Optimal. Leaf size=29 \[ \frac {1}{2} x^2 (A c+b B)+A b \log (x)+\frac {1}{4} B c x^4 \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1584, 446, 76} \[ \frac {1}{2} x^2 (A c+b B)+A b \log (x)+\frac {1}{4} B c x^4 \]
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x^3} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b B+A c+\frac {A b}{x}+B c x\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} (b B+A c) x^2+\frac {1}{4} B c x^4+A b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ \frac {1}{2} x^2 (A c+b B)+A b \log (x)+\frac {1}{4} B c x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 25, normalized size = 0.86 \[ \frac {1}{4} \, B c x^{4} + \frac {1}{2} \, {\left (B b + A c\right )} x^{2} + A b \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 30, normalized size = 1.03 \[ \frac {1}{4} \, B c x^{4} + \frac {1}{2} \, B b x^{2} + \frac {1}{2} \, A c x^{2} + \frac {1}{2} \, A b \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.97 \[ \frac {B c \,x^{4}}{4}+\frac {A c \,x^{2}}{2}+\frac {B b \,x^{2}}{2}+A b \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 28, normalized size = 0.97 \[ \frac {1}{4} \, B c x^{4} + \frac {1}{2} \, {\left (B b + A c\right )} x^{2} + \frac {1}{2} \, A b \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 26, normalized size = 0.90 \[ x^2\,\left (\frac {A\,c}{2}+\frac {B\,b}{2}\right )+\frac {B\,c\,x^4}{4}+A\,b\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 27, normalized size = 0.93 \[ A b \log {\relax (x )} + \frac {B c x^{4}}{4} + x^{2} \left (\frac {A c}{2} + \frac {B b}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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