Optimal. Leaf size=32 \[ \frac {\left (A+B x^2\right )^2}{4 \left (b+c x^2\right )^2 (b B-A c)} \]
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Rubi [A] time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1584, 444, 37} \[ \frac {\left (A+B x^2\right )^2}{4 \left (b+c x^2\right )^2 (b B-A c)} \]
Antiderivative was successfully verified.
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Rule 37
Rule 444
Rule 1584
Rubi steps
\begin {align*} \int \frac {x^7 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {\left (A+B x^2\right )^2}{4 (b B-A c) \left (b+c x^2\right )^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.94 \[ -\frac {c \left (A+2 B x^2\right )+b B}{4 c^2 \left (b+c x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 42, normalized size = 1.31 \[ -\frac {2 \, B c x^{2} + B b + A c}{4 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 28, normalized size = 0.88 \[ -\frac {2 \, B c x^{2} + B b + A c}{4 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 1.22 \[ -\frac {B}{2 \left (c \,x^{2}+b \right ) c^{2}}-\frac {A c -b B}{4 \left (c \,x^{2}+b \right )^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 42, normalized size = 1.31 \[ -\frac {2 \, B c x^{2} + B b + A c}{4 \, {\left (c^{4} x^{4} + 2 \, b c^{3} x^{2} + b^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 44, normalized size = 1.38 \[ -\frac {\frac {A\,c+B\,b}{4\,c^2}+\frac {B\,x^2}{2\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 42, normalized size = 1.31 \[ \frac {- A c - B b - 2 B c x^{2}}{4 b^{2} c^{2} + 8 b c^{3} x^{2} + 4 c^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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