\(\int \frac {(h+i x) (a+b \log (c (d+e x)^n))^3}{f+g x} \, dx\) [230]

Optimal result

Integrand size = 29, antiderivative size = 308 \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx=\frac {6 a b^2 i n^2 x}{g}-\frac {6 b^3 i n^3 x}{g}+\frac {6 b^3 i n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e g}-\frac {3 b i n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e g}+\frac {i (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e g}+\frac {(g h-f i) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \log \left (\frac {e (f+g x)}{e f-d g}\right )}{g^2}+\frac {3 b (g h-f i) n \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}-\frac {6 b^2 (g h-f i) n^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {g (d+e x)}{e f-d g}\right )}{g^2}+\frac {6 b^3 (g h-f i) n^3 \operatorname {PolyLog}\left (4,-\frac {g (d+e x)}{e f-d g}\right )}{g^2} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(799\) vs. \(2(308)=616\).

Time = 0.44 (sec) , antiderivative size = 799, normalized size of antiderivative = 2.59 \[ \int \frac {(h+i x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{f+g x} \, dx =\text {Too large to display} \] Input:

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

Output:

(e*g*i*x*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3 + e*(g*h - f*i)*( 
a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^3*Log[f + g*x] + 3*b*e*g*h*n* 
(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(Log[d + e*x]*Log[(e*(f + 
g*x))/(e*f - d*g)] + PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)]) - 3*b*i*n*( 
a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*(-(g*(d + e*x)*(-1 + Log[d 
+ e*x])) + e*f*(Log[d + e*x]*Log[(e*(f + g*x))/(e*f - d*g)] + PolyLog[2, ( 
g*(d + e*x))/(-(e*f) + d*g)])) + 3*b^2*i*n^2*(a - b*n*Log[d + e*x] + b*Log 
[c*(d + e*x)^n])*(g*(2*e*x - 2*(d + e*x)*Log[d + e*x] + (d + e*x)*Log[d + 
e*x]^2) - e*f*(Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)] + 2*Log[d + e 
*x]*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 2*PolyLog[3, (g*(d + e*x))/ 
(-(e*f) + d*g)])) + 6*b^2*e*g*h*n^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e 
*x)^n])*((Log[d + e*x]^2*Log[(e*(f + g*x))/(e*f - d*g)])/2 + Log[d + e*x]* 
PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - PolyLog[3, (g*(d + e*x))/(-(e*f 
) + d*g)]) + b^3*e*g*h*n^3*(Log[d + e*x]^3*Log[(e*(f + g*x))/(e*f - d*g)] 
+ 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-(e*f) + d*g)] - 6*Log[d + e* 
x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] + 6*PolyLog[4, (g*(d + e*x))/( 
-(e*f) + d*g)]) - b^3*i*n^3*(g*(6*e*x - 6*(d + e*x)*Log[d + e*x] + 3*(d + 
e*x)*Log[d + e*x]^2 - (d + e*x)*Log[d + e*x]^3) + e*f*(Log[d + e*x]^3*Log[ 
(e*(f + g*x))/(e*f - d*g)] + 3*Log[d + e*x]^2*PolyLog[2, (g*(d + e*x))/(-( 
e*f) + d*g)] - 6*Log[d + e*x]*PolyLog[3, (g*(d + e*x))/(-(e*f) + d*g)] ...
 

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 308, 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.

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

\(\Big \downarrow \) 2865

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*int(log((d + e*x)**n*c)**3/(d*f + d*g*x + e*f*x + e*g*x**2),x)*b**3* 
d*e*f*g*i*n + 4*int(log((d + e*x)**n*c)**3/(d*f + d*g*x + e*f*x + e*g*x**2 
),x)*b**3*d*e*g**2*h*n + 4*int(log((d + e*x)**n*c)**3/(d*f + d*g*x + e*f*x 
 + e*g*x**2),x)*b**3*e**2*f**2*i*n - 4*int(log((d + e*x)**n*c)**3/(d*f + d 
*g*x + e*f*x + e*g*x**2),x)*b**3*e**2*f*g*h*n - 12*int(log((d + e*x)**n*c) 
**2/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a*b**2*d*e*f*g*i*n + 12*int(log((d 
 + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a*b**2*d*e*g**2*h*n + 
 12*int(log((d + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a*b**2* 
e**2*f**2*i*n - 12*int(log((d + e*x)**n*c)**2/(d*f + d*g*x + e*f*x + e*g*x 
**2),x)*a*b**2*e**2*f*g*h*n - 12*int(log((d + e*x)**n*c)/(d*f + d*g*x + e* 
f*x + e*g*x**2),x)*a**2*b*d*e*f*g*i*n + 12*int(log((d + e*x)**n*c)/(d*f + 
d*g*x + e*f*x + e*g*x**2),x)*a**2*b*d*e*g**2*h*n + 12*int(log((d + e*x)**n 
*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a**2*b*e**2*f**2*i*n - 12*int(log( 
(d + e*x)**n*c)/(d*f + d*g*x + e*f*x + e*g*x**2),x)*a**2*b*e**2*f*g*h*n - 
4*log(f + g*x)*a**3*e*f*i*n + 4*log(f + g*x)*a**3*e*g*h*n - log((d + e*x)* 
*n*c)**4*b**3*e*f*i + log((d + e*x)**n*c)**4*b**3*e*g*h - 4*log((d + e*x)* 
*n*c)**3*a*b**2*e*f*i + 4*log((d + e*x)**n*c)**3*a*b**2*e*g*h + 4*log((d + 
 e*x)**n*c)**3*b**3*d*g*i*n + 4*log((d + e*x)**n*c)**3*b**3*e*g*i*n*x - 6* 
log((d + e*x)**n*c)**2*a**2*b*e*f*i + 6*log((d + e*x)**n*c)**2*a**2*b*e*g* 
h + 12*log((d + e*x)**n*c)**2*a*b**2*d*g*i*n + 12*log((d + e*x)**n*c)**...