Optimal. Leaf size=20 \[ \text {Int}\left (\frac {1}{x \left (a+b e^{c+d x}\right )^3},x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx &=\int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx\\ \end {align*}
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Mathematica [A] time = 1.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a+b e^{c+d x}\right )^3 x} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{3} x e^{\left (3 \, d x + 3 \, c\right )} + 3 \, a b^{2} x e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a^{2} b x e^{\left (d x + c\right )} + a^{3} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,{\mathrm e}^{d x +c}+a \right )^{3} x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 \, a d x + {\left (2 \, b d x e^{c} + b e^{c}\right )} e^{\left (d x\right )} + a}{2 \, {\left (a^{2} b^{2} d^{2} x^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} x^{2} e^{\left (d x + c\right )} + a^{4} d^{2} x^{2}\right )}} + \int \frac {2 \, d^{2} x^{2} + 3 \, d x + 2}{2 \, {\left (a^{2} b d^{2} x^{3} e^{\left (d x + c\right )} + a^{3} d^{2} x^{3}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{x\,{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {3 a d x + a + \left (2 b d x + b\right ) e^{c + d x}}{2 a^{4} d^{2} x^{2} + 4 a^{3} b d^{2} x^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} x^{2} e^{2 c + 2 d x}} + \frac {\int \frac {3 d x}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac {2 d^{2} x^{2}}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx + \int \frac {2}{a x^{3} + b x^{3} e^{c} e^{d x}}\, dx}{2 a^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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