Optimal. Leaf size=41 \[ \frac {c}{2 a \left (a+b x^2\right )}+\frac {c \log (x)}{a^2}-\frac {c \log \left (a+b x^2\right )}{2 a^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 272, 46}
\begin {gather*} -\frac {c \log \left (a+b x^2\right )}{2 a^2}+\frac {c \log (x)}{a^2}+\frac {c}{2 a \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {a c+b c x^2}{x \left (a+b x^2\right )^3} \, dx &=c \int \frac {1}{x \left (a+b x^2\right )^2} \, dx\\ &=\frac {1}{2} c \text {Subst}\left (\int \frac {1}{x (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} c \text {Subst}\left (\int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {c}{2 a \left (a+b x^2\right )}+\frac {c \log (x)}{a^2}-\frac {c \log \left (a+b x^2\right )}{2 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.83 \begin {gather*} \frac {c \left (\frac {a}{a+b x^2}+2 \log (x)-\log \left (a+b x^2\right )\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 44, normalized size = 1.07
method | result | size |
risch | \(\frac {c}{2 a \left (b \,x^{2}+a \right )}+\frac {c \ln \left (x \right )}{a^{2}}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(38\) |
default | \(c \left (-\frac {b \left (\frac {\ln \left (b \,x^{2}+a \right )}{b}-\frac {a}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{2}}+\frac {\ln \left (x \right )}{a^{2}}\right )\) | \(44\) |
norman | \(\frac {\frac {c}{4}-\frac {b^{2} c \,x^{4}}{4 a^{2}}}{\left (b \,x^{2}+a \right )^{2}}+\frac {c \ln \left (x \right )}{a^{2}}-\frac {c \ln \left (b \,x^{2}+a \right )}{2 a^{2}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 40, normalized size = 0.98 \begin {gather*} \frac {c}{2 \, {\left (a b x^{2} + a^{2}\right )}} - \frac {c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {c \log \left (x^{2}\right )}{2 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.96, size = 54, normalized size = 1.32 \begin {gather*} \frac {a c - {\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (b c x^{2} + a c\right )} \log \left (x\right )}{2 \, {\left (a^{2} b x^{2} + a^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 36, normalized size = 0.88 \begin {gather*} c \left (\frac {1}{2 a^{2} + 2 a b x^{2}} + \frac {\log {\left (x \right )}}{a^{2}} - \frac {\log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 51, normalized size = 1.24 \begin {gather*} \frac {c \log \left (x^{2}\right )}{2 \, a^{2}} - \frac {c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac {b c x^{2} + 2 \, a c}{2 \, {\left (b x^{2} + a\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 37, normalized size = 0.90 \begin {gather*} \frac {c}{2\,a\,\left (b\,x^2+a\right )}-\frac {c\,\ln \left (b\,x^2+a\right )}{2\,a^2}+\frac {c\,\ln \left (x\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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