Optimal. Leaf size=51 \[ -\frac {a^2}{2 x^2}+\frac {1}{2} \left (b^2+2 a c\right ) x^2+\frac {1}{2} b c x^4+\frac {c^2 x^6}{6}+2 a b \log (x) \]
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Rubi [A]
time = 0.03, 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 = {1128, 712}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 712
Rule 1128
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2+c x^4\right )^2}{x^3} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 = 0.01, size = 46, normalized size = 0.90 \begin {gather*} \frac {1}{6} \left (-\frac {3 a^2}{x^2}+3 \left (b^2+2 a c\right ) x^2+3 b c x^4+c^2 x^6+12 a b \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 45, normalized size = 0.88
method | result | size |
default | \(\frac {c^{2} x^{6}}{6}+\frac {b c \,x^{4}}{2}+a c \,x^{2}+\frac {b^{2} x^{2}}{2}-\frac {a^{2}}{2 x^{2}}+2 a b \ln \left (x \right )\) | \(45\) |
risch | \(\frac {c^{2} x^{6}}{6}+\frac {b c \,x^{4}}{2}+a c \,x^{2}+\frac {b^{2} x^{2}}{2}-\frac {a^{2}}{2 x^{2}}+2 a b \ln \left (x \right )\) | \(45\) |
norman | \(\frac {\left (a c +\frac {b^{2}}{2}\right ) x^{4}-\frac {a^{2}}{2}+\frac {c^{2} x^{8}}{6}+\frac {b c \,x^{6}}{2}}{x^{2}}+2 a b \ln \left (x \right )\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 44, normalized size = 0.86 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 47, normalized size = 0.92 \begin {gather*} \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 \left (x\right ) - 3 \, a^{2}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 44, normalized size = 0.86 \begin {gather*} - \frac {a^{2}}{2 x^{2}} + 2 a b \log {\left (x \right )} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{6}}{6} + x^{2} \left (a c + \frac {b^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.87, size = 53, normalized size = 1.04 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} 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 \left (x\right )+\frac {b\,c\,x^4}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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