Optimal. Leaf size=44 \[ \frac {x^2}{2 b^2}-\frac {a^2}{2 b^3 \left (a+b x^2\right )}-\frac {a \log \left (a+b x^2\right )}{b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 44, 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}{2 b^3 \left (a+b x^2\right )}-\frac {a \log \left (a+b x^2\right )}{b^3}+\frac {x^2}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^5}{\left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^2}+\frac {a^2}{b^2 (a+b x)^2}-\frac {2 a}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b^2}-\frac {a^2}{2 b^3 \left (a+b x^2\right )}-\frac {a \log \left (a+b x^2\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 0.86 \begin {gather*} \frac {b x^2-\frac {a^2}{a+b x^2}-2 a \log \left (a+b x^2\right )}{2 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 44, normalized size = 1.00
method | result | size |
risch | \(\frac {x^{2}}{2 b^{2}}-\frac {a^{2}}{2 b^{3} \left (b \,x^{2}+a \right )}-\frac {a \ln \left (b \,x^{2}+a \right )}{b^{3}}\) | \(41\) |
norman | \(\frac {\frac {x^{4}}{2 b}-\frac {a^{2}}{b^{3}}}{b \,x^{2}+a}-\frac {a \ln \left (b \,x^{2}+a \right )}{b^{3}}\) | \(43\) |
default | \(\frac {x^{2}}{2 b^{2}}-\frac {a \left (\frac {a}{b \left (b \,x^{2}+a \right )}+\frac {2 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 b^{2}}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 43, normalized size = 0.98 \begin {gather*} -\frac {a^{2}}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} + \frac {x^{2}}{2 \, b^{2}} - \frac {a \log \left (b x^{2} + a\right )}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 56, normalized size = 1.27 \begin {gather*} \frac {b^{2} x^{4} + a b x^{2} - a^{2} - 2 \, {\left (a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 39, normalized size = 0.89 \begin {gather*} - \frac {a^{2}}{2 a b^{3} + 2 b^{4} x^{2}} - \frac {a \log {\left (a + b x^{2} \right )}}{b^{3}} + \frac {x^{2}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.72, size = 49, normalized size = 1.11 \begin {gather*} \frac {x^{2}}{2 \, b^{2}} - \frac {a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{3}} + \frac {2 \, a b x^{2} + a^{2}}{2 \, {\left (b x^{2} + a\right )} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 45, normalized size = 1.02 \begin {gather*} \frac {x^2}{2\,b^2}-\frac {a^2}{2\,\left (b^4\,x^2+a\,b^3\right )}-\frac {a\,\ln \left (b\,x^2+a\right )}{b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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