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

Optimal. Leaf size=402 \[ \frac{b n (g h-f i)^3 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g^4}+\frac{(h+i x)^2 (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{(g h-f i)^3 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^4}+\frac{(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac{a i x (g h-f i)^2}{g^3}+\frac{b i (d+e x) (g h-f i)^2 \log \left (c (d+e x)^n\right )}{e g^3}-\frac{b n (e h-d i)^2 \log (d+e x) (g h-f i)}{2 e^2 g^2}-\frac{b i n x (e h-d i)^2}{3 e^2 g}-\frac{b n (e h-d i)^3 \log (d+e x)}{3 e^3 g}-\frac{b i n x (e h-d i) (g h-f i)}{2 e g^2}-\frac{b n (h+i x)^2 (e h-d i)}{6 e g}-\frac{b n (h+i x)^2 (g h-f i)}{4 g^2}-\frac{b i n x (g h-f i)^2}{g^3}-\frac{b n (h+i x)^3}{9 g} \]

[Out]

(a*i*(g*h - f*i)^2*x)/g^3 - (b*i*(e*h - d*i)^2*n*x)/(3*e^2*g) - (b*i*(e*h - d*i)*(g*h - f*i)*n*x)/(2*e*g^2) -
(b*i*(g*h - f*i)^2*n*x)/g^3 - (b*(e*h - d*i)*n*(h + i*x)^2)/(6*e*g) - (b*(g*h - f*i)*n*(h + i*x)^2)/(4*g^2) -
(b*n*(h + i*x)^3)/(9*g) - (b*(e*h - d*i)^3*n*Log[d + e*x])/(3*e^3*g) - (b*(e*h - d*i)^2*(g*h - f*i)*n*Log[d +
e*x])/(2*e^2*g^2) + (b*i*(g*h - f*i)^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^3) + ((g*h - f*i)*(h + i*x)^2*(a + b
*Log[c*(d + e*x)^n]))/(2*g^2) + ((h + i*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((g*h - f*i)^3*(a + b*Log[c*(
d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^4 + (b*(g*h - f*i)^3*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))]
)/g^4

________________________________________________________________________________________

Rubi [A]  time = 0.363683, antiderivative size = 402, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2418, 2389, 2295, 2394, 2393, 2391, 2395, 43} \[ \frac{b n (g h-f i)^3 \text{PolyLog}\left (2,-\frac{g (d+e x)}{e f-d g}\right )}{g^4}+\frac{(h+i x)^2 (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{(g h-f i)^3 \log \left (\frac{e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^4}+\frac{(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac{a i x (g h-f i)^2}{g^3}+\frac{b i (d+e x) (g h-f i)^2 \log \left (c (d+e x)^n\right )}{e g^3}-\frac{b n (e h-d i)^2 \log (d+e x) (g h-f i)}{2 e^2 g^2}-\frac{b i n x (e h-d i)^2}{3 e^2 g}-\frac{b n (e h-d i)^3 \log (d+e x)}{3 e^3 g}-\frac{b i n x (e h-d i) (g h-f i)}{2 e g^2}-\frac{b n (h+i x)^2 (e h-d i)}{6 e g}-\frac{b n (h+i x)^2 (g h-f i)}{4 g^2}-\frac{b i n x (g h-f i)^2}{g^3}-\frac{b n (h+i x)^3}{9 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*i*(g*h - f*i)^2*x)/g^3 - (b*i*(e*h - d*i)^2*n*x)/(3*e^2*g) - (b*i*(e*h - d*i)*(g*h - f*i)*n*x)/(2*e*g^2) -
(b*i*(g*h - f*i)^2*n*x)/g^3 - (b*(e*h - d*i)*n*(h + i*x)^2)/(6*e*g) - (b*(g*h - f*i)*n*(h + i*x)^2)/(4*g^2) -
(b*n*(h + i*x)^3)/(9*g) - (b*(e*h - d*i)^3*n*Log[d + e*x])/(3*e^3*g) - (b*(e*h - d*i)^2*(g*h - f*i)*n*Log[d +
e*x])/(2*e^2*g^2) + (b*i*(g*h - f*i)^2*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^3) + ((g*h - f*i)*(h + i*x)^2*(a + b
*Log[c*(d + e*x)^n]))/(2*g^2) + ((h + i*x)^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((g*h - f*i)^3*(a + b*Log[c*(
d + e*x)^n])*Log[(e*(f + g*x))/(e*f - d*g)])/g^4 + (b*(g*h - f*i)^3*n*PolyLog[2, -((g*(d + e*x))/(e*f - d*g))]
)/g^4

