Optimal. Leaf size=43 \[ A b^2 \log (x)+A b c x^2+\frac {1}{4} A c^2 x^4+\frac {B \left (b+c x^2\right )^3}{6 c} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 446, 80, 43} \begin {gather*} A b^2 \log (x)+A b c x^2+\frac {1}{4} A c^2 x^4+\frac {B \left (b+c x^2\right )^3}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^5} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) (b+c x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b+c x^2\right )^3}{6 c}+\frac {1}{2} A \operatorname {Subst}\left (\int \frac {(b+c x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b+c x^2\right )^3}{6 c}+\frac {1}{2} A \operatorname {Subst}\left (\int \left (2 b c+\frac {b^2}{x}+c^2 x\right ) \, dx,x,x^2\right )\\ &=A b c x^2+\frac {1}{4} A c^2 x^4+\frac {B \left (b+c x^2\right )^3}{6 c}+A b^2 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.19 \begin {gather*} A b^2 \log (x)+\frac {1}{4} c x^4 (A c+2 b B)+\frac {1}{2} b x^2 (2 A c+b B)+\frac {1}{6} B c^2 x^6 \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 {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.39, size = 49, normalized size = 1.14 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{4} \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + A b^{2} \log \relax (x) + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b c\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 1.23 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{2} \, B b c x^{4} + \frac {1}{4} \, A c^{2} x^{4} + \frac {1}{2} \, B b^{2} x^{2} + A b c x^{2} + \frac {1}{2} \, A b^{2} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 51, normalized size = 1.19 \begin {gather*} \frac {B \,c^{2} x^{6}}{6}+\frac {A \,c^{2} x^{4}}{4}+\frac {B b c \,x^{4}}{2}+A b c \,x^{2}+\frac {B \,b^{2} x^{2}}{2}+A \,b^{2} \ln \relax (x ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 52, normalized size = 1.21 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{4} \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac {1}{2} \, A b^{2} \log \left (x^{2}\right ) + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b c\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 48, normalized size = 1.12 \begin {gather*} x^2\,\left (\frac {B\,b^2}{2}+A\,c\,b\right )+x^4\,\left (\frac {A\,c^2}{4}+\frac {B\,b\,c}{2}\right )+\frac {B\,c^2\,x^6}{6}+A\,b^2\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 49, normalized size = 1.14 \begin {gather*} A b^{2} \log {\relax (x )} + \frac {B c^{2} x^{6}}{6} + x^{4} \left (\frac {A c^{2}}{4} + \frac {B b c}{2}\right ) + x^{2} \left (A b c + \frac {B b^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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