Optimal. Leaf size=41 \[ \frac {-b c+a d}{2 b^2 \left (a+b x^2\right )}+\frac {d \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {455, 45}
\begin {gather*} \frac {d \log \left (a+b x^2\right )}{2 b^2}-\frac {b c-a d}{2 b^2 \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rubi steps
\begin {align*} \int \frac {x \left (c+d x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {c+d x}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {b c-a d}{b (a+b x)^2}+\frac {d}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b c-a d}{2 b^2 \left (a+b x^2\right )}+\frac {d \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 41, normalized size = 1.00 \begin {gather*} \frac {-b c+a d}{2 b^2 \left (a+b x^2\right )}+\frac {d \log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 38, normalized size = 0.93
method | result | size |
default | \(\frac {d \ln \left (b \,x^{2}+a \right )}{2 b^{2}}-\frac {-a d +b c}{2 b^{2} \left (b \,x^{2}+a \right )}\) | \(38\) |
norman | \(\frac {a d -b c}{2 b^{2} \left (b \,x^{2}+a \right )}+\frac {d \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(38\) |
risch | \(\frac {a d}{2 b^{2} \left (b \,x^{2}+a \right )}-\frac {c}{2 b \left (b \,x^{2}+a \right )}+\frac {d \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 40, normalized size = 0.98 \begin {gather*} -\frac {b c - a d}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {d \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 = 1.15, size = 45, normalized size = 1.10 \begin {gather*} -\frac {b c - a d - {\left (b d x^{2} + a d\right )} \log \left (b x^{2} + a\right )}{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 = 0.18, size = 36, normalized size = 0.88 \begin {gather*} \frac {a d - b c}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {d \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 = 0.74, size = 65, normalized size = 1.59 \begin {gather*} -\frac {d {\left (\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}\right )}}{2 \, b} - \frac {c}{2 \, {\left (b x^{2} + a\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 37, normalized size = 0.90 \begin {gather*} \frac {d\,\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {a\,d-b\,c}{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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