\(\int x \log (d (e+f \sqrt {x})^k) (a+b \log (c x^n)) \, dx\) [122]

Optimal result

Integrand size = 26, antiderivative size = 313 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {5 b e^3 k n \sqrt {x}}{4 f^3}+\frac {3 b e^2 k n x}{8 f^2}-\frac {7 b e k n x^{3/2}}{36 f}+\frac {1}{8} b k n x^2+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right )}{4 f^4}-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^4} \] Output:

-5/4*b*e^3*k*n*x^(1/2)/f^3+3/8*b*e^2*k*n*x/f^2-7/36*b*e*k*n*x^(3/2)/f+1/8* 
b*k*n*x^2+1/4*b*e^4*k*n*ln(e+f*x^(1/2))/f^4-1/4*b*n*x^2*ln(d*(e+f*x^(1/2)) 
^k)+b*e^4*k*n*ln(e+f*x^(1/2))*ln(-f*x^(1/2)/e)/f^4+1/2*e^3*k*x^(1/2)*(a+b* 
ln(c*x^n))/f^3-1/4*e^2*k*x*(a+b*ln(c*x^n))/f^2+1/6*e*k*x^(3/2)*(a+b*ln(c*x 
^n))/f-1/8*k*x^2*(a+b*ln(c*x^n))-1/2*e^4*k*ln(e+f*x^(1/2))*(a+b*ln(c*x^n)) 
/f^4+1/2*x^2*ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n))+b*e^4*k*n*polylog(2,1+f 
*x^(1/2)/e)/f^4
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.07 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {-36 a e^3 f k \sqrt {x}+90 b e^3 f k n \sqrt {x}+18 a e^2 f^2 k x-27 b e^2 f^2 k n x-12 a e f^3 k x^{3/2}+14 b e f^3 k n x^{3/2}+9 a f^4 k x^2-9 b f^4 k n x^2-36 a f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+36 b e^4 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-36 b e^3 f k \sqrt {x} \log \left (c x^n\right )+18 b e^2 f^2 k x \log \left (c x^n\right )-12 b e f^3 k x^{3/2} \log \left (c x^n\right )+9 b f^4 k x^2 \log \left (c x^n\right )-36 b f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+18 e^4 k \log \left (e+f \sqrt {x}\right ) \left (2 a-b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right )+72 b e^4 k n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{72 f^4} \] Input:

Integrate[x*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]),x]
 

Output:

-1/72*(-36*a*e^3*f*k*Sqrt[x] + 90*b*e^3*f*k*n*Sqrt[x] + 18*a*e^2*f^2*k*x - 
 27*b*e^2*f^2*k*n*x - 12*a*e*f^3*k*x^(3/2) + 14*b*e*f^3*k*n*x^(3/2) + 9*a* 
f^4*k*x^2 - 9*b*f^4*k*n*x^2 - 36*a*f^4*x^2*Log[d*(e + f*Sqrt[x])^k] + 18*b 
*f^4*n*x^2*Log[d*(e + f*Sqrt[x])^k] + 36*b*e^4*k*n*Log[1 + (f*Sqrt[x])/e]* 
Log[x] - 36*b*e^3*f*k*Sqrt[x]*Log[c*x^n] + 18*b*e^2*f^2*k*x*Log[c*x^n] - 1 
2*b*e*f^3*k*x^(3/2)*Log[c*x^n] + 9*b*f^4*k*x^2*Log[c*x^n] - 36*b*f^4*x^2*L 
og[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 18*e^4*k*Log[e + f*Sqrt[x]]*(2*a - b* 
n - 2*b*n*Log[x] + 2*b*Log[c*x^n]) + 72*b*e^4*k*n*PolyLog[2, -((f*Sqrt[x]) 
/e)])/f^4
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2823, 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.

\(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx\)

\(\Big \downarrow \) 2823

\(\displaystyle -b n \int \left (-\frac {k \log \left (e+f \sqrt {x}\right ) e^4}{2 f^4 x}+\frac {k e^3}{2 f^3 \sqrt {x}}-\frac {k e^2}{4 f^2}+\frac {k \sqrt {x} e}{6 f}-\frac {k x}{8}+\frac {1}{2} x \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {1}{4} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{f^4}-\frac {e^4 k \log \left (e+f \sqrt {x}\right )}{4 f^4}-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {5 e^3 k \sqrt {x}}{4 f^3}-\frac {3 e^2 k x}{8 f^2}+\frac {7 e k x^{3/2}}{36 f}-\frac {k x^2}{8}\right )\)

Input:

Int[x*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]),x]
 

Output:

(e^3*k*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) - (e^2*k*x*(a + b*Log[c*x^n]))/ 
(4*f^2) + (e*k*x^(3/2)*(a + b*Log[c*x^n]))/(6*f) - (k*x^2*(a + b*Log[c*x^n 
]))/8 - (e^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*f^4) + (x^2*Log[d 
*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/2 - b*n*((5*e^3*k*Sqrt[x])/(4*f^3) 
 - (3*e^2*k*x)/(8*f^2) + (7*e*k*x^(3/2))/(36*f) - (k*x^2)/8 - (e^4*k*Log[e 
 + f*Sqrt[x]])/(4*f^4) + (x^2*Log[d*(e + f*Sqrt[x])^k])/4 - (e^4*k*Log[e + 
 f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^4 - (e^4*k*PolyLog[2, 1 + (f*Sqrt[x]) 
/e])/f^4)
 

