Optimal. Leaf size=50 \[ \frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac {\log (x) (b c-a d)}{a^2}-\frac {c}{2 a x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 50, 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 {(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac {\log (x) (b c-a d)}{a^2}-\frac {c}{2 a x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {c+d x}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c}{a x^2}+\frac {-b c+a d}{a^2 x}-\frac {b (-b c+a d)}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {c}{2 a x^2}-\frac {(b c-a d) \log (x)}{a^2}+\frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.98 \begin {gather*} \frac {(b c-a d) \log \left (a+b x^2\right )}{2 a^2}+\frac {\log (x) (a d-b c)}{a^2}-\frac {c}{2 a x^2} \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 {c+d x^2}{x^3 \left (a+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 48, normalized size = 0.96 \begin {gather*} \frac {{\left (b c - a d\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \, {\left (b c - a d\right )} x^{2} \log \relax (x) - a c}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 72, normalized size = 1.44 \begin {gather*} -\frac {{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{2} c - a b d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac {b c x^{2} - a d x^{2} - a c}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 1.12 \begin {gather*} \frac {d \ln \relax (x )}{a}-\frac {d \ln \left (b \,x^{2}+a \right )}{2 a}-\frac {b c \ln \relax (x )}{a^{2}}+\frac {b c \ln \left (b \,x^{2}+a \right )}{2 a^{2}}-\frac {c}{2 a \,x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 48, normalized size = 0.96 \begin {gather*} \frac {{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac {{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {c}{2 \, a x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 45, normalized size = 0.90 \begin {gather*} \frac {\ln \relax (x)\,\left (a\,d-b\,c\right )}{a^2}-\frac {\ln \left (b\,x^2+a\right )\,\left (a\,d-b\,c\right )}{2\,a^2}-\frac {c}{2\,a\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.69, size = 41, normalized size = 0.82 \begin {gather*} - \frac {c}{2 a x^{2}} + \frac {\left (a d - b c\right ) \log {\relax (x )}}{a^{2}} - \frac {\left (a d - b c\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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