Optimal. Leaf size=299 \[ \frac{3 g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac{3 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^4 n}+\frac{g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.452357, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{3 g^2 e^{-\frac{3 a}{b n}} (d+e x)^3 (e f-d g) \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac{3 g e^{-\frac{2 a}{b n}} (d+e x)^2 (e f-d g)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g)^3 \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^4 n}+\frac{g^3 e^{-\frac{4 a}{b n}} (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2399
Rule 2389
Rule 2300
Rule 2178
Rule 2390
Rule 2310
Rubi steps
\begin{align*} \int \frac{(f+g x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac{(e f-d g)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{3 g (e f-d g)^2 (d+e x)}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{3 g^2 (e f-d g) (d+e x)^2}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{g^3 (d+e x)^3}{e^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=\frac{g^3 \int \frac{(d+e x)^3}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}+\frac{\left (3 g^2 (e f-d g)\right ) \int \frac{(d+e x)^2}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}+\frac{\left (3 g (e f-d g)^2\right ) \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}+\frac{(e f-d g)^3 \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e^3}\\ &=\frac{g^3 \operatorname{Subst}\left (\int \frac{x^3}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}+\frac{\left (3 g (e f-d g)^2\right ) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}+\frac{(e f-d g)^3 \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^4}\\ &=\frac{\left (g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{4 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{3 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left (3 g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{a+b x} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^4 n}+\frac{\left ((e f-d g)^3 (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 )}{e^4 n}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g)^3 (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 )}{b e^4 n}+\frac{3 e^{-\frac{2 a}{b n}} g (e f-d g)^2 (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac{3 e^{-\frac{3 a}{b n}} g^2 (e f-d g) (d+e x)^3 \left (c (d+e x)^n\right )^{-3/n} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}+\frac{e^{-\frac{4 a}{b n}} g^3 (d+e x)^4 \left (c (d+e x)^n\right )^{-4/n} \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^4 n}\\ \end{align*}
Mathematica [A] time = 0.939547, size = 266, normalized size = 0.89 \[ \frac{e^{-\frac{4 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-4/n} \left (e^{\frac{3 a}{b n}} (e f-d g)^3 \left (c (d+e x)^n\right )^{3/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )+g (d+e x) \left (3 e^{\frac{2 a}{b n}} (e f-d g)^2 \left (c (d+e x)^n\right )^{2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-g (d+e x) \left (-3 e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Ei}\left (\frac{3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )-g (d+e x) \text{Ei}\left (\frac{4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )\right )\right )}{b e^4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.63, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( gx+f \right ) ^{3}}{a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) }}\, dx \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*} \int \frac{{\left (g x + f\right )}^{3}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.99384, size = 726, normalized size = 2.43 \begin{align*} \frac{{\left (g^{3} \logintegral \left ({\left (e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}\right )} e^{\left (\frac{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + 3 \,{\left (e f g^{2} - d g^{3}\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} \logintegral \left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} e^{\left (\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) + 3 \,{\left (e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )} \logintegral \left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} e^{\left (\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}\right ) +{\left (e^{3} f^{3} - 3 \, d e^{2} f^{2} g + 3 \, d^{2} e f g^{2} - d^{3} g^{3}\right )} e^{\left (\frac{3 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\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{4 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b e^{4} n} \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{\left (f + g x\right )^{3}}{a + b \log{\left (c \left (d + e x\right )^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.34008, size = 786, normalized size = 2.63 \begin{align*} -\frac{d^{3} g^{3}{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n} - 4\right )}}{b c^{\left (\frac{1}{n}\right )} n} + \frac{3 \, d^{2} f g^{2}{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n} - 3\right )}}{b c^{\left (\frac{1}{n}\right )} n} + \frac{3 \, d^{2} g^{3}{\rm Ei}\left (\frac{2 \, \log \left (c\right )}{n} + \frac{2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{2 \, a}{b n} - 4\right )}}{b c^{\frac{2}{n}} n} - \frac{3 \, d f^{2} g{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n} - 2\right )}}{b c^{\left (\frac{1}{n}\right )} n} - \frac{6 \, d f g^{2}{\rm Ei}\left (\frac{2 \, \log \left (c\right )}{n} + \frac{2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{2 \, a}{b n} - 3\right )}}{b c^{\frac{2}{n}} n} - \frac{3 \, d g^{3}{\rm Ei}\left (\frac{3 \, \log \left (c\right )}{n} + \frac{3 \, a}{b n} + 3 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{3 \, a}{b n} - 4\right )}}{b c^{\frac{3}{n}} n} + \frac{f^{3}{\rm Ei}\left (\frac{\log \left (c\right )}{n} + \frac{a}{b n} + \log \left (x e + d\right )\right ) e^{\left (-\frac{a}{b n} - 1\right )}}{b c^{\left (\frac{1}{n}\right )} n} + \frac{3 \, f^{2} g{\rm Ei}\left (\frac{2 \, \log \left (c\right )}{n} + \frac{2 \, a}{b n} + 2 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{2 \, a}{b n} - 2\right )}}{b c^{\frac{2}{n}} n} + \frac{3 \, f g^{2}{\rm Ei}\left (\frac{3 \, \log \left (c\right )}{n} + \frac{3 \, a}{b n} + 3 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{3 \, a}{b n} - 3\right )}}{b c^{\frac{3}{n}} n} + \frac{g^{3}{\rm Ei}\left (\frac{4 \, \log \left (c\right )}{n} + \frac{4 \, a}{b n} + 4 \, \log \left (x e + d\right )\right ) e^{\left (-\frac{4 \, a}{b n} - 4\right )}}{b c^{\frac{4}{n}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]