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

Optimal result

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

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*l 
n(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
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.94 \[ \int \frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x} \, dx=\frac {6 b d^2 g^2 i^2 (-9 e g h+3 e f i+2 d g i) n \log (d+e x)+e \left (g i x \left (6 a e^2 \left (6 f^2 i^2-3 f g i (6 h+i x)+g^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )-b n \left (12 d^2 g^2 i^2-6 d e g i (9 g h-3 f i+g i x)+e^2 \left (36 f^2 i^2-9 f g i (12 h+i x)+g^2 \left (108 h^2+27 h i x+4 i^2 x^2\right )\right )\right )\right )+36 a e^2 (g h-f i)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )+6 b e \log \left (c (d+e x)^n\right ) \left (g i \left (6 d \left (3 g^2 h^2-3 f g h i+f^2 i^2\right )+e x \left (6 f^2 i^2-3 f g i (6 h+i x)+g^2 \left (18 h^2+9 h i x+2 i^2 x^2\right )\right )\right )+6 e (g h-f i)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )\right )\right )+36 b e^3 (g h-f i)^3 n \operatorname {PolyLog}\left (2,\frac {g (d+e x)}{-e f+d g}\right )}{36 e^3 g^4} \] Input:

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

Output:

(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*(36* 
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)
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2865, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2865

\(\displaystyle \int \left (\frac {(g h-f i)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3 (f+g x)}+\frac {i (g h-f i)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^3}+\frac {i (h+i x) (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac {i (h+i x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(g h-f i)^3 \log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^4}+\frac {(h+i x)^2 (g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^2}+\frac {(h+i x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {a i x (g h-f i)^2}{g^3}+\frac {b i (d+e x) (g h-f i)^2 \log \left (c (d+e x)^n\right )}{e g^3}-\frac {b n (e h-d i)^3 \log (d+e x)}{3 e^3 g}-\frac {b n (e h-d i)^2 \log (d+e x) (g h-f i)}{2 e^2 g^2}-\frac {b i n x (e h-d i)^2}{3 e^2 g}+\frac {b n (g h-f i)^3 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^4}-\frac {b i n x (e h-d i) (g h-f i)}{2 e g^2}-\frac {b n (h+i x)^2 (e h-d i)}{6 e g}-\frac {b i n x (g h-f i)^2}{g^3}-\frac {b n (h+i x)^2 (g h-f i)}{4 g^2}-\frac {b n (h+i x)^3}{9 g}\)

Input:

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

Output:

(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
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> 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] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

Maple [C] (warning: unable to verify)

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

Time = 3.99 (sec) , antiderivative size = 1208, normalized size of antiderivative = 3.00

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

Input:

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

Output:

-1/2*b/e*n/g^2*i^3*d*f*x+3/2*b/e*n/g*i^2*d*h*x+b/e*n/g^3*i^3*d*ln((g*x+f)* 
e+d*g-e*f)*f^2-3/2*b/e^2*n/g*i^2*d^2*ln((g*x+f)*e+d*g-e*f)*h-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/e*n/g*i*d*ln((g*x+f 
)*e+d*g-e*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*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-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*ln((e*x+d)^n)*i/g*x*h^2+b*n/g^4*dilog(((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-b*ln((e*x+d) 
^n)/g^4*ln(g*x+f)*f^3*i^3-3*b*n/g*i*h^2*x+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+3/2*b/e*n/g^2*i^2*d*f*h-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+1/3*b/e^3*n/g*i^3*d^3*ln((g*x+f)*e+d*g-e*f)-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+1/6*b/e*n/g*i^3*d*x^2-49/36*b*n/g^4* 
i^3*f^3+3*b*n/g^2*i^2*f*h*x-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^2*n/g*i^3*d^2* 
x-3*b/e*n/g^2*i^2*d*ln((g*x+f)*e+d*g-e*f)*f*h-3*b*n/g^2*i*f*h^2+15/4*b*n/g 
^3*i^2*f^2*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+(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*(e*x+d)^n...
 

Fricas [F]

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

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

Output:

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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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*l 
og(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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(36*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b*d*e**3*f 
**3*g*i*n - 108*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x 
)*b*d*e**3*f**2*g**2*h*n - 108*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f* 
x + e*g*x**2),x)*b*d*e**3*f*g**3*h**2*i*n + 36*int(log((d + e*x)**n*c)/(d* 
f + d*g*x + e*f*x + e*g*x**2),x)*b*d*e**3*g**4*h**3*n - 36*int(log((d + e* 
x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b*e**4*f**4*i*n + 108*int(log 
((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b*e**4*f**3*g*h*n + 1 
08*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b*e**4*f**2 
*g**2*h**2*i*n - 36*int(log((d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x** 
2),x)*b*e**4*f*g**3*h**3*n + 36*log(f + g*x)*a*e**3*f**3*i*n - 108*log(f + 
 g*x)*a*e**3*f**2*g*h*n - 108*log(f + g*x)*a*e**3*f*g**2*h**2*i*n + 36*log 
(f + g*x)*a*e**3*g**3*h**3*n + 18*log((d + e*x)**n*c)**2*b*e**3*f**3*i - 5 
4*log((d + e*x)**n*c)**2*b*e**3*f**2*g*h - 54*log((d + e*x)**n*c)**2*b*e** 
3*f*g**2*h**2*i + 18*log((d + e*x)**n*c)**2*b*e**3*g**3*h**3 - 12*log((d + 
 e*x)**n*c)*b*d**3*g**3*i*n - 18*log((d + e*x)**n*c)*b*d**2*e*f*g**2*i*n + 
 54*log((d + e*x)**n*c)*b*d**2*e*g**3*h*n - 36*log((d + e*x)**n*c)*b*d*e** 
2*f**2*g*i*n + 108*log((d + e*x)**n*c)*b*d*e**2*f*g**2*h*n + 108*log((d + 
e*x)**n*c)*b*d*e**2*g**3*h**2*i*n - 36*log((d + e*x)**n*c)*b*e**3*f**2*g*i 
*n*x + 108*log((d + e*x)**n*c)*b*e**3*f*g**2*h*n*x + 18*log((d + e*x)**n*c 
)*b*e**3*f*g**2*i*n*x**2 + 108*log((d + e*x)**n*c)*b*e**3*g**3*h**2*i*n...