Optimal. Leaf size=27 \[ -\frac {b^2}{2 x^2}+\frac {c^2 x^2}{2}+2 b c \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1598, 272, 45}
\begin {gather*} -\frac {b^2}{2 x^2}+2 b c \log (x)+\frac {c^2 x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^7} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(b+c x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (c^2+\frac {b^2}{x^2}+\frac {2 b c}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^2}{2 x^2}+\frac {c^2 x^2}{2}+2 b c \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} -\frac {b^2}{2 x^2}+\frac {c^2 x^2}{2}+2 b c \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 24, normalized size = 0.89
method | result | size |
default | \(-\frac {b^{2}}{2 x^{2}}+\frac {c^{2} x^{2}}{2}+2 b c \ln \left (x \right )\) | \(24\) |
risch | \(-\frac {b^{2}}{2 x^{2}}+\frac {c^{2} x^{2}}{2}+2 b c \ln \left (x \right )\) | \(24\) |
norman | \(\frac {-\frac {1}{2} b^{2} x^{4}+\frac {1}{2} c^{2} x^{8}}{x^{6}}+2 b c \ln \left (x \right )\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 24, normalized size = 0.89 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac {b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 27, normalized size = 1.00 \begin {gather*} \frac {c^{2} x^{4} + 4 \, b c x^{2} \log \left (x\right ) - b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 24, normalized size = 0.89 \begin {gather*} - \frac {b^{2}}{2 x^{2}} + 2 b c \log {\left (x \right )} + \frac {c^{2} x^{2}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.86, size = 32, normalized size = 1.19 \begin {gather*} \frac {1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac {2 \, b c x^{2} + b^{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 = 23, normalized size = 0.85 \begin {gather*} \frac {c^2\,x^2}{2}-\frac {b^2}{2\,x^2}+2\,b\,c\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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