^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^3}-\frac {b^3}{a^3 (b+a x)^3}+\frac {3 b^2}{a^3 (b+a x)^2}-\frac {3 b}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 a^3}+\frac {b^3}{4 a^4 \left (b+a x^2\right )^2}-\frac {3 b^2}{2 a^4 \left (b+a x^2\right )}-\frac {3 b \log \left (b+a x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 48, normalized size = 0.74 \begin {gather*} -\frac {-2 a x^2+\frac {b^2 \left (5 b+6 a x^2\right )}{\left (b+a x^2\right )^2}+6 b \log \left (b+a x^2\right )}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 62, normalized size = 0.95
method | result | size |
norman | \(\frac {-\frac {3 b^{2} x^{2}}{a^{3}}+\frac {x^{6}}{2 a}-\frac {9 b^{3}}{4 a^{4}}}{\left (a \,x^{2}+b \right )^{2}}-\frac {3 b \ln \left (a \,x^{2}+b \right )}{2 a^{4}}\) | \(54\) |
risch | \(\frac {x^{2}}{2 a^{3}}+\frac {-\frac {3 b^{2} x^{2}}{2}-\frac {5 b^{3}}{4 a}}{a^{3} \left (a \,x^{2}+b \right )^{2}}-\frac {3 b \ln \left (a \,x^{2}+b \right )}{2 a^{4}}\) | \(54\) |
default | \(\frac {x^{2}}{2 a^{3}}-\frac {b \left (-\frac {b^{2}}{2 a \left (a \,x^{2}+b \right )^{2}}+\frac {3 b}{a \left (a \,x^{2}+b \right )}+\frac {3 \ln \left (a \,x^{2}+b \right )}{a}\right )}{2 a^{3}}\) | \(62\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 66, normalized size = 1.02 \begin {gather*} -\frac {6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} + \frac {x^{2}}{2 \, a^{3}} - \frac {3 \, b \log \left (a x^{2} + b\right )}{2 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 91, normalized size = 1.40 \begin {gather*} \frac {2 \, a^{3} x^{6} + 4 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} - 5 \, b^{3} - 6 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 68, normalized size = 1.05 \begin {gather*} \frac {- 6 a b^{2} x^{2} - 5 b^{3}}{4 a^{6} x^{4} + 8 a^{5} b x^{2} + 4 a^{4} b^{2}} + \frac {x^{2}}{2 a^{3}} - \frac {3 b \log {\left (a x^{2} + b \right )}}{2 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 53, normalized size = 0.82 \begin {gather*} \frac {x^{2}}{2 \, a^{3}} - \frac {3 \, b \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} - \frac {6 \, a b^{2} x^{2} + 5 \, b^{3}}{4 \, {\left (a x^{2} + b\right )}^{2} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.11, size = 68, normalized size = 1.05 \begin {gather*} \frac {x^2}{2\,a^3}-\frac {\frac {5\,b^3}{4\,a}+\frac {3\,b^2\,x^2}{2}}{a^5\,x^4+2\,a^4\,b\,x^2+a^3\,b^2}-\frac {3\,b\,\ln \left (a\,x^2+b\right )}{2\,a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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