Integrand size = 15, antiderivative size = 159 \[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=-\frac {1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}-\frac {3 x}{2 a^3 d}+\frac {x}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac {x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x^2}{2 a^3}+\frac {3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}-\frac {x \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a^3 d}-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2} \]
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Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {2216, 2215, 2221, 2317, 2438, 2222, 2320, 36, 29, 31, 46} \[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=-\frac {\operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a^3 d^2}+\frac {3 \log \left (a+b e^{c+d x}\right )}{2 a^3 d^2}-\frac {x \log \left (\frac {b e^{c+d x}}{a}+1\right )}{a^3 d}-\frac {3 x}{2 a^3 d}+\frac {x^2}{2 a^3}-\frac {1}{2 a^2 d^2 \left (a+b e^{c+d x}\right )}+\frac {x}{a^2 d \left (a+b e^{c+d x}\right )}+\frac {x}{2 a d \left (a+b e^{c+d x}\right )^2} \]
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 {\text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,e^{c+d x}\right )}{a^2 d^2}-\frac {\text {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 {\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^2}+\frac {\text {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 \text {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*}
Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75 \[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {\frac {a d x}{\left (a+b e^{c+d x}\right )^2}+\frac {-1+2 d x}{a+b e^{c+d x}}+\frac {-3 d x+3 \log \left (1+\frac {b e^{c+d x}}{a}\right )}{a}+\frac {d x \left (d x-2 \log \left (1+\frac {b e^{c+d x}}{a}\right )\right )-2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a}\right )}{a}}{2 a^2 d^2} \]
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Time = 0.05 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {2 x b d \,{\mathrm e}^{d x +c}+3 x a d -b \,{\mathrm e}^{d x +c}-a}{2 a^{2} d^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}+\frac {3 \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{2 a^{3} d^{2}}-\frac {3 \ln \left ({\mathrm e}^{d x +c}\right )}{2 d^{2} a^{3}}+\frac {c \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{d^{2} a^{3}}-\frac {c \ln \left ({\mathrm e}^{d x +c}\right )}{d^{2} a^{3}}+\frac {x^{2}}{2 a^{3}}+\frac {c x}{d \,a^{3}}+\frac {c^{2}}{2 d^{2} a^{3}}-\frac {x \ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d}-\frac {\ln \left (1+\frac {b \,{\mathrm e}^{d x +c}}{a}\right ) c}{d^{2} a^{3}}-\frac {\operatorname {Li}_{2}\left (-\frac {b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3} d^{2}}\) | \(216\) |
derivativedivides | \(\frac {-\frac {\operatorname {dilog}\left (\frac {a +b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3}}-\frac {\left (d x +c \right ) \ln \left (\frac {a +b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3}}+\frac {3 \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{2 a^{3}}-\frac {1}{2 a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )}-\frac {b^{2} \left (d x +c \right ) {\mathrm e}^{2 d x +2 c}}{2 a^{3} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {b \left (d x +c \right ) {\mathrm e}^{d x +c}}{a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {b \left (d x +c \right ) {\mathrm e}^{d x +c}}{a^{3} \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {\left (d x +c \right )^{2}}{2 a^{3}}-c \left (-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a^{3}}\right )}{d^{2}}\) | \(251\) |
default | \(\frac {-\frac {\operatorname {dilog}\left (\frac {a +b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3}}-\frac {\left (d x +c \right ) \ln \left (\frac {a +b \,{\mathrm e}^{d x +c}}{a}\right )}{a^{3}}+\frac {3 \ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{2 a^{3}}-\frac {1}{2 a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )}-\frac {b^{2} \left (d x +c \right ) {\mathrm e}^{2 d x +2 c}}{2 a^{3} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {b \left (d x +c \right ) {\mathrm e}^{d x +c}}{a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}-\frac {b \left (d x +c \right ) {\mathrm e}^{d x +c}}{a^{3} \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {\left (d x +c \right )^{2}}{2 a^{3}}-c \left (-\frac {\ln \left (a +b \,{\mathrm e}^{d x +c}\right )}{a^{3}}+\frac {1}{a^{2} \left (a +b \,{\mathrm e}^{d x +c}\right )}+\frac {1}{2 a \left (a +b \,{\mathrm e}^{d x +c}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}\right )}{a^{3}}\right )}{d^{2}}\) | \(251\) |
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Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (142) = 284\).
Time = 0.26 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.13 \[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {a^{2} d^{2} x^{2} - a^{2} c^{2} - 3 \, a^{2} c - a^{2} - 2 \, {\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )} + a}{a} + 1\right ) + {\left (b^{2} d^{2} x^{2} - b^{2} c^{2} - 3 \, b^{2} d x - 3 \, b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (2 \, a b d^{2} x^{2} - 2 \, a b c^{2} - 4 \, a b d x - 6 \, a b c - a b\right )} e^{\left (d x + c\right )} + {\left (2 \, a^{2} c + 3 \, a^{2} + {\left (2 \, b^{2} c + 3 \, b^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (2 \, a b c + 3 \, a b\right )} e^{\left (d x + c\right )}\right )} \log \left (b e^{\left (d x + c\right )} + a\right ) - 2 \, {\left (a^{2} d x + a^{2} c + {\left (b^{2} d x + b^{2} c\right )} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, {\left (a b d x + a b c\right )} e^{\left (d x + c\right )}\right )} \log \left (\frac {b e^{\left (d x + c\right )} + a}{a}\right )}{2 \, {\left (a^{3} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d^{2} e^{\left (d x + c\right )} + a^{5} d^{2}\right )}} \]
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\[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {3 a d x - a + \left (2 b d x - b\right ) e^{c + d x}}{2 a^{4} d^{2} + 4 a^{3} b d^{2} e^{c + d x} + 2 a^{2} b^{2} d^{2} e^{2 c + 2 d x}} + \frac {\int \frac {2 d x}{a + b e^{c} e^{d x}}\, dx + \int \left (- \frac {3}{a + b e^{c} e^{d x}}\right )\, dx}{2 a^{2} d} \]
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Time = 0.22 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.94 \[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=\frac {3 \, a d x + {\left (2 \, b d x e^{c} - b e^{c}\right )} e^{\left (d x\right )} - a}{2 \, {\left (a^{2} b^{2} d^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b d^{2} e^{\left (d x + c\right )} + a^{4} d^{2}\right )}} + \frac {x^{2}}{2 \, a^{3}} - \frac {3 \, x}{2 \, a^{3} d} - \frac {d x \log \left (\frac {b e^{\left (d x + c\right )}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b e^{\left (d x + c\right )}}{a}\right )}{a^{3} d^{2}} + \frac {3 \, \log \left (b e^{\left (d x + c\right )} + a\right )}{2 \, a^{3} d^{2}} \]
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\[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=\int { \frac {x}{{\left (b e^{\left (d x + c\right )} + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {x}{\left (a+b e^{c+d x}\right )^3} \, dx=\int \frac {x}{{\left (a+b\,{\mathrm {e}}^{c+d\,x}\right )}^3} \,d x \]
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