\(\int \frac {(h+i x)^3 (a+b \log (c (d+e x)^n))}{f+g x} \, dx\) [217]

Optimal result

Integrand size = 29, antiderivative size = 402 \[ \int \frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {a i (g h-f i)^2 x}{g^3}-\frac {b i (e h-d i)^2 n x}{3 e^2 g}-\frac {b i (e h-d i) (g h-f i) n x}{2 e g^2}-\frac {b i (g h-f i)^2 n x}{g^3}-\frac {b (e h-d i) n (h+i x)^2}{6 e g}-\frac {b (g h-f i) n (h+i x)^2}{4 g^2}-\frac {b n (h+i x)^3}{9 g}-\frac {b (e h-d i)^3 n \log (d+e x)}{3 e^3 g}-\frac {b (e h-d i)^2 (g h-f i) n \log (d+e x)}{2 e^2 g^2}+\frac {b i (g h-f i)^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g^3}+\frac {(g h-f i) (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {(g h-f i)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^4}+\frac {b (g h-f i)^3 n \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^4} \]

[Out]

a*i*(-f*i+g*h)^2*x/g^3-1/3*b*i*(-d*i+e*h)^2*n*x/e^2/g-1/2*b*i*(-d*i+e*h)*(-f*i+g*h)*n*x/e/g^2-b*i*(-f*i+g*h)^2
*n*x/g^3-1/6*b*(-d*i+e*h)*n*(i*x+h)^2/e/g-1/4*b*(-f*i+g*h)*n*(i*x+h)^2/g^2-1/9*b*n*(i*x+h)^3/g-1/3*b*(-d*i+e*h
)^3*n*ln(e*x+d)/e^3/g-1/2*b*(-d*i+e*h)^2*(-f*i+g*h)*n*ln(e*x+d)/e^2/g^2+b*i*(-f*i+g*h)^2*(e*x+d)*ln(c*(e*x+d)^
n)/e/g^3+1/2*(-f*i+g*h)*(i*x+h)^2*(a+b*ln(c*(e*x+d)^n))/g^2+1/3*(i*x+h)^3*(a+b*ln(c*(e*x+d)^n))/g+(-f*i+g*h)^3
*(a+b*ln(c*(e*x+d)^n))*ln(e*(g*x+f)/(-d*g+e*f))/g^4+b*(-f*i+g*h)^3*n*polylog(2,-g*(e*x+d)/(-d*g+e*f))/g^4

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2465, 2436, 2332, 2441, 2440, 2438, 2442, 45} \[ \int \frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {(g h-f i)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^4}+\frac {(h+i x)^2 (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {a i x (g h-f i)^2}{g^3}+\frac {b i (d+e x) (g h-f i)^2 \log \left (c (d+e x)^n\right )}{e g^3}-\frac {b n (e h-d i)^3 \log (d+e x)}{3 e^3 g}-\frac {b n (e h-d i)^2 \log (d+e x) (g h-f i)}{2 e^2 g^2}-\frac {b i n x (e h-d i)^2}{3 e^2 g}+\frac {b n (g h-f i)^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^4}-\frac {b i n x (e h-d i) (g h-f i)}{2 e g^2}-\frac {b n (h+i x)^2 (e h-d i)}{6 e g}-\frac {b i n x (g h-f i)^2}{g^3}-\frac {b n (h+i x)^2 (g h-f i)}{4 g^2}-\frac {b n (h+i x)^3}{9 g} \]

[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 45

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])

Rule 2332

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

Rule 2436

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 2438

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 2440

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 2441

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 2442

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 2465

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]

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.94 \[ \int \frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {6 b d^2 g^2 i^2 (-9 e g h+3 e f i+2 d g i) n \log (d+e x)+e \left (g i x \left (6 a e^2 \left (6 f^2 i^2-3 f g i (6 h+i x)+g^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )-b n \left (12 d^2 g^2 i^2-6 d e g i (9 g h-3 f i+g i x)+e^2 \left (36 f^2 i^2-9 f g i (12 h+i x)+g^2 \left (108 h^2+27 h i x+4 i^2 x^2\right )\right )\right )\right )+36 a e^2 (g h-f i)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )+6 b e \log \left (c (d+e x)^n\right ) \left (g i \left (6 d \left (3 g^2 h^2-3 f g h i+f^2 i^2\right )+e x \left (6 f^2 i^2-3 f g i (6 h+i x)+g^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right )+6 e (g h-f i)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )\right )+36 b e^3 (g h-f i)^3 n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{36 e^3 g^4} \]

[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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.03 (sec) , antiderivative size = 1208, normalized size of antiderivative = 3.00

method	result	size
risch	\(\text {Expression too large to display}\)	\(1208\)

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\int \frac {{\left (h+i\,x\right )}^3\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{f+g\,x} \,d x \]

[In]

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

[Out]

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