Optimal. Leaf size=48 \[ -\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{3 x^3}-\frac {b (2 a B+A b)}{x}+b^2 B x \]
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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} \begin {gather*} -\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{3 x^3}-\frac {b (2 a B+A b)}{x}+b^2 B x \end {gather*}
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^6} \, dx &=\int \left (b^2 B+\frac {a^2 A}{x^6}+\frac {a (2 A b+a B)}{x^4}+\frac {b (A b+2 a B)}{x^2}\right ) \, dx\\ &=-\frac {a^2 A}{5 x^5}-\frac {a (2 A b+a B)}{3 x^3}-\frac {b (A b+2 a B)}{x}+b^2 B x\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 1.00 \begin {gather*} -\frac {a^2 A}{5 x^5}-\frac {a (a B+2 A b)}{3 x^3}-\frac {b (2 a B+A b)}{x}+b^2 B x \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.42, size = 53, normalized size = 1.10 \begin {gather*} \frac {15 \, B b^{2} x^{6} - 15 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} - 3 \, A a^{2} - 5 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 53, normalized size = 1.10 \begin {gather*} B b^{2} x - \frac {30 \, B a b x^{4} + 15 \, A b^{2} x^{4} + 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} + 3 \, A a^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 45, normalized size = 0.94 \begin {gather*} B \,b^{2} x -\frac {\left (A b +2 B a \right ) b}{x}-\frac {A \,a^{2}}{5 x^{5}}-\frac {\left (2 A b +B a \right ) a}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 51, normalized size = 1.06 \begin {gather*} B b^{2} x - \frac {15 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 5 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 50, normalized size = 1.04 \begin {gather*} B\,b^2\,x-\frac {x^2\,\left (\frac {B\,a^2}{3}+\frac {2\,A\,b\,a}{3}\right )+x^4\,\left (A\,b^2+2\,B\,a\,b\right )+\frac {A\,a^2}{5}}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.60, size = 54, normalized size = 1.12 \begin {gather*} B b^{2} x + \frac {- 3 A a^{2} + x^{4} \left (- 15 A b^{2} - 30 B a b\right ) + x^{2} \left (- 10 A a b - 5 B a^{2}\right )}{15 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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