\(\int \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n)) \, dx\) [54]

Optimal result

Integrand size = 25, antiderivative size = 172 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {3 b n \sqrt {x}}{d f}+b n x-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {b n \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {2 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2} \] Output:

-3*b*n*x^(1/2)/d/f+b*n*x-b*n*x*ln(d*(1/d+f*x^(1/2)))+b*n*ln(1+d*f*x^(1/2)) 
/d^2/f^2+x^(1/2)*(a+b*ln(c*x^n))/d/f-1/2*x*(a+b*ln(c*x^n))+x*ln(d*(1/d+f*x 
^(1/2)))*(a+b*ln(c*x^n))-ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/d^2/f^2-2*b*n*p 
olylog(2,-d*f*x^(1/2))/d^2/f^2
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.68 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {-2 \left (-1+d^2 f^2 x\right ) \log \left (1+d f \sqrt {x}\right ) \left (a-b n+b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-2 a+6 b n+a d f \sqrt {x}-2 b d f n \sqrt {x}+b \left (-2+d f \sqrt {x}\right ) \log \left (c x^n\right )\right )+4 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{2 d^2 f^2} \] Input:

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

Output:

-1/2*(-2*(-1 + d^2*f^2*x)*Log[1 + d*f*Sqrt[x]]*(a - b*n + b*Log[c*x^n]) + 
d*f*Sqrt[x]*(-2*a + 6*b*n + a*d*f*Sqrt[x] - 2*b*d*f*n*Sqrt[x] + b*(-2 + d* 
f*Sqrt[x])*Log[c*x^n]) + 4*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2817, 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.

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

\(\Big \downarrow \) 2817

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right )}{d^2 f^2}+\frac {3 \sqrt {x}}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-x\right )\)

Input:

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

Output:

(Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) - (x*(a + b*Log[c*x^n]))/2 + x*Log[d*(d 
^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[ 
c*x^n]))/(d^2*f^2) - b*n*((3*Sqrt[x])/(d*f) - x + x*Log[d*(d^(-1) + f*Sqrt 
[x])] - Log[1 + d*f*Sqrt[x]]/(d^2*f^2) + (2*PolyLog[2, -(d*f*Sqrt[x])])/(d 
^2*f^2))
 

Defintions of rubi rules used

rule 2009

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

rule 2817

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. 
)]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], x]}, 
Simp[(a + b*Log[c*x^n])^p   u, x] - Simp[b*n*p   Int[(a + b*Log[c*x^n])^(p 
- 1)/x   u, x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] 
&& RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 
 1] && EqQ[m, 1] && EqQ[d*e, 1]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

(b*x*log(x^n) - (b*(n - log(c)) - a)*x)*log(d*f*sqrt(x) + 1) - 1/9*(3*b*d* 
f*x^2*log(x^n) + (3*a*d*f - (5*d*f*n - 3*d*f*log(c))*b)*x^2)/sqrt(x) + int 
egrate(1/2*(b*d^2*f^2*x*log(x^n) + (a*d^2*f^2 - (d^2*f^2*n - d^2*f^2*log(c 
))*b)*x)/(d*f*sqrt(x) + 1), x)
 

Giac [F]

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

integrate(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)*log((f*sqrt(x) + 1/d)*d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b d f +2 \sqrt {x}\, a d f -6 \sqrt {x}\, b d f n -2 \left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b n +2 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b d f n +2 \mathrm {log}\left (\sqrt {x}\, d f +1\right )^{2} b n +2 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} x -2 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b +2 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a \,d^{2} f^{2} x -2 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a -2 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,d^{2} f^{2} n x +2 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b n -\mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} x -a \,d^{2} f^{2} x +2 b \,d^{2} f^{2} n x}{2 d^{2} f^{2}} \] Input:

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

Output:

(2*sqrt(x)*log(x**n*c)*b*d*f + 2*sqrt(x)*a*d*f - 6*sqrt(x)*b*d*f*n - 2*int 
(log(sqrt(x)*d*f + 1)/(d**2*f**2*x**2 - x),x)*b*n + 2*int((sqrt(x)*log(sqr 
t(x)*d*f + 1))/(d**2*f**2*x**2 - x),x)*b*d*f*n + 2*log(sqrt(x)*d*f + 1)**2 
*b*n + 2*log(sqrt(x)*d*f + 1)*log(x**n*c)*b*d**2*f**2*x - 2*log(sqrt(x)*d* 
f + 1)*log(x**n*c)*b + 2*log(sqrt(x)*d*f + 1)*a*d**2*f**2*x - 2*log(sqrt(x 
)*d*f + 1)*a - 2*log(sqrt(x)*d*f + 1)*b*d**2*f**2*n*x + 2*log(sqrt(x)*d*f 
+ 1)*b*n - log(x**n*c)*b*d**2*f**2*x - a*d**2*f**2*x + 2*b*d**2*f**2*n*x)/ 
(2*d**2*f**2)