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]

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(h+217 x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx &=\int \left (\frac{217 (-217 f+g h)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac{217 (-217 f+g h) (h+217 x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{217 (h+217 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{(-217 f+g h)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}\right ) \, dx\\ &=\frac{217 \int (h+217 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}-\frac{(217 (217 f-g h)) \int (h+217 x) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}+\frac{\left (217 (217 f-g h)^2\right ) \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^3}-\frac{(217 f-g h)^3 \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{g^3}\\ &=\frac{217 a (217 f-g h)^2 x}{g^3}-\frac{(217 f-g h) (h+217 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{(h+217 x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{(217 f-g h)^3 \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^4}+\frac{\left (217 b (217 f-g h)^2\right ) \int \log \left (c (d+e x)^n\right ) \, dx}{g^3}-\frac{(b e n) \int \frac{(h+217 x)^3}{d+e x} \, dx}{3 g}+\frac{(b e (217 f-g h) n) \int \frac{(h+217 x)^2}{d+e x} \, dx}{2 g^2}+\frac{\left (b e (217 f-g h)^3 n\right ) \int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{g^4}\\ &=\frac{217 a (217 f-g h)^2 x}{g^3}-\frac{(217 f-g h) (h+217 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{(h+217 x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{(217 f-g h)^3 \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^4}+\frac{\left (217 b (217 f-g h)^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^3}-\frac{(b e n) \int \left (\frac{217 (-217 d+e h)^2}{e^3}+\frac{217 (-217 d+e h) (h+217 x)}{e^2}+\frac{217 (h+217 x)^2}{e}+\frac{(-217 d+e h)^3}{e^3 (d+e x)}\right ) \, dx}{3 g}+\frac{(b e (217 f-g h) n) \int \left (\frac{217 (-217 d+e h)}{e^2}+\frac{217 (h+217 x)}{e}+\frac{(-217 d+e h)^2}{e^2 (d+e x)}\right ) \, dx}{2 g^2}+\frac{\left (b (217 f-g h)^3 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 )}{g^4}\\ &=\frac{217 a (217 f-g h)^2 x}{g^3}-\frac{217 b (217 d-e h)^2 n x}{3 e^2 g}-\frac{217 b (217 d-e h) (217 f-g h) n x}{2 e g^2}-\frac{217 b (217 f-g h)^2 n x}{g^3}+\frac{b (217 d-e h) n (h+217 x)^2}{6 e g}+\frac{b (217 f-g h) n (h+217 x)^2}{4 g^2}-\frac{b n (h+217 x)^3}{9 g}+\frac{b (217 d-e h)^3 n \log (d+e x)}{3 e^3 g}+\frac{b (217 d-e h)^2 (217 f-g h) n \log (d+e x)}{2 e^2 g^2}+\frac{217 b (217 f-g h)^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^3}-\frac{(217 f-g h) (h+217 x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac{(h+217 x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{(217 f-g h)^3 \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^4}-\frac{b (217 f-g h)^3 n \text{Li}_2\left (-\frac{g (d+e x)}{e f-d g}\right )}{g^4}\\ \end{align*}

