Optimal. Leaf size=35 \[ \frac {a c}{2 b^2 \left (a+b x^2\right )}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A] time = 0.03, antiderivative size = 35, 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, 266, 43} \begin {gather*} \frac {a c}{2 b^2 \left (a+b x^2\right )}+\frac {c \log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx &=c \int \frac {x^3}{\left (a+b x^2\right )^2} \, dx\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} c \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {a c}{2 b^2 \left (a+b x^2\right )}+\frac {c \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.80 \begin {gather*} \frac {c \left (\frac {a}{a+b x^2}+\log \left (a+b x^2\right )\right )}{2 b^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 {x^3 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.60, size = 40, normalized size = 1.14 \begin {gather*} \frac {a c + {\left (b c x^{2} + a c\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 32, normalized size = 0.91 \begin {gather*} \frac {c \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} + \frac {a c}{2 \, {\left (b x^{2} + a\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.91 \begin {gather*} \frac {a c}{2 \left (b \,x^{2}+a \right ) b^{2}}+\frac {c \ln \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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maxima [A] time = 1.03, size = 34, normalized size = 0.97 \begin {gather*} \frac {a c}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {c \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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mupad [B] time = 0.09, size = 31, normalized size = 0.89 \begin {gather*} \frac {c\,\ln \left (b\,x^2+a\right )}{2\,b^2}+\frac {a\,c}{2\,b^2\,\left (b\,x^2+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 31, normalized size = 0.89 \begin {gather*} c \left (\frac {a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac {\log {\left (a + b x^{2} \right )}}{2 b^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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