Defintions of rubi rules used

rule 2009

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

rule 2823

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n])   u, x] - Simp[b*n   Int[1/x 
 u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q 
+ 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
 

Maple [F]

\[\int x \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]

Input:

int(x*ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n)),x)
 

Output:

int(x*ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n)),x)
 

Fricas [F]

\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \] Input:

integrate(x*log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n)),x, algorithm="fricas")
 

Output:

integral((b*x*log(c*x^n) + a*x)*log((f*sqrt(x) + e)^k*d), x)
 

Sympy [F(-1)]

Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:

integrate(x*ln(d*(e+f*x**(1/2))**k)*(a+b*ln(c*x**n)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \] Input:

integrate(x*log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n)),x, algorithm="maxima")
 

Output:

1/100*(50*b*e*x^2*log(d)*log(x^n) + 25*(2*a*e*log(d) - (e*n*log(d) - 2*e*l 
og(c)*log(d))*b)*x^2 + 25*(2*b*e*x^2*log(x^n) - ((e*n - 2*e*log(c))*b - 2* 
a*e)*x^2)*log((f*sqrt(x) + e)^k) - (10*b*f*k*x^3*log(x^n) + (10*a*f*k - (9 
*f*k*n - 10*f*k*log(c))*b)*x^3)/sqrt(x))/e + integrate(1/8*(2*b*f^2*k*x^2* 
log(x^n) + (2*a*f^2*k - (f^2*k*n - 2*f^2*k*log(c))*b)*x^2)/(e*f*sqrt(x) + 
e^2), x)
 

Giac [F]

\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \] Input:

integrate(x*log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n)),x, algorithm="giac")
 

Output:

integrate((b*log(c*x^n) + a)*x*log((f*sqrt(x) + e)^k*d), x)
 

Mupad [F(-1)]

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

int(x*log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)),x)
 

Output:

int(x*log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)), x)
 

Reduce [F]

\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {36 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,e^{3} f \,k^{2}+12 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b e \,f^{3} k^{2} x +36 \sqrt {x}\, a \,e^{3} f \,k^{2}+12 \sqrt {x}\, a e \,f^{3} k^{2} x -90 \sqrt {x}\, b \,e^{3} f \,k^{2} n -14 \sqrt {x}\, b e \,f^{3} k^{2} n x +36 \left (\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b \,e^{6} k n -36 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b \,e^{5} f k n -36 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) a \,e^{4} k^{2}+18 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) b \,e^{4} k^{2} n +36 \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )^{2} b \,e^{4} n -36 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,e^{4} k +36 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,f^{4} k \,x^{2}+36 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) a \,f^{4} k \,x^{2}-18 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,f^{4} k n \,x^{2}-18 \,\mathrm {log}\left (x^{n} c \right ) b \,e^{2} f^{2} k^{2} x -9 \,\mathrm {log}\left (x^{n} c \right ) b \,f^{4} k^{2} x^{2}-18 a \,e^{2} f^{2} k^{2} x -9 a \,f^{4} k^{2} x^{2}+27 b \,e^{2} f^{2} k^{2} n x +9 b \,f^{4} k^{2} n \,x^{2}}{72 f^{4} k} \] Input:

int(x*log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n)),x)
 

Output:

(36*sqrt(x)*log(x**n*c)*b*e**3*f*k**2 + 12*sqrt(x)*log(x**n*c)*b*e*f**3*k* 
*2*x + 36*sqrt(x)*a*e**3*f*k**2 + 12*sqrt(x)*a*e*f**3*k**2*x - 90*sqrt(x)* 
b*e**3*f*k**2*n - 14*sqrt(x)*b*e*f**3*k**2*n*x + 36*int(log((sqrt(x)*f + e 
)**k*d)/(e**2*x - f**2*x**2),x)*b*e**6*k*n - 36*int((sqrt(x)*log((sqrt(x)* 
f + e)**k*d))/(e**2*x - f**2*x**2),x)*b*e**5*f*k*n - 36*log(sqrt(x)*f + e) 
*a*e**4*k**2 + 18*log(sqrt(x)*f + e)*b*e**4*k**2*n + 36*log((sqrt(x)*f + e 
)**k*d)**2*b*e**4*n - 36*log((sqrt(x)*f + e)**k*d)*log(x**n*c)*b*e**4*k + 
36*log((sqrt(x)*f + e)**k*d)*log(x**n*c)*b*f**4*k*x**2 + 36*log((sqrt(x)*f 
 + e)**k*d)*a*f**4*k*x**2 - 18*log((sqrt(x)*f + e)**k*d)*b*f**4*k*n*x**2 - 
 18*log(x**n*c)*b*e**2*f**2*k**2*x - 9*log(x**n*c)*b*f**4*k**2*x**2 - 18*a 
*e**2*f**2*k**2*x - 9*a*f**4*k**2*x**2 + 27*b*e**2*f**2*k**2*n*x + 9*b*f** 
4*k**2*n*x**2)/(72*f**4*k)