Optimal. Leaf size=40 \[ -\frac {b^3}{2 x^2}+\frac {3}{2} b c^2 x^2+\frac {c^3 x^4}{4}+3 b^2 c \log (x) \]
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Rubi [A]
time = 0.02, 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 = {1598, 272, 45}
\begin {gather*} -\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} \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 )^3}{x^9} \, dx &=\int \frac {\left (b+c x^2\right )^3}{x^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(b+c x)^3}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 35, normalized size = 0.88
method | result | size |
default | \(-\frac {b^{3}}{2 x^{2}}+\frac {3 b \,c^{2} x^{2}}{2}+\frac {c^{3} x^{4}}{4}+3 b^{2} c \ln \left (x \right )\) | \(35\) |
norman | \(\frac {-\frac {1}{2} b^{3} x^{6}+\frac {1}{4} c^{3} x^{12}+\frac {3}{2} b \,c^{2} x^{10}}{x^{8}}+3 b^{2} c \ln \left (x \right )\) | \(40\) |
risch | \(\frac {c^{3} x^{4}}{4}+\frac {3 b \,c^{2} x^{2}}{2}+\frac {9 b^{2} c}{4}-\frac {b^{3}}{2 x^{2}}+3 b^{2} c \ln \left (x \right )\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 36, normalized size = 0.90 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 38, normalized size = 0.95 \begin {gather*} \frac {c^{3} x^{6} + 6 \, b c^{2} x^{4} + 12 \, b^{2} c x^{2} \log \left (x\right ) - 2 \, b^{3}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 37, normalized size = 0.92 \begin {gather*} - \frac {b^{3}}{2 x^{2}} + 3 b^{2} c \log {\left (x \right )} + \frac {3 b c^{2} x^{2}}{2} + \frac {c^{3} x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.63, size = 46, normalized size = 1.15 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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 \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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