Optimal. Leaf size=35 \[ \frac {d x^2}{2 b}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, 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 {(b c-a d) \log \left (a+b x^2\right )}{2 b^2}+\frac {d x^2}{2 b} \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 )}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {c+d x}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {d x^2}{2 b}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.89 \begin {gather*} \frac {b d x^2+(b c-a 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 = 32, normalized size = 0.91
method | result | size |
default | \(\frac {d \,x^{2}}{2 b}+\frac {\left (-a d +b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(32\) |
norman | \(\frac {d \,x^{2}}{2 b}-\frac {\left (a d -b c \right ) \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(32\) |
risch | \(\frac {d \,x^{2}}{2 b}-\frac {\ln \left (b \,x^{2}+a \right ) a d}{2 b^{2}}+\frac {c \ln \left (b \,x^{2}+a \right )}{2 b}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 31, normalized size = 0.89 \begin {gather*} \frac {d x^{2}}{2 \, b} + \frac {{\left (b c - a d\right )} \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 = 1.25, size = 29, normalized size = 0.83 \begin {gather*} \frac {b d x^{2} + {\left (b c - a d\right )} \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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Sympy [A]
time = 0.12, size = 27, normalized size = 0.77 \begin {gather*} \frac {d x^{2}}{2 b} - \frac {\left (a d - b c\right ) \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 = 0.99, size = 32, normalized size = 0.91 \begin {gather*} \frac {d x^{2}}{2 \, b} + \frac {{\left (b c - a d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 31, normalized size = 0.89 \begin {gather*} \frac {d\,x^2}{2\,b}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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