Mathematica [A]  time = 0.580326, size = 379, normalized size = 0.94 \[ \frac{36 b e^3 n (g h-f i)^3 \text{PolyLog}\left (2,\frac{g (d+e x)}{d g-e f}\right )+e \left (g i x \left (6 a e^2 \left (6 f^2 i^2-3 f g i (6 h+i x)+g^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )-b n \left (12 d^2 g^2 i^2-6 d e g i (-3 f i+9 g h+g i x)+e^2 \left (36 f^2 i^2-9 f g i (12 h+i x)+g^2 \left (108 h^2+27 h i x+4 i^2 x^2\right )\right )\right )\right )+36 a e^2 (g h-f i)^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )+6 b e \log \left (c (d+e x)^n\right ) \left (g i \left (6 d \left (f^2 i^2-3 f g h i+3 g^2 h^2\right )+e x \left (6 f^2 i^2-3 f g i (6 h+i x)+g^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right )+6 e (g h-f i)^3 \log \left (\frac{e (f+g x)}{e f-d g}\right )\right )\right )+6 b d^2 g^2 i^2 n \log (d+e x) (2 d g i+3 e f i-9 e g h)}{36 e^3 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b*d^2*g^2*i^2*(-9*e*g*h + 3*e*f*i + 2*d*g*i)*n*Log[d + e*x] + e*(g*i*x*(6*a*e^2*(6*f^2*i^2 - 3*f*g*i*(6*h +
i*x) + g^2*(18*h^2 + 9*h*i*x + 2*i^2*x^2)) - b*n*(12*d^2*g^2*i^2 - 6*d*e*g*i*(9*g*h - 3*f*i + g*i*x) + e^2*(3
6*f^2*i^2 - 9*f*g*i*(12*h + i*x) + g^2*(108*h^2 + 27*h*i*x + 4*i^2*x^2)))) + 36*a*e^2*(g*h - f*i)^3*Log[(e*(f
+ g*x))/(e*f - d*g)] + 6*b*e*Log[c*(d + e*x)^n]*(g*i*(6*d*(3*g^2*h^2 - 3*f*g*h*i + f^2*i^2) + e*x*(6*f^2*i^2 -
3*f*g*i*(6*h + i*x) + g^2*(18*h^2 + 9*h*i*x + 2*i^2*x^2))) + 6*e*(g*h - f*i)^3*Log[(e*(f + g*x))/(e*f - d*g)]
)) + 36*b*e^3*(g*h - f*i)^3*n*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)])/(36*e^3*g^4)

