Optimal. Leaf size=34 \[ \frac {A \log (x)}{a}-\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a b} \]
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Rubi [A] time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 72} \begin {gather*} \frac {A \log (x)}{a}-\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a b} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{a x}+\frac {-A b+a B}{a (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {A \log (x)}{a}-\frac {(A b-a B) \log \left (a+b x^2\right )}{2 a b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {(a B-A b) \log \left (a+b x^2\right )}{2 a b}+\frac {A \log (x)}{a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{x \left (a+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.47, size = 32, normalized size = 0.94 \begin {gather*} \frac {2 \, A b \log \relax (x) + {\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 36, normalized size = 1.06 \begin {gather*} \frac {A \log \left (x^{2}\right )}{2 \, a} + \frac {{\left (B a - A b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 37, normalized size = 1.09 \begin {gather*} \frac {A \ln \relax (x )}{a}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a}+\frac {B \ln \left (b \,x^{2}+a \right )}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 35, normalized size = 1.03 \begin {gather*} \frac {A \log \left (x^{2}\right )}{2 \, a} + \frac {{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 32, normalized size = 0.94 \begin {gather*} \frac {A\,\ln \relax (x)}{a}-\frac {\ln \left (b\,x^2+a\right )\,\left (A\,b-B\,a\right )}{2\,a\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.70, size = 26, normalized size = 0.76 \begin {gather*} \frac {A \log {\relax (x )}}{a} + \frac {\left (- A b + B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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