Optimal. Leaf size=43 \[ a^2 A \log (x)+a A b x^2+\frac {B \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} A b^2 x^4 \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {446, 80, 43} \[ a^2 A \log (x)+a A b x^2+\frac {B \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} A b^2 x^4 \]
Antiderivative was successfully verified.
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Rule 43
Rule 80
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (A+B x)}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (a+b x^2\right )^3}{6 b}+\frac {1}{2} A \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (a+b x^2\right )^3}{6 b}+\frac {1}{2} A \operatorname {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^2\right )\\ &=a A b x^2+\frac {1}{4} A b^2 x^4+\frac {B \left (a+b x^2\right )^3}{6 b}+a^2 A \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.19 \[ a^2 A \log (x)+\frac {1}{4} b x^4 (2 a B+A b)+\frac {1}{2} a x^2 (a B+2 A b)+\frac {1}{6} b^2 B x^6 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 49, normalized size = 1.14 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{4} \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + A a^{2} \log \relax (x) + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 53, normalized size = 1.23 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{2} \, B a b x^{4} + \frac {1}{4} \, A b^{2} x^{4} + \frac {1}{2} \, B a^{2} x^{2} + A a b x^{2} + \frac {1}{2} \, A a^{2} \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 51, normalized size = 1.19 \[ \frac {B \,b^{2} x^{6}}{6}+\frac {A \,b^{2} x^{4}}{4}+\frac {B a b \,x^{4}}{2}+A a b \,x^{2}+\frac {B \,a^{2} x^{2}}{2}+A \,a^{2} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 52, normalized size = 1.21 \[ \frac {1}{6} \, B b^{2} x^{6} + \frac {1}{4} \, {\left (2 \, B a b + A b^{2}\right )} x^{4} + \frac {1}{2} \, A a^{2} \log \left (x^{2}\right ) + \frac {1}{2} \, {\left (B a^{2} + 2 \, A a b\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 48, normalized size = 1.12 \[ x^2\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+x^4\,\left (\frac {A\,b^2}{4}+\frac {B\,a\,b}{2}\right )+\frac {B\,b^2\,x^6}{6}+A\,a^2\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 49, normalized size = 1.14 \[ A a^{2} \log {\relax (x )} + \frac {B b^{2} x^{6}}{6} + x^{4} \left (\frac {A b^{2}}{4} + \frac {B a b}{2}\right ) + x^{2} \left (A a b + \frac {B a^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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