Optimal. Leaf size=51 \[ -\frac {a^2 c}{2 x^2}+\frac {1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac {1}{4} b^2 d x^4 \]
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Rubi [A] time = 0.04, 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 = {446, 76} \[ -\frac {a^2 c}{2 x^2}+\frac {1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac {1}{4} b^2 d x^4 \]
Antiderivative was successfully verified.
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Rule 76
Rule 446
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b (b c+2 a d)+\frac {a^2 c}{x^2}+\frac {a (2 b c+a d)}{x}+b^2 d x\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2 c}{2 x^2}+\frac {1}{2} b (b c+2 a d) x^2+\frac {1}{4} b^2 d x^4+a (2 b c+a d) \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.96 \[ \frac {1}{4} \left (-\frac {2 a^2 c}{x^2}+2 b x^2 (2 a d+b c)+4 a \log (x) (a d+2 b c)+b^2 d x^4\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 54, normalized size = 1.06 \[ \frac {b^{2} d x^{6} + 2 \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} + 4 \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \log \relax (x) - 2 \, a^{2} c}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 70, normalized size = 1.37 \[ \frac {1}{4} \, b^{2} d x^{4} + \frac {1}{2} \, b^{2} c x^{2} + a b d x^{2} + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} \log \left (x^{2}\right ) - \frac {2 \, a b c x^{2} + a^{2} d x^{2} + a^{2} c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 50, normalized size = 0.98 \[ \frac {b^{2} d \,x^{4}}{4}+a b d \,x^{2}+\frac {b^{2} c \,x^{2}}{2}+a^{2} d \ln \relax (x )+2 a b c \ln \relax (x )-\frac {a^{2} c}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 52, normalized size = 1.02 \[ \frac {1}{4} \, b^{2} d x^{4} + \frac {1}{2} \, {\left (b^{2} c + 2 \, a b d\right )} x^{2} + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} \log \left (x^{2}\right ) - \frac {a^{2} c}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 48, normalized size = 0.94 \[ x^2\,\left (\frac {c\,b^2}{2}+a\,d\,b\right )+\ln \relax (x)\,\left (d\,a^2+2\,b\,c\,a\right )-\frac {a^2\,c}{2\,x^2}+\frac {b^2\,d\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 48, normalized size = 0.94 \[ - \frac {a^{2} c}{2 x^{2}} + a \left (a d + 2 b c\right ) \log {\relax (x )} + \frac {b^{2} d x^{4}}{4} + x^{2} \left (a b d + \frac {b^{2} c}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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