Optimal. Leaf size=51 \[ -\frac {a^2}{6 x^6}-\frac {a b}{2 x^4}-\frac {b^2+2 a c}{2 x^2}+\frac {c^2 x^2}{2}+2 b c \log (x) \]
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Rubi [A]
time = 0.02, 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}{6 x^6}-\frac {2 a c+b^2}{2 x^2}-\frac {a b}{2 x^4}+2 b c \log (x)+\frac {c^2 x^2}{2} \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^7} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\left (a+b x+c x^2\right )^2}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (c^2+\frac {a^2}{x^4}+\frac {2 a b}{x^3}+\frac {b^2+2 a c}{x^2}+\frac {2 b c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{6 x^6}-\frac {a b}{2 x^4}-\frac {b^2+2 a c}{2 x^2}+\frac {c^2 x^2}{2}+2 b c \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 0.98 \begin {gather*} -\frac {a^2+3 a b x^2+3 b^2 x^4+6 a c x^4-3 c^2 x^8-12 b c x^6 \log (x)}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 44, normalized size = 0.86
method | result | size |
default | \(\frac {c^{2} x^{2}}{2}-\frac {a b}{2 x^{4}}-\frac {2 a c +b^{2}}{2 x^{2}}+2 b c \ln \left (x \right )-\frac {a^{2}}{6 x^{6}}\) | \(44\) |
norman | \(\frac {\left (-a c -\frac {b^{2}}{2}\right ) x^{4}-\frac {a^{2}}{6}+\frac {c^{2} x^{8}}{2}-\frac {a b \,x^{2}}{2}}{x^{6}}+2 b c \ln \left (x \right )\) | \(47\) |
risch | \(\frac {c^{2} x^{2}}{2}+\frac {\left (-a c -\frac {b^{2}}{2}\right ) x^{4}-\frac {a b \,x^{2}}{2}-\frac {a^{2}}{6}}{x^{6}}+2 b c \ln \left (x \right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 45, normalized size = 0.88 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac {3 \, {\left (b^{2} + 2 \, a c\right )} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 48, normalized size = 0.94 \begin {gather*} \frac {3 \, c^{2} x^{8} + 12 \, b c x^{6} \log \left (x\right ) - 3 \, {\left (b^{2} + 2 \, a c\right )} x^{4} - 3 \, a b x^{2} - a^{2}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 48, normalized size = 0.94 \begin {gather*} 2 b c \log {\left (x \right )} + \frac {c^{2} x^{2}}{2} + \frac {- a^{2} - 3 a b x^{2} + x^{4} \left (- 6 a c - 3 b^{2}\right )}{6 x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.51, size = 54, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac {11 \, b c x^{6} + 3 \, b^{2} x^{4} + 6 \, a c x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.14, size = 46, normalized size = 0.90 \begin {gather*} \frac {c^2\,x^2}{2}-\frac {\frac {a^2}{6}+x^4\,\left (\frac {b^2}{2}+a\,c\right )+\frac {a\,b\,x^2}{2}}{x^6}+2\,b\,c\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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