Optimal. Leaf size=33 \[ \frac {1}{4} x^4 (A c+b B)+\frac {1}{2} A b x^2+\frac {1}{6} B c x^6 \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1584, 444, 43} \[ \frac {1}{4} x^4 (A c+b B)+\frac {1}{2} A b x^2+\frac {1}{6} B c x^6 \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )}{x} \, dx &=\int x \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int (A+B x) (b+c x) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (A b+(b B+A c) x+B c x^2\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} A b x^2+\frac {1}{4} (b B+A c) x^4+\frac {1}{6} B c x^6\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {1}{4} x^4 (A c+b B)+\frac {1}{2} A b x^2+\frac {1}{6} B c x^6 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 27, normalized size = 0.82 \[ \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, {\left (B b + A c\right )} x^{4} + \frac {1}{2} \, A b x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.88 \[ \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, B b x^{4} + \frac {1}{4} \, A c x^{4} + \frac {1}{2} \, A b x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 28, normalized size = 0.85 \[ \frac {B c \,x^{6}}{6}+\frac {A b \,x^{2}}{2}+\frac {\left (A c +b B \right ) x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 27, normalized size = 0.82 \[ \frac {1}{6} \, B c x^{6} + \frac {1}{4} \, {\left (B b + A c\right )} x^{4} + \frac {1}{2} \, A b x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 28, normalized size = 0.85 \[ \frac {B\,c\,x^6}{6}+\left (\frac {A\,c}{4}+\frac {B\,b}{4}\right )\,x^4+\frac {A\,b\,x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 29, normalized size = 0.88 \[ \frac {A b x^{2}}{2} + \frac {B c x^{6}}{6} + x^{4} \left (\frac {A c}{4} + \frac {B b}{4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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