Optimal. Leaf size=57 \[ -\frac {b x^2}{a^3}+\frac {x^4}{4 a^2}+\frac {b^3}{2 a^4 \left (b+a x^2\right )}+\frac {3 b^2 \log \left (b+a x^2\right )}{2 a^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, 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^3}{2 a^4 \left (a x^2+b\right )}+\frac {3 b^2 \log \left (a x^2+b\right )}{2 a^4}-\frac {b x^2}{a^3}+\frac {x^4}{4 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^3}{\left (a+\frac {b}{x^2}\right )^2} \, dx &=\int \frac {x^7}{\left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3}{(b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {2 b}{a^3}+\frac {x}{a^2}-\frac {b^3}{a^3 (b+a x)^2}+\frac {3 b^2}{a^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b x^2}{a^3}+\frac {x^4}{4 a^2}+\frac {b^3}{2 a^4 \left (b+a x^2\right )}+\frac {3 b^2 \log \left (b+a x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 0.86 \begin {gather*} \frac {-4 a b x^2+a^2 x^4+\frac {2 b^3}{b+a x^2}+6 b^2 \log \left (b+a x^2\right )}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 54, normalized size = 0.95
method | result | size |
default | \(\frac {\left (a \,x^{2}-2 b \right )^{2}}{4 a^{4}}+\frac {b^{2} \left (\frac {b}{a \left (a \,x^{2}+b \right )}+\frac {3 \ln \left (a \,x^{2}+b \right )}{a}\right )}{2 a^{3}}\) | \(54\) |
norman | \(\frac {\frac {x^{6}}{4 a}-\frac {3 b \,x^{4}}{4 a^{2}}+\frac {3 b^{3}}{2 a^{4}}}{a \,x^{2}+b}+\frac {3 b^{2} \ln \left (a \,x^{2}+b \right )}{2 a^{4}}\) | \(54\) |
risch | \(\frac {x^{4}}{4 a^{2}}-\frac {b \,x^{2}}{a^{3}}+\frac {b^{2}}{a^{4}}+\frac {b^{3}}{2 a^{4} \left (a \,x^{2}+b \right )}+\frac {3 b^{2} \ln \left (a \,x^{2}+b \right )}{2 a^{4}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 54, normalized size = 0.95 \begin {gather*} \frac {b^{3}}{2 \, {\left (a^{5} x^{2} + a^{4} b\right )}} + \frac {3 \, b^{2} \log \left (a x^{2} + b\right )}{2 \, a^{4}} + \frac {a x^{4} - 4 \, b x^{2}}{4 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 70, normalized size = 1.23 \begin {gather*} \frac {a^{3} x^{6} - 3 \, a^{2} b x^{4} - 4 \, a b^{2} x^{2} + 2 \, b^{3} + 6 \, {\left (a b^{2} x^{2} + b^{3}\right )} \log \left (a x^{2} + b\right )}{4 \, {\left (a^{5} x^{2} + a^{4} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 53, normalized size = 0.93 \begin {gather*} \frac {b^{3}}{2 a^{5} x^{2} + 2 a^{4} b} + \frac {x^{4}}{4 a^{2}} - \frac {b x^{2}}{a^{3}} + \frac {3 b^{2} \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 = 0.64, size = 55, normalized size = 0.96 \begin {gather*} \frac {3 \, b^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{4}} + \frac {b^{3}}{2 \, {\left (a x^{2} + b\right )} a^{4}} + \frac {a^{2} x^{4} - 4 \, a b x^{2}}{4 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 57, normalized size = 1.00 \begin {gather*} \frac {x^4}{4\,a^2}+\frac {b^3}{2\,a\,\left (a^4\,x^2+b\,a^3\right )}-\frac {b\,x^2}{a^3}+\frac {3\,b^2\,\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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