Optimal. Leaf size=43 \[ a^2 c \log (x)+a b c x^2+\frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} b^2 c x^4 \]
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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} \[ a^2 c \log (x)+a b c x^2+\frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{4} b^2 c x^4 \]
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 (c+d x^2\right )}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2 (c+d x)}{x} \, dx,x,x^2\right )\\ &=\frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {d \left (a+b x^2\right )^3}{6 b}+\frac {1}{2} c \operatorname {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+b^2 x\right ) \, dx,x,x^2\right )\\ &=a b c x^2+\frac {1}{4} b^2 c x^4+\frac {d \left (a+b x^2\right )^3}{6 b}+a^2 c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.19 \[ a^2 c \log (x)+\frac {1}{4} b x^4 (2 a d+b c)+\frac {1}{2} a x^2 (a d+2 b c)+\frac {1}{6} b^2 d x^6 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 49, normalized size = 1.14 \[ \frac {1}{6} \, b^{2} d x^{6} + \frac {1}{4} \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c \log \relax (x) + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 53, normalized size = 1.23 \[ \frac {1}{6} \, b^{2} d x^{6} + \frac {1}{4} \, b^{2} c x^{4} + \frac {1}{2} \, a b d x^{4} + a b c x^{2} + \frac {1}{2} \, a^{2} d x^{2} + \frac {1}{2} \, a^{2} c \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 51, normalized size = 1.19 \[ \frac {b^{2} d \,x^{6}}{6}+\frac {a b d \,x^{4}}{2}+\frac {b^{2} c \,x^{4}}{4}+\frac {a^{2} d \,x^{2}}{2}+a b c \,x^{2}+a^{2} c \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 52, normalized size = 1.21 \[ \frac {1}{6} \, b^{2} d x^{6} + \frac {1}{4} \, {\left (b^{2} c + 2 \, a b d\right )} x^{4} + \frac {1}{2} \, a^{2} c \log \left (x^{2}\right ) + \frac {1}{2} \, {\left (2 \, a b c + a^{2} d\right )} x^{2} \]
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 \[ x^2\,\left (\frac {d\,a^2}{2}+b\,c\,a\right )+x^4\,\left (\frac {c\,b^2}{4}+\frac {a\,d\,b}{2}\right )+\frac {b^2\,d\,x^6}{6}+a^2\,c\,\ln \relax (x) \]
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 \[ a^{2} c \log {\relax (x )} + \frac {b^{2} d x^{6}}{6} + x^{4} \left (\frac {a b d}{2} + \frac {b^{2} c}{4}\right ) + x^{2} \left (\frac {a^{2} d}{2} + a b c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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