________________________________________________________________________________________

Maple [C]  time = 0.602, size = 2801, normalized size = 7. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*ln((e*x+d)^n)*i^3/g*x^3+b*ln((e*x+d)^n)/g*ln(g*x+f)*h^3+a*i^3/g^3*f^2*x+3*a*i/g*h^2*x-a/g^4*ln(g*x+f)*f^
3*i^3-1/2*a*i^3/g^2*x^2*f+1/3*b*ln(c)*i^3/g*x^3+b*ln(c)/g*ln(g*x+f)*h^3+3/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^
n)^2/g^3*ln(g*x+f)*f^2*h*i^2-3*b/e*n/g^2*i^2*d*ln((g*x+f)*e+d*g-f*e)*f*h+1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)*i^3/g^2*x^2*f-3/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g^2*ln(g*x+f)*f*h^2*i-1/2*I*b*P
i*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g*ln(g*x+f)*h^3-3/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^3*ln(g*x+
f)*f^2*h*i^2-3/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i^2/g^2*f*h*x+3/2*I*b*Pi*csgn(I*(e*x+d)^n)*csg
n(I*c*(e*x+d)^n)^2/g^3*ln(g*x+f)*f^2*h*i^2+3/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i/g*h^2*x-3/2*I*
b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g^2*ln(g*x+f)*f*h^2*i+3*b*ln((e*x+d)^n)/g^3*ln(g*x+f)*f^2*h*i^2-3
*b*ln((e*x+d)^n)/g^2*ln(g*x+f)*f*h^2*i+1/6*b/e*n/g*i^3*d*x^2+3*b*n/g^2*i^2*h*f*x+1/3*b/e^3*n/g*i^3*d^3*ln((g*x
+f)*e+d*g-f*e)+b*n/g^4*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f^3*i^3-3*b*n/g^3*dilog(((g*x+f)*e+d*g-f*e)
/(d*g-e*f))*f^2*h*i^2+3*b*n/g^2*dilog(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f*h^2*i-1/9*b*n/g*i^3*x^3-49/36*b*n/g^4*i
^3*f^3-b*n/g*dilog(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*h^3+3/2*a*i^2/g*x^2*h+3*b*ln(c)/g^3*ln(g*x+f)*f^2*h*i^2-3*b*
ln(c)/g^2*ln(g*x+f)*f*h^2*i-3*b*ln(c)*i^2/g^2*f*h*x+b/e*n/g^3*i^3*d*ln((g*x+f)*e+d*g-f*e)*f^2+3/2*b/e*n/g*i^2*
d*h*x-1/2*b/e*n/g^2*i^3*d*f*x-3/2*b/e^2*n/g*i^2*d^2*ln((g*x+f)*e+d*g-f*e)*h+1/2*b/e^2*n/g^2*i^3*d^2*ln((g*x+f)
*e+d*g-f*e)*f-3*b*n/g^3*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f^2*h*i^2+3/2*b/e*n/g^2*i^2*d*h*f+1/6*I*b*
Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i^3/g*x^3+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i^3/g^3*f^2*x-
1/6*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i^3/g*x^3-1/2*b*ln((e*x+d)^n)*i^3/g^2*x^2*f+3/2*b*l
n((e*x+d)^n)*i^2/g*x^2*h+b*ln((e*x+d)^n)*i^3/g^3*f^2*x+3*b*ln((e*x+d)^n)*i/g*h^2*x-b*ln((e*x+d)^n)/g^4*ln(g*x+
f)*f^3*i^3-2/3*b/e*n/g^3*i^3*d*f^2-1/3*b/e^2*n/g^2*i^3*d^2*f-1/2*b*ln(c)*i^3/g^2*x^2*f+3/2*b*ln(c)*i^2/g*x^2*h
+b*ln(c)*i^3/g^3*f^2*x-1/3*b/e^2*n/g*i^3*d^2*x+a/g*ln(g*x+f)*h^3+1/3*a*i^3/g*x^3+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n
)^3*i^3/g^2*x^2*f+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^4*ln(g*x+f)*f^3*i^3+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^3+1/6*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i^3/g*x^3-1/2*I*b*Pi*csgn(I*c*(e*x+d)
^n)^3*i^3/g^3*f^2*x-3/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i^2/g*x^2*h-3/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i/g*h^2*x+1/
2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g*ln(g*x+f)*h^3+3*b/e*n/g*i*d*ln((g*x+f)*e+d*g-f*e)*h^2+3*b*n/g^2*ln(
g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))*f*h^2*i+3*a/g^3*ln(g*x+f)*f^2*h*i^2-3*a/g^2*ln(g*x+f)*f*h^2*i-3*a*i^2
/g^2*f*h*x+3*b*ln(c)*i/g*h^2*x-b*ln(c)/g^4*ln(g*x+f)*f^3*i^3-b*n/g*ln(g*x+f)*ln(((g*x+f)*e+d*g-f*e)/(d*g-e*f))
*h^3-3*b*n/g*i*h^2*x+1/4*b*n/g^2*i^3*x^2*f-b*n/g^3*i^3*x*f^2-3/4*b*n/g*i^2*h*x^2+b*n/g^4*dilog(((g*x+f)*e+d*g-
f*e)/(d*g-e*f))*f^3*i^3+15/4*b*n/g^3*i^2*h*f^2-3*b*n/g^2*i*h^2*f-3*b*ln((e*x+d)^n)*i^2/g^2*f*h*x-1/2*I*b*Pi*cs
gn(I*c*(e*x+d)^n)^3/g*ln(g*x+f)*h^3-1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i^3/g*x^3+3/4*I*b*Pi*csgn(I*(e*x+d)^n)*cs
gn(I*c*(e*x+d)^n)^2*i^2/g*x^2*h+3/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i^2/g*x^2*h-1/2*I*b*Pi*csgn(I*(e*x+
d)^n)*csgn(I*c*(e*x+d)^n)^2/g^4*ln(g*x+f)*f^3*i^3+3/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*i^2/g^2*f*h*x-1/4*I*b*Pi*cs
gn(I*c)*csgn(I*c*(e*x+d)^n)^2*i^3/g^2*x^2*f-3/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i/g*h^2
*x-3/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i^2/g^2*f*h*x+3/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*ln(g*x+f)*f*h
^2*i-3/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i^2/g*x^2*h-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d
)^n)*csgn(I*c*(e*x+d)^n)*i^3/g^3*f^2*x+3/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*i^2/g^2*f*h*
x-3/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g^3*ln(g*x+f)*f^2*h*i^2+3/2*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*h^2*i+1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d
)^n)/g^4*ln(g*x+f)*f^3*i^3-1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i^3/g^2*x^2*f-1/2*I*b*Pi*csgn(I*
c)*csgn(I*c*(e*x+d)^n)^2/g^4*ln(g*x+f)*f^3*i^3+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*i^3/g^3*f^2*
x+3/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*i/g*h^2*x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 3 \, a h^{2} i{\left (\frac{x}{g} - \frac{f \log \left (g x + f\right )}{g^{2}}\right )} - \frac{1}{6} \, a i^{3}{\left (\frac{6 \, f^{3} \log \left (g x + f\right )}{g^{4}} - \frac{2 \, g^{2} x^{3} - 3 \, f g x^{2} + 6 \, f^{2} x}{g^{3}}\right )} + \frac{3}{2} \, a h i^{2}{\left (\frac{2 \, f^{2} \log \left (g x + f\right )}{g^{3}} + \frac{g x^{2} - 2 \, f x}{g^{2}}\right )} + \frac{a h^{3} \log \left (g x + f\right )}{g} + \int \frac{b i^{3} x^{3} \log \left (c\right ) + 3 \, b h i^{2} x^{2} \log \left (c\right ) + 3 \, b h^{2} i x \log \left (c\right ) + b h^{3} \log \left (c\right ) +{\left (b i^{3} x^{3} + 3 \, b h i^{2} x^{2} + 3 \, b h^{2} i x + b h^{3}\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)^3*(a+b*log(c*(e*x+d)^n))/(g*x+f),x, algorithm="maxima")

