Optimal. Leaf size=135 \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0803857, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2389, 2297, 2300, 2178} \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2389
Rule 2297
Rule 2300
Rule 2178
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^3} \, dx,x,d+e x\right )}{e}\\ &=-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c x^n\right )\right )^2} \, dx,x,d+e x\right )}{2 b e n}\\ &=-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 b^2 e n^2}\\ &=-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{\left ((d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{2 b^2 e n^3}\\ &=\frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{d+e x}{2 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^2}-\frac{d+e x}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.0527516, size = 118, normalized size = 0.87 \[ \frac{e^{-\frac{a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{2 b^3 e n^3}-\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )+b n\right )}{2 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.091, size = 735, normalized size = 5.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (d n + d \log \left (c\right )\right )} b + a d +{\left ({\left (e n + e \log \left (c\right )\right )} b + a e\right )} x +{\left (b e x + b d\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{2 \,{\left (b^{4} e n^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{4} e n^{2} \log \left (c\right )^{2} + 2 \, a b^{3} e n^{2} \log \left (c\right ) + a^{2} b^{2} e n^{2} + 2 \,{\left (b^{4} e n^{2} \log \left (c\right ) + a b^{3} e n^{2}\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )}} + \int \frac{1}{2 \,{\left (b^{3} n^{2} \log \left ({\left (e x + d\right )}^{n}\right ) + b^{3} n^{2} \log \left (c\right ) + a b^{2} n^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.15283, size = 632, normalized size = 4.68 \begin{align*} -\frac{{\left ({\left (b^{2} d n^{2} + a b d n +{\left (b^{2} e n^{2} + a b e n\right )} x +{\left (b^{2} e n^{2} x + b^{2} d n^{2}\right )} \log \left (e x + d\right ) +{\left (b^{2} e n x + b^{2} d n\right )} \log \left (c\right )\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} -{\left (b^{2} n^{2} \log \left (e x + d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (e x + d\right )\right )} \logintegral \left ({\left (e x + d\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right )\right )} e^{\left (-\frac{b \log \left (c\right ) + a}{b n}\right )}}{2 \,{\left (b^{5} e n^{5} \log \left (e x + d\right )^{2} + b^{5} e n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} e n^{3} \log \left (c\right ) + a^{2} b^{3} e n^{3} + 2 \,{\left (b^{5} e n^{4} \log \left (c\right ) + a b^{4} e n^{4}\right )} \log \left (e x + d\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \log{\left (c \left (d + e x\right )^{n} \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.36717, size = 1785, normalized size = 13.22 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]