Optimal. Leaf size=53 \[ -\frac {a^2 A}{7 x^7}-\frac {a (a B+2 A b)}{5 x^5}-\frac {b (2 a B+A b)}{3 x^3}-\frac {b^2 B}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {448} \[ -\frac {a^2 A}{7 x^7}-\frac {a (a B+2 A b)}{5 x^5}-\frac {b (2 a B+A b)}{3 x^3}-\frac {b^2 B}{x} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^8} \, dx &=\int \left (\frac {a^2 A}{x^8}+\frac {a (2 A b+a B)}{x^6}+\frac {b (A b+2 a B)}{x^4}+\frac {b^2 B}{x^2}\right ) \, dx\\ &=-\frac {a^2 A}{7 x^7}-\frac {a (2 A b+a B)}{5 x^5}-\frac {b (A b+2 a B)}{3 x^3}-\frac {b^2 B}{x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 1.06 \[ -\frac {3 a^2 \left (5 A+7 B x^2\right )+14 a b x^2 \left (3 A+5 B x^2\right )+35 b^2 x^4 \left (A+3 B x^2\right )}{105 x^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 53, normalized size = 1.00 \[ -\frac {105 \, B b^{2} x^{6} + 35 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 15 \, A a^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 55, normalized size = 1.04 \[ -\frac {105 \, B b^{2} x^{6} + 70 \, B a b x^{4} + 35 \, A b^{2} x^{4} + 21 \, B a^{2} x^{2} + 42 \, A a b x^{2} + 15 \, A a^{2}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 48, normalized size = 0.91 \[ -\frac {B \,b^{2}}{x}-\frac {\left (A b +2 B a \right ) b}{3 x^{3}}-\frac {A \,a^{2}}{7 x^{7}}-\frac {\left (2 A b +B a \right ) a}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 53, normalized size = 1.00 \[ -\frac {105 \, B b^{2} x^{6} + 35 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 15 \, A a^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 52, normalized size = 0.98 \[ -\frac {x^2\,\left (\frac {B\,a^2}{5}+\frac {2\,A\,b\,a}{5}\right )+x^4\,\left (\frac {A\,b^2}{3}+\frac {2\,B\,a\,b}{3}\right )+\frac {A\,a^2}{7}+B\,b^2\,x^6}{x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 58, normalized size = 1.09 \[ \frac {- 15 A a^{2} - 105 B b^{2} x^{6} + x^{4} \left (- 35 A b^{2} - 70 B a b\right ) + x^{2} \left (- 42 A a b - 21 B a^{2}\right )}{105 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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