Optimal. Leaf size=69 \[ \frac{1}{a^2 d \left (a+b e^{c+d x}\right )}-\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d}+\frac{x}{a^3}+\frac{1}{2 a d \left (a+b e^{c+d x}\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0418567, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2282, 44} \[ \frac{1}{a^2 d \left (a+b e^{c+d x}\right )}-\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d}+\frac{x}{a^3}+\frac{1}{2 a d \left (a+b e^{c+d x}\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b e^{c+d x}\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+b x)^3} \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x}-\frac{b}{a (a+b x)^3}-\frac{b}{a^2 (a+b x)^2}-\frac{b}{a^3 (a+b x)}\right ) \, dx,x,e^{c+d x}\right )}{d}\\ &=\frac{1}{2 a d \left (a+b e^{c+d x}\right )^2}+\frac{1}{a^2 d \left (a+b e^{c+d x}\right )}+\frac{x}{a^3}-\frac{\log \left (a+b e^{c+d x}\right )}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.059463, size = 62, normalized size = 0.9 \[ \frac{\frac{a^2}{\left (a+b e^{c+d x}\right )^2}+\frac{2 a}{a+b e^{c+d x}}-2 \log \left (a+b e^{c+d x}\right )+2 d x}{2 a^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 74, normalized size = 1.1 \begin{align*}{\frac{\ln \left ({{\rm e}^{dx+c}} \right ) }{d{a}^{3}}}-{\frac{\ln \left ( a+b{{\rm e}^{dx+c}} \right ) }{d{a}^{3}}}+{\frac{1}{{a}^{2}d \left ( a+b{{\rm e}^{dx+c}} \right ) }}+{\frac{1}{2\,ad \left ( a+b{{\rm e}^{dx+c}} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05004, size = 113, normalized size = 1.64 \begin{align*} \frac{2 \, b e^{\left (d x + c\right )} + 3 \, a}{2 \,{\left (a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} b e^{\left (d x + c\right )} + a^{4}\right )} d} + \frac{d x + c}{a^{3} d} - \frac{\log \left (b e^{\left (d x + c\right )} + a\right )}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.52389, size = 300, normalized size = 4.35 \begin{align*} \frac{2 \, b^{2} d x e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{2} d x + 3 \, a^{2} + 2 \,{\left (2 \, a b d x + a b\right )} e^{\left (d x + c\right )} - 2 \,{\left (b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b e^{\left (d x + c\right )} + a^{2}\right )} \log \left (b e^{\left (d x + c\right )} + a\right )}{2 \,{\left (a^{3} b^{2} d e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{4} b d e^{\left (d x + c\right )} + a^{5} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.335967, size = 76, normalized size = 1.1 \begin{align*} \frac{3 a + 2 b e^{c + d x}}{2 a^{4} d + 4 a^{3} b d e^{c + d x} + 2 a^{2} b^{2} d e^{2 c + 2 d x}} + \frac{x}{a^{3}} - \frac{\log{\left (\frac{a}{b} + e^{c + d x} \right )}}{a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21957, size = 93, normalized size = 1.35 \begin{align*} \frac{d x + c}{a^{3} d} - \frac{\log \left ({\left | b e^{\left (d x + c\right )} + a \right |}\right )}{a^{3} d} + \frac{2 \, a b e^{\left (d x + c\right )} + 3 \, a^{2}}{2 \,{\left (b e^{\left (d x + c\right )} + a\right )}^{2} a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]