3.223 \(\int \frac{a+b \log (c (d+e x)^n)}{(f+g x) (h+i x)^3} \, dx\)

Optimal. Leaf size=402 \[ \frac{b g^2 n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac{b g^2 n \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right )}{(g h-f i)^3}+\frac{g^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}-\frac{g^2 \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)^2}+\frac{a+b \log \left (c (d+e x)^n\right )}{2 (h+i x)^2 (g h-f i)}-\frac{b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac{b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}-\frac{b e n}{2 (h+i x) (e h-d i) (g h-f i)}-\frac{b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}+\frac{b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2} \]

[Out]

-(b*e*n)/(2*(e*h - d*i)*(g*h - f*i)*(h + i*x)) - (b*e*g*n*Log[d + e*x])/((e*h - d*i)*(g*h - f*i)^2) - (b*e^2*n
*Log[d + e*x])/(2*(e*h - d*i)^2*(g*h - f*i)) + (a + b*Log[c*(d + e*x)^n])/(2*(g*h - f*i)*(h + i*x)^2) + (g*(a
+ b*Log[c*(d + e*x)^n]))/((g*h - f*i)^2*(h + i*x)) + (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f -
d*g)])/(g*h - f*i)^3 + (b*e*g*n*Log[h + i*x])/((e*h - d*i)*(g*h - f*i)^2) + (b*e^2*n*Log[h + i*x])/(2*(e*h - d
*i)^2*(g*h - f*i)) - (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^3 + (b*g^2*n*
PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^3 - (b*g^2*n*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(
g*h - f*i)^3

________________________________________________________________________________________

Rubi [A]  time = 0.371058, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2418, 2394, 2393, 2391, 2395, 44, 36, 31} \[ \frac{b g^2 n \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{(g h-f i)^3}-\frac{b g^2 n \text{PolyLog}\left (2,-\frac{i (d+e x)}{e h-d i}\right )}{(g h-f i)^3}+\frac{g^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}-\frac{g^2 \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(g h-f i)^3}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x) (g h-f i)^2}+\frac{a+b \log \left (c (d+e x)^n\right )}{2 (h+i x)^2 (g h-f i)}-\frac{b e^2 n \log (d+e x)}{2 (e h-d i)^2 (g h-f i)}+\frac{b e^2 n \log (h+i x)}{2 (e h-d i)^2 (g h-f i)}-\frac{b e n}{2 (h+i x) (e h-d i) (g h-f i)}-\frac{b e g n \log (d+e x)}{(e h-d i) (g h-f i)^2}+\frac{b e g n \log (h+i x)}{(e h-d i) (g h-f i)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e*x)^n])/((f + g*x)*(h + i*x)^3),x]

[Out]

-(b*e*n)/(2*(e*h - d*i)*(g*h - f*i)*(h + i*x)) - (b*e*g*n*Log[d + e*x])/((e*h - d*i)*(g*h - f*i)^2) - (b*e^2*n
*Log[d + e*x])/(2*(e*h - d*i)^2*(g*h - f*i)) + (a + b*Log[c*(d + e*x)^n])/(2*(g*h - f*i)*(h + i*x)^2) + (g*(a
+ b*Log[c*(d + e*x)^n]))/((g*h - f*i)^2*(h + i*x)) + (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f -
d*g)])/(g*h - f*i)^3 + (b*e*g*n*Log[h + i*x])/((e*h - d*i)*(g*h - f*i)^2) + (b*e^2*n*Log[h + i*x])/(2*(e*h - d
*i)^2*(g*h - f*i)) - (g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)])/(g*h - f*i)^3 + (b*g^2*n*
PolyLog[2, -((g*(d + e*x))/(e*f - d*g))])/(g*h - f*i)^3 - (b*g^2*n*PolyLog[2, -((i*(d + e*x))/(e*h - d*i))])/(
g*h - f*i)^3

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 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]

