Optimal. Leaf size=48 \[ -\frac {a^2 A}{3 x^3}+b x (2 a B+A b)-\frac {a (a B+2 A b)}{x}+\frac {1}{3} b^2 B x^3 \]
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Rubi [A] time = 0.03, antiderivative size = 48, 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} \[ -\frac {a^2 A}{3 x^3}+b x (2 a B+A b)-\frac {a (a B+2 A b)}{x}+\frac {1}{3} b^2 B x^3 \]
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^4} \, dx &=\int \left (b (A b+2 a B)+\frac {a^2 A}{x^4}+\frac {a (2 A b+a B)}{x^2}+b^2 B x^2\right ) \, dx\\ &=-\frac {a^2 A}{3 x^3}-\frac {a (2 A b+a B)}{x}+b (A b+2 a B) x+\frac {1}{3} b^2 B x^3\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.04 \[ \frac {a^2 (-B)-2 a A b}{x}-\frac {a^2 A}{3 x^3}+b x (2 a B+A b)+\frac {1}{3} b^2 B x^3 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 52, normalized size = 1.08 \[ \frac {B b^{2} x^{6} + 3 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - A a^{2} - 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 50, normalized size = 1.04 \[ \frac {1}{3} \, B b^{2} x^{3} + 2 \, B a b x + A b^{2} x - \frac {3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + A a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 46, normalized size = 0.96 \[ \frac {B \,b^{2} x^{3}}{3}+A \,b^{2} x +2 B a b x -\frac {A \,a^{2}}{3 x^{3}}-\frac {\left (2 A b +B a \right ) a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 50, normalized size = 1.04 \[ \frac {1}{3} \, B b^{2} x^{3} + {\left (2 \, B a b + A b^{2}\right )} x - \frac {A a^{2} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 50, normalized size = 1.04 \[ x\,\left (A\,b^2+2\,B\,a\,b\right )-\frac {x^2\,\left (B\,a^2+2\,A\,b\,a\right )+\frac {A\,a^2}{3}}{x^3}+\frac {B\,b^2\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 51, normalized size = 1.06 \[ \frac {B b^{2} x^{3}}{3} + x \left (A b^{2} + 2 B a b\right ) + \frac {- A a^{2} + x^{2} \left (- 6 A a b - 3 B a^{2}\right )}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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