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

Optimal. Leaf size=119 \[ \frac{b n (g h-f i) \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g^2}+\frac{(g h-f i) \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{a i x}{g}+\frac{b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac{b i n x}{g} \]

[Out]

(a*i*x)/g - (b*i*n*x)/g + (b*i*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) + ((g*h - f*i)*(a + b*Log[c*(d + e*x)^n])*L
og[(e*(f + g*x))/(e*f - d*g)])/g^2 + (b*(g*h - f*i)*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2

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Rubi [A]  time = 0.138921, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2418, 2389, 2295, 2394, 2393, 2391} \[ \frac{b n (g h-f i) \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g^2}+\frac{(g h-f i) \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{a i x}{g}+\frac{b i (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac{b i n x}{g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*i*x)/g - (b*i*n*x)/g + (b*i*(d + e*x)*Log[c*(d + e*x)^n])/(e*g) + ((g*h - f*i)*(a + b*Log[c*(d + e*x)^n])*L
og[(e*(f + g*x))/(e*f - d*g)])/g^2 + (b*(g*h - f*i)*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/g^2

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2394

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

Rule 2393

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(h+219 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx &=\int \left (\frac{219 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{(-219 f+g h) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g (f+g x)}\right ) \, dx\\ &=\frac{219 \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}+\frac{(-219 f+g h) \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g}\\ &=\frac{219 a x}{g}-\frac{(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g^2}+\frac{(219 b) \int \log \left (c (d+e x)^n\right ) \, dx}{g}+\frac{(b e (219 f-g h) n) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^2}\\ &=\frac{219 a x}{g}-\frac{(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g^2}+\frac{(219 b) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g}+\frac{(b (219 f-g h) n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac{219 a x}{g}-\frac{219 b n x}{g}+\frac{219 b (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac{(219 f-g h) \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{g^2}-\frac{b (219 f-g h) n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^2}\\ \end{align*}

Mathematica [A]  time = 0.109202, size = 110, normalized size = 0.92 \[ \frac{b n (g h-f i) \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+(g h-f i) \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+a g i x+\frac{b g i (d+e x) \log \left (c (d+e x)^n\right )}{e}-b g i n x}{g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*g*i*x - b*g*i*n*x + (b*g*i*(d + e*x)*Log[c*(d + e*x)^n])/e + (g*h - f*i)*(a + b*Log[c*(d + e*x)^n])*Log[(e*
(f + g*x))/(e*f - d*g)] + b*(g*h - f*i)*n*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)])/g^2

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Maple [C]  time = 0.599, size = 750, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*ln(g*x+f)*f*i+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i/g*x+1/2*I*b*Pi
*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g^2*ln(g*x+f)*f*i-1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+
d)^n)^2/g^2*ln(g*x+f)*f*i-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g*ln(g*x+f)*h-1/2*I*b*Pi*
csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i/g*x-a/g^2*ln(g*x+f)*f*i+b*ln(c)/g*ln(g*x+f)*h+b*ln(c)*i/g*x+
1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g*ln(g*x+f)*h+b*ln((e*x+d)^n)/g*ln(g*x+f)*h+b*ln((e*x+d)^n)*i/g*x-1
/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g^2*ln(g*x+f)*f*i+a/g*ln(g*x+f)*h-b*n/g^2*i*f-b*n/g*dilog(((g*x+f)*e
+d*g-f*e)/(d*g-e*f))*h+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g*ln(g*x+f)*h+1/2*I*b*Pi*csgn(I*(e*x
+d)^n)*csgn(I*c*(e*x+d)^n)^2*i/g*x-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g*ln(g*x+f)*h+b/e*n/g*i*d*ln((g*x+f)*e+d*g
-f*e)+b*n/g^2*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f*i-b*ln((e*x+d)^n)/g^2*ln(g*x+f)*f*i-b*ln(c)/g^2*ln
(g*x+f)*f*i+b*n/g^2*dilog(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f*i-b*n/g*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))
*h-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i/g*x+a*i*x/g-b*i*n*x/g

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a i{\left (\frac{x}{g} - \frac{f \log \left (g x + f\right )}{g^{2}}\right )} + \frac{a h \log \left (g x + f\right )}{g} + \int \frac{b i x \log \left (c\right ) + b h \log \left (c\right ) +{\left (b i x + b h\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*i*(x/g - f*log(g*x + f)/g^2) + a*h*log(g*x + f)/g + integrate((b*i*x*log(c) + b*h*log(c) + (b*i*x + b*h)*log
((e*x + d)^n))/(g*x + f), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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