[Out]

3*a*h^2*i*(x/g - f*log(g*x + f)/g^2) - 1/6*a*i^3*(6*f^3*log(g*x + f)/g^4 - (2*g^2*x^3 - 3*f*g*x^2 + 6*f^2*x)/g
^3) + 3/2*a*h*i^2*(2*f^2*log(g*x + f)/g^3 + (g*x^2 - 2*f*x)/g^2) + a*h^3*log(g*x + f)/g + integrate((b*i^3*x^3
*log(c) + 3*b*h*i^2*x^2*log(c) + 3*b*h^2*i*x*log(c) + b*h^3*log(c) + (b*i^3*x^3 + 3*b*h*i^2*x^2 + 3*b*h^2*i*x
+ b*h^3)*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^{3} x^{3} + 3 \, a h i^{2} x^{2} + 3 \, a h^{2} i x + a h^{3} +{\left (b i^{3} x^{3} + 3 \, b h i^{2} x^{2} + 3 \, b h^{2} i x + b h^{3}\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)^3*(a+b*log(c*(e*x+d)^n))/(g*x+f),x, algorithm="fricas")

[Out]

integral((a*i^3*x^3 + 3*a*h*i^2*x^2 + 3*a*h^2*i*x + a*h^3 + (b*i^3*x^3 + 3*b*h*i^2*x^2 + 3*b*h^2*i*x + b*h^3)*
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 )^{3}}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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