3.90 \(\int \frac{f+g x}{a+b \log (c (d+e x)^n)} \, dx\)

Optimal. Leaf size=139 \[ \frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac{g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n} \]

[Out]

((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e^2*E^(a/(b*n))*n*(c*(d + e*x)^n)^n
^(-1)) + (g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^2*E^((2*a)/(b*n))*n*(c*(d +
e*x)^n)^(2/n))

________________________________________________________________________________________

Rubi [A]  time = 0.161926, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2399, 2389, 2300, 2178, 2390, 2310} \[ \frac{e^{-\frac{a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )}{b e^2 n}+\frac{g e^{-\frac{2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )}{b e^2 n} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)/(a + b*Log[c*(d + e*x)^n]),x]

[Out]

((e*f - d*g)*(d + e*x)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)])/(b*e^2*E^(a/(b*n))*n*(c*(d + e*x)^n)^n
^(-1)) + (g*(d + e*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)])/(b*e^2*E^((2*a)/(b*n))*n*(c*(d +
e*x)^n)^(2/n))

Rule 2399

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2300

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rubi steps

\begin{align*} \int \frac{f+g x}{a+b \log \left (c (d+e x)^n\right )} \, dx &=\int \left (\frac{e f-d g}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}+\frac{g (d+e x)}{e \left (a+b \log \left (c (d+e x)^n\right )\right )}\right ) \, dx\\ &=\frac{g \int \frac{d+e x}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e}+\frac{(e f-d g) \int \frac{1}{a+b \log \left (c (d+e x)^n\right )} \, dx}{e}\\ &=\frac{g \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2}+\frac{(e f-d g) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2}\\ &=\frac{\left (g (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^2 n}+\frac{\left ((e f-d g) (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^2 n}\\ &=\frac{e^{-\frac{a}{b n}} (e f-d g) (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^2 n}+\frac{e^{-\frac{2 a}{b n}} g (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^2 n}\\ \end{align*}

Mathematica [A]  time = 0.162296, size = 126, normalized size = 0.91 \[ \frac{e^{-\frac{2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (e^{\frac{a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac{1}{n}} \text{Ei}\left (\frac{a+b \log \left (c (d+e x)^n\right )}{b n}\right )+g (d+e x) \text{Ei}\left (\frac{2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{b n}\right )\right )}{b e^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)/(a + b*Log[c*(d + e*x)^n]),x]

[Out]

((d + e*x)*(E^(a/(b*n))*(e*f - d*g)*(c*(d + e*x)^n)^n^(-1)*ExpIntegralEi[(a + b*Log[c*(d + e*x)^n])/(b*n)] + g
*(d + e*x)*ExpIntegralEi[(2*(a + b*Log[c*(d + e*x)^n]))/(b*n)]))/(b*e^2*E^((2*a)/(b*n))*n*(c*(d + e*x)^n)^(2/n
))

________________________________________________________________________________________

Maple [F]  time = 0.407, size = 0, normalized size = 0. \begin{align*} \int{\frac{gx+f}{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]

int((g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

[Out]

int((g*x+f)/(a+b*ln(c*(e*x+d)^n)),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{g x + f}{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]

integrate((g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="maxima")

[Out]

integrate((g*x + f)/(b*log((e*x + d)^n*c) + a), x)

________________________________________________________________________________________

Fricas [A]  time = 2.02464, size = 267, normalized size = 1.92 \begin{align*} \frac{{\left ({\left (e f - d g\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )} \logintegral \left ({\left (e x + d\right )} e^{\left (\frac{b \log \left (c\right ) + a}{b n}\right )}\right ) + g \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 )\right )} e^{\left (-\frac{2 \,{\left (b \log \left (c\right ) + a\right )}}{b n}\right )}}{b e^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="fricas")

[Out]

((e*f - d*g)*e^((b*log(c) + a)/(b*n))*log_integral((e*x + d)*e^((b*log(c) + a)/(b*n))) + g*log_integral((e^2*x
^2 + 2*d*e*x + d^2)*e^(2*(b*log(c) + a)/(b*n))))*e^(-2*(b*log(c) + a)/(b*n))/(b*e^2*n)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{f + g x}{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]

integrate((g*x+f)/(a+b*ln(c*(e*x+d)**n)),x)

[Out]

Integral((f + g*x)/(a + b*log(c*(d + e*x)**n)), x)

________________________________________________________________________________________

Giac [A]  time = 1.30581, size = 215, normalized size = 1.55 \begin{align*} -\frac{d 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{f{\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{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)/(a+b*log(c*(e*x+d)^n)),x, algorithm="giac")

[Out]

-d*g*Ei(log(c)/n + a/(b*n) + log(x*e + d))*e^(-a/(b*n) - 2)/(b*c^(1/n)*n) + f*Ei(log(c)/n + a/(b*n) + log(x*e
+ d))*e^(-a/(b*n) - 1)/(b*c^(1/n)*n) + g*Ei(2*log(c)/n + 2*a/(b*n) + 2*log(x*e + d))*e^(-2*a/(b*n) - 2)/(b*c^(
2/n)*n)