Optimal. Leaf size=35 \[ \frac {B x^2}{2 c}-\frac {(b B-A c) \log \left (b+c x^2\right )}{2 c^2} \]
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Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1584, 444, 43} \[ \frac {B x^2}{2 c}-\frac {(b B-A c) \log \left (b+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {x \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{b+c x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {B}{c}+\frac {-b B+A c}{c (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac {B x^2}{2 c}-\frac {(b B-A c) \log \left (b+c x^2\right )}{2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 31, normalized size = 0.89 \[ \frac {(A c-b B) \log \left (b+c x^2\right )+B c x^2}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 30, normalized size = 0.86 \[ \frac {B c x^{2} - {\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 32, normalized size = 0.91 \[ \frac {B x^{2}}{2 \, c} - \frac {{\left (B b - A c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 1.14 \[ \frac {B \,x^{2}}{2 c}+\frac {A \ln \left (c \,x^{2}+b \right )}{2 c}-\frac {B b \ln \left (c \,x^{2}+b \right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 31, normalized size = 0.89 \[ \frac {B x^{2}}{2 \, c} - \frac {{\left (B b - A c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 31, normalized size = 0.89 \[ \frac {B\,x^2}{2\,c}+\frac {\ln \left (c\,x^2+b\right )\,\left (A\,c-B\,b\right )}{2\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 27, normalized size = 0.77 \[ \frac {B x^{2}}{2 c} - \frac {\left (- A c + B b\right ) \log {\left (b + c x^{2} \right )}}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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