Rule 2395

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c (d+e x)^n\right )}{(h+223 x)^3 (f+g x)} \, dx &=\int \left (\frac{223 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h) (h+223 x)^3}-\frac{223 g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)^2}+\frac{223 g^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3 (h+223 x)}-\frac{g^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3 (f+g x)}\right ) \, dx\\ &=\frac{\left (223 g^2\right ) \int \frac{a+b \log \left (c (d+e x)^n\right )}{h+223 x} \, dx}{(223 f-g h)^3}-\frac{g^3 \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{(223 f-g h)^3}-\frac{(223 g) \int \frac{a+b \log \left (c (d+e x)^n\right )}{(h+223 x)^2} \, dx}{(223 f-g h)^2}+\frac{223 \int \frac{a+b \log \left (c (d+e x)^n\right )}{(h+223 x)^3} \, dx}{223 f-g h}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 (223 f-g h) (h+223 x)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)}+\frac{g^2 \log \left (-\frac{e (h+223 x)}{223 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3}-\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(223 f-g h)^3}-\frac{\left (b e g^2 n\right ) \int \frac{\log \left (\frac{e (h+223 x)}{-223 d+e h}\right )}{d+e x} \, dx}{(223 f-g h)^3}+\frac{\left (b e g^2 n\right ) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{(223 f-g h)^3}-\frac{(b e g n) \int \frac{1}{(h+223 x) (d+e x)} \, dx}{(223 f-g h)^2}+\frac{(b e n) \int \frac{1}{(h+223 x)^2 (d+e x)} \, dx}{2 (223 f-g h)}\\ &=-\frac{a+b \log \left (c (d+e x)^n\right )}{2 (223 f-g h) (h+223 x)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)}+\frac{g^2 \log \left (-\frac{e (h+223 x)}{223 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3}-\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(223 f-g h)^3}+\frac{\left (b g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{(223 f-g h)^3}-\frac{\left (b g^2 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{223 x}{-223 d+e h}\right )}{x} \, dx,x,d+e x\right )}{(223 f-g h)^3}-\frac{(223 b e g n) \int \frac{1}{h+223 x} \, dx}{(223 d-e h) (223 f-g h)^2}+\frac{\left (b e^2 g n\right ) \int \frac{1}{d+e x} \, dx}{(223 d-e h) (223 f-g h)^2}+\frac{(b e n) \int \left (\frac{223}{(223 d-e h) (h+223 x)^2}-\frac{223 e}{(223 d-e h)^2 (h+223 x)}+\frac{e^2}{(223 d-e h)^2 (d+e x)}\right ) \, dx}{2 (223 f-g h)}\\ &=-\frac{b e n}{2 (223 d-e h) (223 f-g h) (h+223 x)}-\frac{b e g n \log (h+223 x)}{(223 d-e h) (223 f-g h)^2}-\frac{b e^2 n \log (h+223 x)}{2 (223 d-e h)^2 (223 f-g h)}+\frac{b e g n \log (d+e x)}{(223 d-e h) (223 f-g h)^2}+\frac{b e^2 n \log (d+e x)}{2 (223 d-e h)^2 (223 f-g h)}-\frac{a+b \log \left (c (d+e x)^n\right )}{2 (223 f-g h) (h+223 x)^2}+\frac{g \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^2 (h+223 x)}+\frac{g^2 \log \left (-\frac{e (h+223 x)}{223 d-e h}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{(223 f-g h)^3}-\frac{g^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e (f+g x)}{e f-d g}\right )}{(223 f-g h)^3}-\frac{b g^2 n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{(223 f-g h)^3}+\frac{b g^2 n \text{Li}_2\left (\frac{223 (d+e x)}{223 d-e h}\right )}{(223 f-g h)^3}\\ \end{align*}

Mathematica [A]  time = 0.420305, size = 311, normalized size = 0.77 \[ \frac{2 b g^2 n \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )-2 b g^2 n \text{PolyLog}\left (2,\frac{i (d+e x)}{d i-e h}\right )+2 g^2 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{2 g (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{h+i x}+\frac{(g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{(h+i x)^2}-2 g^2 \log \left (\frac{e (h+i x)}{e h-d i}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{2 b e g n (g h-f i) (\log (d+e x)-\log (h+i x))}{e h-d i}-\frac{b e n (g h-f i)^2 (e (h+i x) \log (d+e x)-d i-e (h+i x) \log (h+i x)+e h)}{(h+i x) (e h-d i)^2}}{2 (g h-f i)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])/((f + g*x)*(h + i*x)^3),x]

