Optimal. Leaf size=23 \[ b c x^2+\frac {c^2 x^4}{4}+b^2 \log (x) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, 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*} b^2 \log (x)+b c x^2+\frac {c^2 x^4}{4} \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^5} \, dx &=\int \frac {\left (b+c x^2\right )^2}{x} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(b+c x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (2 b c+\frac {b^2}{x}+c^2 x\right ) \, dx,x,x^2\right )\\ &=b c x^2+\frac {c^2 x^4}{4}+b^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} b c x^2+\frac {c^2 x^4}{4}+b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 22, normalized size = 0.96
method | result | size |
default | \(b c \,x^{2}+\frac {c^{2} x^{4}}{4}+b^{2} \ln \left (x \right )\) | \(22\) |
risch | \(\frac {c^{2} x^{4}}{4}+b c \,x^{2}+b^{2}+b^{2} \ln \left (x \right )\) | \(25\) |
norman | \(\frac {b c \,x^{6}+\frac {1}{4} c^{2} x^{8}}{x^{4}}+b^{2} \ln \left (x \right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, b^{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 = 0.35, size = 21, normalized size = 0.91 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + b^{2} \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 20, normalized size = 0.87 \begin {gather*} b^{2} \log {\left (x \right )} + b c x^{2} + \frac {c^{2} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.34, size = 24, normalized size = 1.04 \begin {gather*} \frac {1}{4} \, c^{2} x^{4} + b c x^{2} + \frac {1}{2} \, b^{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 = 0.03, size = 21, normalized size = 0.91 \begin {gather*} b^2\,\ln \left (x\right )+\frac {c^2\,x^4}{4}+b\,c\,x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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