Optimal. Leaf size=68 \[ -\frac {A \log \left (a+b x^2\right )}{2 a^3}+\frac {A \log (x)}{a^3}+\frac {A}{2 a^2 \left (a+b x^2\right )}+\frac {A b-a B}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.06, antiderivative size = 68, 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, 77} \begin {gather*} \frac {A}{2 a^2 \left (a+b x^2\right )}-\frac {A \log \left (a+b x^2\right )}{2 a^3}+\frac {A \log (x)}{a^3}+\frac {A b-a B}{4 a b \left (a+b x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x \left (a+b x^2\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{a^3 x}+\frac {-A b+a B}{a (a+b x)^3}-\frac {A b}{a^2 (a+b x)^2}-\frac {A b}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {A b-a B}{4 a b \left (a+b x^2\right )^2}+\frac {A}{2 a^2 \left (a+b x^2\right )}+\frac {A \log (x)}{a^3}-\frac {A \log \left (a+b x^2\right )}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 0.87 \begin {gather*} \frac {\frac {a \left (a^2 (-B)+3 a A b+2 A b^2 x^2\right )}{b \left (a+b x^2\right )^2}-2 A \log \left (a+b x^2\right )+4 A \log (x)}{4 a^3} \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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.45, size = 119, normalized size = 1.75 \begin {gather*} \frac {2 \, A a b^{2} x^{2} - B a^{3} + 3 \, A a^{2} b - 2 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (A b^{3} x^{4} + 2 \, A a b^{2} x^{2} + A a^{2} b\right )} \log \relax (x)}{4 \, {\left (a^{3} b^{3} x^{4} + 2 \, a^{4} b^{2} x^{2} + a^{5} b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 76, normalized size = 1.12 \begin {gather*} \frac {A \log \left (x^{2}\right )}{2 \, a^{3}} - \frac {A \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} + \frac {3 \, A b^{3} x^{4} + 8 \, A a b^{2} x^{2} - B a^{3} + 6 \, A a^{2} b}{4 \, {\left (b x^{2} + a\right )}^{2} a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 1.00 \begin {gather*} \frac {A}{4 \left (b \,x^{2}+a \right )^{2} a}-\frac {B}{4 \left (b \,x^{2}+a \right )^{2} b}+\frac {A}{2 \left (b \,x^{2}+a \right ) a^{2}}+\frac {A \ln \relax (x )}{a^{3}}-\frac {A \ln \left (b \,x^{2}+a \right )}{2 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 77, normalized size = 1.13 \begin {gather*} \frac {2 \, A b^{2} x^{2} - B a^{2} + 3 \, A a b}{4 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} - \frac {A \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac {A \log \left (x^{2}\right )}{2 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 71, normalized size = 1.04 \begin {gather*} \frac {\frac {3\,A\,b-B\,a}{4\,a\,b}+\frac {A\,b\,x^2}{2\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {A\,\ln \left (b\,x^2+a\right )}{2\,a^3}+\frac {A\,\ln \relax (x)}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 75, normalized size = 1.10 \begin {gather*} \frac {A \log {\relax (x )}}{a^{3}} - \frac {A \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{3}} + \frac {3 A a b + 2 A b^{2} x^{2} - B a^{2}}{4 a^{4} b + 8 a^{3} b^{2} x^{2} + 4 a^{2} b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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