Optimal. Leaf size=27 \[ -\frac {b^2}{2 x^2}+2 b c \log (x)+\frac {c^2 x^2}{2} \]
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Rubi [A] time = 0.02, 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 = {1584, 266, 43} \[ -\frac {b^2}{2 x^2}+2 b c \log (x)+\frac {c^2 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
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} \operatorname {Subst}\left (\int \frac {(b+c x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {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 \[ -\frac {b^2}{2 x^2}+2 b c \log (x)+\frac {c^2 x^2}{2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 27, normalized size = 1.00 \[ \frac {c^{2} x^{4} + 4 \, b c x^{2} \log \relax (x) - b^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 32, normalized size = 1.19 \[ \frac {1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac {2 \, b c x^{2} + b^{2}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.89 \[ \frac {c^{2} x^{2}}{2}+2 b c \ln \relax (x )-\frac {b^{2}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 24, normalized size = 0.89 \[ \frac {1}{2} \, c^{2} x^{2} + b c \log \left (x^{2}\right ) - \frac {b^{2}}{2 \, x^{2}} \]
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 \[ \frac {c^2\,x^2}{2}-\frac {b^2}{2\,x^2}+2\,b\,c\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 24, normalized size = 0.89 \[ - \frac {b^{2}}{2 x^{2}} + 2 b c \log {\relax (x )} + \frac {c^{2} x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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