Optimal. Leaf size=40 \[ -\frac {b^3}{2 x^2}+3 b^2 c \log (x)+\frac {3}{2} b c^2 x^2+\frac {c^3 x^4}{4} \]
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Rubi [A] time = 0.03, antiderivative size = 40, 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} \[ 3 b^2 c \log (x)-\frac {b^3}{2 x^2}+\frac {3}{2} b c^2 x^2+\frac {c^3 x^4}{4} \]
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 )^3}{x^9} \, dx &=\int \frac {\left (b+c x^2\right )^3}{x^3} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+c x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (3 b c^2+\frac {b^3}{x^2}+\frac {3 b^2 c}{x}+c^3 x\right ) \, dx,x,x^2\right )\\ &=-\frac {b^3}{2 x^2}+\frac {3}{2} b c^2 x^2+\frac {c^3 x^4}{4}+3 b^2 c \log (x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 40, normalized size = 1.00 \[ -\frac {b^3}{2 x^2}+3 b^2 c \log (x)+\frac {3}{2} b c^2 x^2+\frac {c^3 x^4}{4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 38, normalized size = 0.95 \[ \frac {c^{3} x^{6} + 6 \, b c^{2} x^{4} + 12 \, b^{2} c x^{2} \log \relax (x) - 2 \, b^{3}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 1.15 \[ \frac {1}{4} \, c^{3} x^{4} + \frac {3}{2} \, b c^{2} x^{2} + \frac {3}{2} \, b^{2} c \log \left (x^{2}\right ) - \frac {3 \, b^{2} c x^{2} + b^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 35, normalized size = 0.88 \[ \frac {c^{3} x^{4}}{4}+\frac {3 b \,c^{2} x^{2}}{2}+3 b^{2} c \ln \relax (x )-\frac {b^{3}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 36, normalized size = 0.90 \[ \frac {1}{4} \, c^{3} x^{4} + \frac {3}{2} \, b c^{2} x^{2} + \frac {3}{2} \, b^{2} c \log \left (x^{2}\right ) - \frac {b^{3}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 34, normalized size = 0.85 \[ \frac {c^3\,x^4}{4}-\frac {b^3}{2\,x^2}+\frac {3\,b\,c^2\,x^2}{2}+3\,b^2\,c\,\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 37, normalized size = 0.92 \[ - \frac {b^{3}}{2 x^{2}} + 3 b^{2} c \log {\relax (x )} + \frac {3 b c^{2} x^{2}}{2} + \frac {c^{3} x^{4}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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