Optimal. Leaf size=33 \[ \frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} \frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{\left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a}{2 b^2 \left (a+b x^2\right )}+\frac {\log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.82 \begin {gather*} \frac {\frac {a}{a+b x^2}+\log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 30, normalized size = 0.91
method | result | size |
default | \(\frac {a}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(30\) |
norman | \(\frac {a}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(30\) |
risch | \(\frac {a}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {\ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 32, normalized size = 0.97 \begin {gather*} \frac {a}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {\log \left (b x^{2} + a\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.09, size = 35, normalized size = 1.06 \begin {gather*} \frac {{\left (b x^{2} + a\right )} \log \left (b x^{2} + a\right ) + a}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 29, normalized size = 0.88 \begin {gather*} \frac {a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.20, size = 48, normalized size = 1.45 \begin {gather*} -\frac {\frac {\log \left (\frac {{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2} {\left | b \right |}}\right )}{b} - \frac {a}{{\left (b x^{2} + a\right )} b}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 29, normalized size = 0.88 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {a}{2\,b^2\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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