Optimal. Leaf size=47 \[ a b x^2+\frac {1}{4} \left (b^2+2 a c\right ) x^4+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8}+a^2 \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 47, 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*} a^2 \log (x)+\frac {1}{4} x^4 \left (2 a c+b^2\right )+a b x^2+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8} \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} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (2 a b+\frac {a^2}{x}+\left (b^2+2 a c\right ) x+2 b c x^2+c^2 x^3\right ) \, dx,x,x^2\right )\\ &=a b x^2+\frac {1}{4} \left (b^2+2 a c\right ) x^4+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8}+a^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 47, normalized size = 1.00 \begin {gather*} a b x^2+\frac {1}{4} \left (b^2+2 a c\right ) x^4+\frac {1}{3} b c x^6+\frac {c^2 x^8}{8}+a^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 44, normalized size = 0.94
method | result | size |
norman | \(\left (\frac {a c}{2}+\frac {b^{2}}{4}\right ) x^{4}+a b \,x^{2}+\frac {c^{2} x^{8}}{8}+\frac {b c \,x^{6}}{3}+a^{2} \ln \left (x \right )\) | \(43\) |
default | \(\frac {c^{2} x^{8}}{8}+\frac {b c \,x^{6}}{3}+\frac {a c \,x^{4}}{2}+\frac {b^{2} x^{4}}{4}+a b \,x^{2}+a^{2} \ln \left (x \right )\) | \(44\) |
risch | \(\frac {a c \,x^{4}}{2}+a b \,x^{2}+\frac {a \,b^{2}}{2 c}+\frac {b c \,x^{6}}{3}+\frac {b^{2} x^{4}}{4}-\frac {b^{4}}{24 c^{2}}+\frac {c^{2} x^{8}}{8}+a^{2} \ln \left (x \right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 44, normalized size = 0.94 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, {\left (b^{2} + 2 \, a c\right )} x^{4} + a b x^{2} + \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 41, normalized size = 0.87 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, {\left (b^{2} + 2 \, a c\right )} x^{4} + a b x^{2} + a^{2} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 42, normalized size = 0.89 \begin {gather*} a^{2} \log {\left (x \right )} + a b x^{2} + \frac {b c x^{6}}{3} + \frac {c^{2} x^{8}}{8} + x^{4} \left (\frac {a c}{2} + \frac {b^{2}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.84, size = 46, normalized size = 0.98 \begin {gather*} \frac {1}{8} \, c^{2} x^{8} + \frac {1}{3} \, b c x^{6} + \frac {1}{4} \, b^{2} x^{4} + \frac {1}{2} \, a c x^{4} + a b x^{2} + \frac {1}{2} \, a^{2} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 42, normalized size = 0.89 \begin {gather*} a^2\,\ln \left (x\right )+x^4\,\left (\frac {b^2}{4}+\frac {a\,c}{2}\right )+\frac {c^2\,x^8}{8}+a\,b\,x^2+\frac {b\,c\,x^6}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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