[Out]

(((g*h - f*i)^2*(a + b*Log[c*(d + e*x)^n]))/(h + i*x)^2 + (2*g*(g*h - f*i)*(a + b*Log[c*(d + e*x)^n]))/(h + i*
x) + 2*g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)] - (2*b*e*g*(g*h - f*i)*n*(Log[d + e*x] -
Log[h + i*x]))/(e*h - d*i) - (b*e*(g*h - f*i)^2*n*(e*h - d*i + e*(h + i*x)*Log[d + e*x] - e*(h + i*x)*Log[h +
i*x]))/((e*h - d*i)^2*(h + i*x)) - 2*g^2*(a + b*Log[c*(d + e*x)^n])*Log[(e*(h + i*x))/(e*h - d*i)] + 2*b*g^2*n
*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*b*g^2*n*PolyLog[2, (i*(d + e*x))/(-(e*h) + d*i)])/(2*(g*h - f*i)
^3)

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Maple [C]  time = 0.695, size = 1468, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(e*x+d)^n))/(g*x+f)/(i*x+h)^3,x)

[Out]

b*ln((e*x+d)^n)*g^2/(f*i-g*h)^3*ln(i*x+h)+b*ln((e*x+d)^n)*g/(f*i-g*h)^2/(i*x+h)-b*ln((e*x+d)^n)*g^2/(f*i-g*h)^
3*ln(g*x+f)+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*g/(f*i-g*h)^2/(i*x+h)+b*e*n/(f*i-g*h)^2/(d*i-e*
h)^2*ln(e*x+d)*d*g*i-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g/(f*i-g*h)^2/(i*x+h)+1/2*I*b*
Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g^2/(f*i-g*h)^3*ln(g*x+f)+b*n*g^2/(f*i-g*h)^3*ln(g*x+f)*ln(
((g*x+f)*e+d*g-f*e)/(d*g-e*f))-b*n*g^2/(f*i-g*h)^3*ln(i*x+h)*ln(((i*x+h)*e+d*i-e*h)/(d*i-e*h))+1/4*I*b*Pi*csgn
(I*c*(e*x+d)^n)^3/(f*i-g*h)/(i*x+h)^2+b*ln(c)*g/(f*i-g*h)^2/(i*x+h)-b*ln(c)*g^2/(f*i-g*h)^3*ln(g*x+f)+b*ln(c)*
g^2/(f*i-g*h)^3*ln(i*x+h)+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*g^2/(f*i-g*h)^3*ln(i*x+h)-1/2*I*b
*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*g^2/(f*i-g*h)^3*ln(i*x+h)+a*g^2/(f*i-g*h)^3*ln(i*x+h)+a*g/
(f*i-g*h)^2/(i*x+h)-a*g^2/(f*i-g*h)^3*ln(g*x+f)-1/2*b*ln(c)/(f*i-g*h)/(i*x+h)^2-3/2*b*e^2*n/(f*i-g*h)^2/(d*i-e
*h)^2*ln(e*x+d)*g*h-1/2*b*e^2*n/(f*i-g*h)^2/(d*i-e*h)^2*ln(i*x+h)*f*i+3/2*b*e^2*n/(f*i-g*h)^2/(d*i-e*h)^2*ln(i
*x+h)*g*h-1/2*a/(f*i-g*h)/(i*x+h)^2-1/2*b*ln((e*x+d)^n)/(f*i-g*h)/(i*x+h)^2+b*n*g^2/(f*i-g*h)^3*dilog(((g*x+f)
*e+d*g-f*e)/(d*g-e*f))-b*n*g^2/(f*i-g*h)^3*dilog(((i*x+h)*e+d*i-e*h)/(d*i-e*h))+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*
(e*x+d)^n)^2*g^2/(f*i-g*h)^3*ln(i*x+h)+1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/(f*i-g*h)/(i
*x+h)^2-1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*g^2/(f*i-g*h)^3*ln(g*x+f)+1/2*b*e*n/(f*i-g*h)^2/(d*
i-e*h)/(i*x+h)*g*h-1/2*b*e*n/(f*i-g*h)^2/(d*i-e*h)/(i*x+h)*f*i+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g/(f
*i-g*h)^2/(i*x+h)-1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*g^2/(f*i-g*h)^3*ln(g*x+f)+1/2*b*e^2*n/(f*i-g*h)^2
/(d*i-e*h)^2*ln(e*x+d)*f*i-b*e*n/(f*i-g*h)^2/(d*i-e*h)^2*ln(i*x+h)*d*g*i+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g^2/
(f*i-g*h)^3*ln(g*x+f)-1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/(f*i-g*h)/(i*x+h)^2-1/2*I*b*Pi*csgn(I*c*(e*x+
d)^n)^3*g^2/(f*i-g*h)^3*ln(i*x+h)-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*g/(f*i-g*h)^2/(i*x+h)-1/4*I*b*Pi*csgn(I*(e*
x+d)^n)*csgn(I*c*(e*x+d)^n)^2/(f*i-g*h)/(i*x+h)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (\frac{2 \, g^{2} \log \left (g x + f\right )}{g^{3} h^{3} - 3 \, f g^{2} h^{2} i + 3 \, f^{2} g h i^{2} - f^{3} i^{3}} - \frac{2 \, g^{2} \log \left (i x + h\right )}{g^{3} h^{3} - 3 \, f g^{2} h^{2} i + 3 \, f^{2} g h i^{2} - f^{3} i^{3}} + \frac{2 \, g i x + 3 \, g h - f i}{g^{2} h^{4} - 2 \, f g h^{3} i + f^{2} h^{2} i^{2} +{\left (g^{2} h^{2} i^{2} - 2 \, f g h i^{3} + f^{2} i^{4}\right )} x^{2} + 2 \,{\left (g^{2} h^{3} i - 2 \, f g h^{2} i^{2} + f^{2} h i^{3}\right )} x}\right )} a + b \int \frac{\log \left ({\left (e x + d\right )}^{n}\right ) + \log \left (c\right )}{g i^{3} x^{4} + f h^{3} +{\left (3 \, g h i^{2} + f i^{3}\right )} x^{3} + 3 \,{\left (g h^{2} i + f h i^{2}\right )} x^{2} +{\left (g h^{3} + 3 \, f h^{2} i\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))/(g*x+f)/(i*x+h)^3,x, algorithm="maxima")

