Optimal. Leaf size=68 \[ -\frac {A \log \left (b+c x^2\right )}{2 b^3}+\frac {A \log (x)}{b^3}+\frac {A}{2 b^2 \left (b+c x^2\right )}-\frac {b B-A c}{4 b c \left (b+c x^2\right )^2} \]
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Rubi [A] time = 0.07, antiderivative size = 68, 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, 446, 77} \[ \frac {A}{2 b^2 \left (b+c x^2\right )}-\frac {A \log \left (b+c x^2\right )}{2 b^3}+\frac {A \log (x)}{b^3}-\frac {b B-A c}{4 b c \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {A+B x^2}{x \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x (b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {A}{b^3 x}+\frac {b B-A c}{b (b+c x)^3}-\frac {A c}{b^2 (b+c x)^2}-\frac {A c}{b^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b B-A c}{4 b c \left (b+c x^2\right )^2}+\frac {A}{2 b^2 \left (b+c x^2\right )}+\frac {A \log (x)}{b^3}-\frac {A \log \left (b+c x^2\right )}{2 b^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 59, normalized size = 0.87 \[ \frac {\frac {b \left (3 A b c+2 A c^2 x^2+b^2 (-B)\right )}{c \left (b+c x^2\right )^2}-2 A \log \left (b+c x^2\right )+4 A \log (x)}{4 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 119, normalized size = 1.75 \[ \frac {2 \, A b c^{2} x^{2} - B b^{3} + 3 \, A b^{2} c - 2 \, {\left (A c^{3} x^{4} + 2 \, A b c^{2} x^{2} + A b^{2} c\right )} \log \left (c x^{2} + b\right ) + 4 \, {\left (A c^{3} x^{4} + 2 \, A b c^{2} x^{2} + A b^{2} c\right )} \log \relax (x)}{4 \, {\left (b^{3} c^{3} x^{4} + 2 \, b^{4} c^{2} x^{2} + b^{5} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 76, normalized size = 1.12 \[ \frac {A \log \left (x^{2}\right )}{2 \, b^{3}} - \frac {A \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3}} + \frac {3 \, A c^{3} x^{4} + 8 \, A b c^{2} x^{2} - B b^{3} + 6 \, A b^{2} c}{4 \, {\left (c x^{2} + b\right )}^{2} b^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 68, normalized size = 1.00 \[ \frac {A}{4 \left (c \,x^{2}+b \right )^{2} b}-\frac {B}{4 \left (c \,x^{2}+b \right )^{2} c}+\frac {A}{2 \left (c \,x^{2}+b \right ) b^{2}}+\frac {A \ln \relax (x )}{b^{3}}-\frac {A \ln \left (c \,x^{2}+b \right )}{2 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 77, normalized size = 1.13 \[ \frac {2 \, A c^{2} x^{2} - B b^{2} + 3 \, A b c}{4 \, {\left (b^{2} c^{3} x^{4} + 2 \, b^{3} c^{2} x^{2} + b^{4} c\right )}} - \frac {A \log \left (c x^{2} + b\right )}{2 \, b^{3}} + \frac {A \log \left (x^{2}\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 71, normalized size = 1.04 \[ \frac {\frac {3\,A\,c-B\,b}{4\,b\,c}+\frac {A\,c\,x^2}{2\,b^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}-\frac {A\,\ln \left (c\,x^2+b\right )}{2\,b^3}+\frac {A\,\ln \relax (x)}{b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 75, normalized size = 1.10 \[ \frac {A \log {\relax (x )}}{b^{3}} - \frac {A \log {\left (\frac {b}{c} + x^{2} \right )}}{2 b^{3}} + \frac {3 A b c + 2 A c^{2} x^{2} - B b^{2}}{4 b^{4} c + 8 b^{3} c^{2} x^{2} + 4 b^{2} c^{3} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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