Optimal. Leaf size=51 \[ -\frac {a^2}{2 x^2}+\frac {1}{2} x^2 \left (2 a c+b^2\right )+2 a b \log (x)+\frac {1}{2} b c x^4+\frac {c^2 x^6}{6} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1114, 698} \[ -\frac {a^2}{2 x^2}+\frac {1}{2} x^2 \left (2 a c+b^2\right )+2 a b \log (x)+\frac {1}{2} b c x^4+\frac {c^2 x^6}{6} \]
Antiderivative was successfully verified.
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Rule 698
Rule 1114
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (b^2 \left (1+\frac {2 a c}{b^2}\right )+\frac {a^2}{x^2}+\frac {2 a b}{x}+2 b c x+c^2 x^2\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{2 x^2}+\frac {1}{2} \left (b^2+2 a c\right ) x^2+\frac {1}{2} b c x^4+\frac {c^2 x^6}{6}+2 a b \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 46, normalized size = 0.90 \[ \frac {1}{6} \left (-\frac {3 a^2}{x^2}+3 x^2 \left (2 a c+b^2\right )+12 a b \log (x)+3 b c x^4+c^2 x^6\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 47, normalized size = 0.92 \[ \frac {c^{2} x^{8} + 3 \, b c x^{6} + 3 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 12 \, a b x^{2} \log \relax (x) - 3 \, a^{2}}{6 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 1.04 \[ \frac {1}{6} \, c^{2} x^{6} + \frac {1}{2} \, b c x^{4} + \frac {1}{2} \, b^{2} x^{2} + a c x^{2} + a b \log \left (x^{2}\right ) - \frac {2 \, a b x^{2} + a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 45, normalized size = 0.88 \[ \frac {c^{2} x^{6}}{6}+\frac {b c \,x^{4}}{2}+a c \,x^{2}+\frac {b^{2} x^{2}}{2}+2 a b \ln \relax (x )-\frac {a^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 44, normalized size = 0.86 \[ \frac {1}{6} \, c^{2} x^{6} + \frac {1}{2} \, b c x^{4} + \frac {1}{2} \, {\left (b^{2} + 2 \, a c\right )} x^{2} + a b \log \left (x^{2}\right ) - \frac {a^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 43, normalized size = 0.84 \[ x^2\,\left (\frac {b^2}{2}+a\,c\right )-\frac {a^2}{2\,x^2}+\frac {c^2\,x^6}{6}+2\,a\,b\,\ln \relax (x)+\frac {b\,c\,x^4}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 44, normalized size = 0.86 \[ - \frac {a^{2}}{2 x^{2}} + 2 a b \log {\relax (x )} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{6}}{6} + x^{2} \left (a c + \frac {b^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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