Optimal. Leaf size=159 \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac{1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac{3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d}-\frac{3 x}{2 a^3 d}+\frac{x^2}{2 a^3}+\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2} \]
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Rubi [A] time = 0.331502, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.733, Rules used = {2185, 2184, 2190, 2279, 2391, 2191, 2282, 36, 29, 31, 44} \[ -\frac{\text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}-\frac{1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac{3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}-\frac{x \log \left (\frac{b e^{c+d x}}{a}+1\right )}{a^3 d}-\frac{3 x}{2 a^3 d}+\frac{x^2}{2 a^3}+\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
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Rule 2185
Rule 2184
Rule 2190
Rule 2279
Rule 2391
Rule 2191
Rule 2282
Rule 36
Rule 29
Rule 31
Rule 44
Rubi steps
\begin{align*} \int \frac{x}{\left (a+b e^{c+d x}\right )^3} \, dx &=\frac{\int \frac{x}{\left (a+b e^{c+d x}\right )^2} \, dx}{a}-\frac{b \int \frac{e^{c+d x} x}{\left (a+b e^{c+d x}\right )^3} \, dx}{a}\\ &=\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac{\int \frac{x}{a+b e^{c+d x}} \, dx}{a^2}-\frac{b \int \frac{e^{c+d x} x}{\left (a+b e^{c+d x}\right )^2} \, dx}{a^2}-\frac{\int \frac{1}{\left (a+b e^{c+d x}\right )^2} \, dx}{2 a d}\\ &=\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x^2}{2 a^3}-\frac{b \int \frac{e^{c+d x} x}{a+b e^{c+d x}} \, dx}{a^3}-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,e^{c+d x}\right )}{2 a d^2}-\frac{\int \frac{1}{a+b e^{c+d x}} \, dx}{a^2 d}\\ &=\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x^2}{2 a^3}-\frac{x \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,e^{c+d x}\right )}{a^2 d^2}-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,e^{c+d x}\right )}{2 a d^2}+\frac{\int \log \left (1+\frac{b e^{c+d x}}{a}\right ) \, dx}{a^3 d}\\ &=-\frac{1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac{x}{2 a^3 d}+\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x^2}{2 a^3}+\frac{\log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}-\frac{x \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}\\ &=-\frac{1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac{3 x}{2 a^3 d}+\frac{x}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac{x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x^2}{2 a^3}+\frac{3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}-\frac{x \log \left (1+\frac{b e^{c+d x}}{a}\right )}{a^3 d}-\frac{\text{Li}_2\left (-\frac{b e^{c+d x}}{a}\right )}{a^3 d^2}\\ \end{align*}
Mathematica [A] time = 0.134123, size = 120, normalized size = 0.75 \[ \frac{\frac{d x \left (d x-2 \log \left (\frac{b e^{c+d x}}{a}+1\right )\right )-2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a}\right )}{a}+\frac{a d x}{\left (a+b e^{c+d x}\right )^2}+\frac{2 d x-1}{a+b e^{c+d x}}+\frac{3 \log \left (\frac{b e^{c+d x}}{a}+1\right )-3 d x}{a}}{2 a^2 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.026, size = 393, normalized size = 2.5 \begin{align*}{\frac{3\,\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{2\,{a}^{3}{d}^{2}}}-{\frac{1}{2\,{a}^{2}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{{b}^{2} \left ({{\rm e}^{dx+c}} \right ) ^{2}x}{2\,d{a}^{3} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}-{\frac{{b}^{2} \left ({{\rm e}^{dx+c}} \right ) ^{2}c}{2\,{a}^{3}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}-{\frac{b{{\rm e}^{dx+c}}x}{{a}^{2}d \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}-{\frac{b{{\rm e}^{dx+c}}c}{{a}^{2}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}}+{\frac{{x}^{2}}{2\,{a}^{3}}}+{\frac{cx}{d{a}^{3}}}+{\frac{{c}^{2}}{2\,{a}^{3}{d}^{2}}}-{\frac{1}{{a}^{3}{d}^{2}}{\it dilog} \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{x}{d{a}^{3}}\ln \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{c}{{a}^{3}{d}^{2}}\ln \left ({\frac{a+b{{\rm e}^{dx+c}}}{a}} \right ) }-{\frac{b{{\rm e}^{dx+c}}x}{d{a}^{3} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{b{{\rm e}^{dx+c}}c}{{a}^{3}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{c\ln \left ({{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{2}}}+{\frac{c\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{{a}^{3}{d}^{2}}}-{\frac{c}{{a}^{2}{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) }}-{\frac{c}{2\,a{d}^{2} \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10557, size = 201, normalized size = 1.26 \begin{align*} \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} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac{x^{2}}{2 \, a^{3}} - \frac{3 \, x}{2 \, a^{3} d} - \frac{d x \log \left (\frac{b e^{\left (d x + c\right )}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )}}{a}\right )}{a^{3} d^{2}} + \frac{3 \, \log \left (b e^{\left (d x + c\right )} + a\right )}{2 \, a^{3} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.47681, size = 760, normalized size = 4.78 \begin{align*} \frac{a^{2} d^{2} x^{2} - a^{2} c^{2} - 3 \, a^{2} c - a^{2} - 2 \,{\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )}{\rm Li}_2\left (-\frac{b e^{\left (d x + c\right )} + a}{a} + 1\right ) +{\left (b^{2} d^{2} x^{2} - b^{2} c^{2} - 3 \, b^{2} d x - 3 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} +{\left (2 \, a b d^{2} x^{2} - 2 \, a b c^{2} - 4 \, a b d x - 6 \, a b c - a b\right )} e^{\left (d x + c\right )} +{\left (2 \, a^{2} c + 3 \, a^{2} +{\left (2 \, b^{2} c + 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (2 \, a b c + 3 \, a b\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \,{\left (a^{2} d x + a^{2} c +{\left (b^{2} d x + b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \,{\left (a b d x + a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac{b e^{\left (d x + c\right )} + a}{a}\right )}{2 \,{\left (a^{3} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{2} e^{\left (d x + c\right )} + a^{5} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 a d x - a + \left (2 b d x - b\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac{\int \frac{2 d x}{a + b e^{c} e^{d x}}\, dx + \int - \frac{3}{a + b e^{c} e^{d x}}\, dx}{2 a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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