3.194 \(\int \frac{h+i x}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx\)

Optimal. Leaf size=71 \[ \frac{i e^{-\frac{a}{b}} \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^2}+\frac{(f h-e i) \log (a+b \log (c (e+f x)))}{b d f^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.21831, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {2411, 12, 2353, 2299, 2178, 2302, 29} \[ \frac{i e^{-\frac{a}{b}} \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^2}+\frac{(f h-e i) \log (a+b \log (c (e+f x)))}{b d f^2} \]

Antiderivative was successfully verified.

[In]

Int[(h + i*x)/((d*e + d*f*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2353

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2299

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

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 2302

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{h+194 x}{(d e+d f x) (a+b \log (c (e+f x)))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\frac{-194 e+f h}{f}+\frac{194 x}{f}}{d x (a+b \log (c x))} \, dx,x,e+f x\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\frac{-194 e+f h}{f}+\frac{194 x}{f}}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{194}{f (a+b \log (c x))}+\frac{-194 e+f h}{f x (a+b \log (c x))}\right ) \, dx,x,e+f x\right )}{d f}\\ &=\frac{194 \operatorname{Subst}\left (\int \frac{1}{a+b \log (c x)} \, dx,x,e+f x\right )}{d f^2}-\frac{(194 e-f h) \operatorname{Subst}\left (\int \frac{1}{x (a+b \log (c x))} \, dx,x,e+f x\right )}{d f^2}\\ &=\frac{194 \operatorname{Subst}\left (\int \frac{e^x}{a+b x} \, dx,x,\log (c (e+f x))\right )}{c d f^2}-\frac{(194 e-f h) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a+b \log (c (e+f x))\right )}{b d f^2}\\ &=\frac{194 e^{-\frac{a}{b}} \text{Ei}\left (\frac{a+b \log (c (e+f x))}{b}\right )}{b c d f^2}-\frac{(194 e-f h) \log (a+b \log (c (e+f x)))}{b d f^2}\\ \end{align*}

Mathematica [A]  time = 0.14342, size = 76, normalized size = 1.07 \[ \frac{i e^{-\frac{a}{b}} \text{Ei}\left (\frac{a}{b}+\log (c (e+f x))\right )+c f h \log (f (a+b \log (c (e+f x))))-c e i \log (a+b \log (c (e+f x)))}{b c d f^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(h + i*x)/((d*e + d*f*x)*(a + b*Log[c*(e + f*x)])),x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.529, size = 0, normalized size = 0. \begin{align*} \int{\frac{ix+h}{ \left ( dfx+de \right ) \left ( a+b\ln \left ( c \left ( fx+e \right ) \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x+h)/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x)

[Out]

int((i*x+h)/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} i \int \frac{x}{b d e \log \left (c\right ) + a d e +{\left (b d f \log \left (c\right ) + a d f\right )} x +{\left (b d f x + b d e\right )} \log \left (f x + e\right )}\,{d x} + \frac{h \log \left (\frac{b \log \left (f x + e\right ) + b \log \left (c\right ) + a}{b}\right )}{b d f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(d*f*x+d*e)/(a+b*log(c*(f*x+e))),x, algorithm="maxima")

[Out]

i*integrate(x/(b*d*e*log(c) + a*d*e + (b*d*f*log(c) + a*d*f)*x + (b*d*f*x + b*d*e)*log(f*x + e)), x) + h*log((
b*log(f*x + e) + b*log(c) + a)/b)/(b*d*f)

________________________________________________________________________________________

Fricas [A]  time = 1.68376, size = 157, normalized size = 2.21 \begin{align*} \frac{{\left ({\left (c f h - c e i\right )} e^{\frac{a}{b}} \log \left (b \log \left (c f x + c e\right ) + a\right ) + i \logintegral \left ({\left (c f x + c e\right )} e^{\frac{a}{b}}\right )\right )} e^{\left (-\frac{a}{b}\right )}}{b c d f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(d*f*x+d*e)/(a+b*log(c*(f*x+e))),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{h}{a e + a f x + b e \log{\left (c e + c f x \right )} + b f x \log{\left (c e + c f x \right )}}\, dx + \int \frac{i x}{a e + a f x + b e \log{\left (c e + c f x \right )} + b f x \log{\left (c e + c f x \right )}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(d*f*x+d*e)/(a+b*ln(c*(f*x+e))),x)

[Out]

(Integral(h/(a*e + a*f*x + b*e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x)), x) + Integral(i*x/(a*e + a*f*x + b*
e*log(c*e + c*f*x) + b*f*x*log(c*e + c*f*x)), x))/d

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{i x + h}{{\left (d f x + d e\right )}{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x+h)/(d*f*x+d*e)/(a+b*log(c*(f*x+e))),x, algorithm="giac")

[Out]

integrate((i*x + h)/((d*f*x + d*e)*(b*log((f*x + e)*c) + a)), x)