Optimal. Leaf size=51 \[ -\frac {a^2 A}{2 x^2}+\frac {1}{2} b (A b+2 a B) x^2+\frac {1}{4} b^2 B x^4+a (2 A b+a B) \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 77}
\begin {gather*} -\frac {a^2 A}{2 x^2}+\frac {1}{2} b x^2 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac {1}{4} b^2 B x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^2 (A+B x)}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (b (A b+2 a B)+\frac {a^2 A}{x^2}+\frac {a (2 A b+a B)}{x}+b^2 B x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2 A}{2 x^2}+\frac {1}{2} b (A b+2 a B) x^2+\frac {1}{4} b^2 B x^4+a (2 A b+a B) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.96 \begin {gather*} \frac {1}{4} \left (-\frac {2 a^2 A}{x^2}+2 b (A b+2 a B) x^2+b^2 B x^4+4 a (2 A b+a B) \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 48, normalized size = 0.94
method | result | size |
default | \(\frac {b^{2} B \,x^{4}}{4}+\frac {A \,b^{2} x^{2}}{2}+B a b \,x^{2}-\frac {a^{2} A}{2 x^{2}}+a \left (2 A b +B a \right ) \ln \left (x \right )\) | \(48\) |
norman | \(\frac {\left (\frac {1}{2} b^{2} A +a b B \right ) x^{4}-\frac {a^{2} A}{2}+\frac {b^{2} B \,x^{6}}{4}}{x^{2}}+\left (2 a b A +a^{2} B \right ) \ln \left (x \right )\) | \(51\) |
risch | \(\frac {b^{2} B \,x^{4}}{4}+\frac {A \,b^{2} x^{2}}{2}+B a b \,x^{2}+\frac {A^{2} b^{2}}{4 B}+a b A +a^{2} B -\frac {a^{2} A}{2 x^{2}}+2 A \ln \left (x \right ) a b +B \ln \left (x \right ) a^{2}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 52, normalized size = 1.02 \begin {gather*} \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, {\left (2 \, B a b + A b^{2}\right )} x^{2} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} \log \left (x^{2}\right ) - \frac {A a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 54, normalized size = 1.06 \begin {gather*} \frac {B b^{2} x^{6} + 2 \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + 4 \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \log \left (x\right ) - 2 \, A a^{2}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 48, normalized size = 0.94 \begin {gather*} - \frac {A a^{2}}{2 x^{2}} + \frac {B b^{2} x^{4}}{4} + a \left (2 A b + B a\right ) \log {\left (x \right )} + x^{2} \left (\frac {A b^{2}}{2} + B a b\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.18, size = 70, normalized size = 1.37 \begin {gather*} \frac {1}{4} \, B b^{2} x^{4} + B a b x^{2} + \frac {1}{2} \, A b^{2} x^{2} + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} \log \left (x^{2}\right ) - \frac {B a^{2} x^{2} + 2 \, A a b x^{2} + A a^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 48, normalized size = 0.94 \begin {gather*} x^2\,\left (\frac {A\,b^2}{2}+B\,a\,b\right )+\ln \left (x\right )\,\left (B\,a^2+2\,A\,b\,a\right )-\frac {A\,a^2}{2\,x^2}+\frac {B\,b^2\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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