[Out]

1/2*(2*g^2*log(g*x + f)/(g^3*h^3 - 3*f*g^2*h^2*i + 3*f^2*g*h*i^2 - f^3*i^3) - 2*g^2*log(i*x + h)/(g^3*h^3 - 3*
f*g^2*h^2*i + 3*f^2*g*h*i^2 - f^3*i^3) + (2*g*i*x + 3*g*h - f*i)/(g^2*h^4 - 2*f*g*h^3*i + f^2*h^2*i^2 + (g^2*h
^2*i^2 - 2*f*g*h*i^3 + f^2*i^4)*x^2 + 2*(g^2*h^3*i - 2*f*g*h^2*i^2 + f^2*h*i^3)*x))*a + b*integrate((log((e*x
+ d)^n) + log(c))/(g*i^3*x^4 + f*h^3 + (3*g*h*i^2 + f*i^3)*x^3 + 3*(g*h^2*i + f*h*i^2)*x^2 + (g*h^3 + 3*f*h^2*
i)*x), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g i^{3} x^{4} + f h^{3} +{\left (3 \, g h i^{2} + f i^{3}\right )} x^{3} + 3 \,{\left (g h^{2} i + f h i^{2}\right )} x^{2} +{\left (g h^{3} + 3 \, f h^{2} i\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))/(g*x+f)/(i*x+h)^3,x, algorithm="fricas")

[Out]

integral((b*log((e*x + d)^n*c) + a)/(g*i^3*x^4 + f*h^3 + (3*g*h*i^2 + f*i^3)*x^3 + 3*(g*h^2*i + f*h*i^2)*x^2 +
(g*h^3 + 3*f*h^2*i)*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(e*x+d)**n))/(g*x+f)/(i*x+h)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )}{\left (i x + h\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(e*x+d)^n))/(g*x+f)/(i*x+h)^3,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)/((g*x + f)*(i*x + h)^3), x)