\(\int x \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n)) \, dx\) [53]

Optimal result

Integrand size = 26, antiderivative size = 268 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {5 b n \sqrt {x}}{4 d^3 f^3}+\frac {3 b n x}{8 d^2 f^2}-\frac {7 b n x^{3/2}}{36 d f}+\frac {1}{8} b n x^2+\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}-\frac {1}{4} b n x^2 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.71 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {18 \left (-1+d^4 f^4 x^2\right ) \log \left (1+d f \sqrt {x}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-3 a \left (-12+6 d f \sqrt {x}-4 d^2 f^2 x+3 d^3 f^3 x^{3/2}\right )+b n \left (-90+27 d f \sqrt {x}-14 d^2 f^2 x+9 d^3 f^3 x^{3/2}\right )-3 b \left (-12+6 d f \sqrt {x}-4 d^2 f^2 x+3 d^3 f^3 x^{3/2}\right ) \log \left (c x^n\right )\right )-72 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{72 d^4 f^4} \] Input:

Integrate[x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]),x]
 

Output:

(18*(-1 + d^4*f^4*x^2)*Log[1 + d*f*Sqrt[x]]*(2*a - b*n + 2*b*Log[c*x^n]) + 
 d*f*Sqrt[x]*(-3*a*(-12 + 6*d*f*Sqrt[x] - 4*d^2*f^2*x + 3*d^3*f^3*x^(3/2)) 
 + b*n*(-90 + 27*d*f*Sqrt[x] - 14*d^2*f^2*x + 9*d^3*f^3*x^(3/2)) - 3*b*(-1 
2 + 6*d*f*Sqrt[x] - 4*d^2*f^2*x + 3*d^3*f^3*x^(3/2))*Log[c*x^n]) - 72*b*n* 
PolyLog[2, -(d*f*Sqrt[x])])/(72*d^4*f^4)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96, 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.

\(\displaystyle \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2823

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

\(\Big \downarrow \) 2009

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

Input:

Int[x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]),x]
 

Output:

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

\[\int x \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integral((b*x*log(c*x^n) + a*x)*log(d*f*sqrt(x) + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)*x*log((f*sqrt(x) + 1/d)*d), x)
 

Giac [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)*x*log((f*sqrt(x) + 1/d)*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {12 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,d^{3} f^{3} x +36 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b d f +12 \sqrt {x}\, a \,d^{3} f^{3} x +36 \sqrt {x}\, a d f -14 \sqrt {x}\, b \,d^{3} f^{3} n x -90 \sqrt {x}\, b d f n -36 \left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b n +36 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b d f n +36 \mathrm {log}\left (\sqrt {x}\, d f +1\right )^{2} b n +36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b \,d^{4} f^{4} x^{2}-36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b +36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a \,d^{4} f^{4} x^{2}-36 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a -18 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,d^{4} f^{4} n \,x^{2}+18 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b n -9 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{4} f^{4} x^{2}-18 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} x -9 a \,d^{4} f^{4} x^{2}-18 a \,d^{2} f^{2} x +9 b \,d^{4} f^{4} n \,x^{2}+27 b \,d^{2} f^{2} n x}{72 d^{4} f^{4}} \] Input:

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

Output:

(12*sqrt(x)*log(x**n*c)*b*d**3*f**3*x + 36*sqrt(x)*log(x**n*c)*b*d*f + 12* 
sqrt(x)*a*d**3*f**3*x + 36*sqrt(x)*a*d*f - 14*sqrt(x)*b*d**3*f**3*n*x - 90 
*sqrt(x)*b*d*f*n - 36*int(log(sqrt(x)*d*f + 1)/(d**2*f**2*x**2 - x),x)*b*n 
 + 36*int((sqrt(x)*log(sqrt(x)*d*f + 1))/(d**2*f**2*x**2 - x),x)*b*d*f*n + 
 36*log(sqrt(x)*d*f + 1)**2*b*n + 36*log(sqrt(x)*d*f + 1)*log(x**n*c)*b*d* 
*4*f**4*x**2 - 36*log(sqrt(x)*d*f + 1)*log(x**n*c)*b + 36*log(sqrt(x)*d*f 
+ 1)*a*d**4*f**4*x**2 - 36*log(sqrt(x)*d*f + 1)*a - 18*log(sqrt(x)*d*f + 1 
)*b*d**4*f**4*n*x**2 + 18*log(sqrt(x)*d*f + 1)*b*n - 9*log(x**n*c)*b*d**4* 
f**4*x**2 - 18*log(x**n*c)*b*d**2*f**2*x - 9*a*d**4*f**4*x**2 - 18*a*d**2* 
f**2*x + 9*b*d**4*f**4*n*x**2 + 27*b*d**2*f**2*n*x)/(72*d**4*f**4)