\(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^{3/2}} \, dx\) [141]

Optimal result

Integrand size = 30, antiderivative size = 199 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=-\frac {4 b f k n \log \left (e+f \sqrt {x}\right )}{e}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{\sqrt {x}}+\frac {4 b f k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e}+\frac {2 b f k n \log (x)}{e}-\frac {b f k n \log ^2(x)}{2 e}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+\frac {f k \log (x) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {4 b f k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e} \] Output:

-4*b*f*k*n*ln(e+f*x^(1/2))/e-4*b*n*ln(d*(e+f*x^(1/2))^k)/x^(1/2)+4*b*f*k*n 
*ln(e+f*x^(1/2))*ln(-f*x^(1/2)/e)/e+2*b*f*k*n*ln(x)/e-1/2*b*f*k*n*ln(x)^2/ 
e-2*f*k*ln(e+f*x^(1/2))*(a+b*ln(c*x^n))/e-2*ln(d*(e+f*x^(1/2))^k)*(a+b*ln( 
c*x^n))/x^(1/2)+f*k*ln(x)*(a+b*ln(c*x^n))/e+4*b*f*k*n*polylog(2,1+f*x^(1/2 
)/e)/e
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.73 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+2 b n+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {2 f k \log \left (e+f \sqrt {x}\right ) \left (a+2 b n-b n \log (x)+b \log \left (c x^n\right )\right )}{e}-\frac {f k \log (x) \left (4 b n \log \left (1+\frac {f \sqrt {x}}{e}\right )+b n \log (x)-2 \left (a+2 b n+b \log \left (c x^n\right )\right )\right )}{2 e}-\frac {4 b f k n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e} \] Input:

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(3/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.

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

\(\Big \downarrow \) 2823

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

\(\Big \downarrow \) 2009

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

Input:

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(3/2),x]
 

Output:

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

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]

\[\int \frac {\ln \left (d \left (e +f \sqrt {x}\right )^{k}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{\frac {3}{2}}}d x\]

Input:

int(ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n))/x^(3/2),x)
 

Output:

int(ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n))/x^(3/2),x)
 

Fricas [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {3}{2}}} \,d x } \] Input:

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(3/2),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(ln(d*(e+f*x**(1/2))**k)*(a+b*ln(c*x**n))/x**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {3}{2}}} \,d x } \] Input:

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*f*k*x*log(x^n) + (a*f*k + (2*f*k*n + f*k*log(c))*b)*x)/x^2, x 
)/e - 1/9*(2*(3*b*f^4*k*x^2*log(x^n) + (3*a*f^4*k + (4*f^4*k*n + 3*f^4*k*l 
og(c))*b)*x^2)/sqrt(x) + 18*(b*e^4*x*log(x^n) + (a*e^4 + (2*e^4*n + e^4*lo 
g(c))*b)*x)*log((f*sqrt(x) + e)^k)/x^(3/2) - 9*(b*e*f^3*k*x^2*log(x^n) + ( 
a*e*f^3*k + (e*f^3*k*n + e*f^3*k*log(c))*b)*x^2)/x + 18*((b*e^2*f^2*k*log( 
c) + a*e^2*f^2*k)*x^2 + (a*e^4*log(d) + (2*e^4*n*log(d) + e^4*log(c)*log(d 
))*b)*x + (b*e^2*f^2*k*x^2 + b*e^4*x*log(d))*log(x^n))/x^(3/2))/e^4 + inte 
grate((b*f^5*k*x*log(x^n) + (a*f^5*k + (2*f^5*k*n + f^5*k*log(c))*b)*x)/(e 
^4*f*sqrt(x) + e^5), x)
 

Giac [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {3}{2}}} \,d x } \] Input:

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{3/2}} \,d x \] Input:

int((log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)))/x^(3/2),x)
 

Output:

int((log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)))/x^(3/2), x)
 

Reduce [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}} \, dx=\frac {4 \sqrt {x}\, \left (\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b \,e^{2} f k \,n^{2}-4 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) b \,e^{3} k \,n^{2}-4 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, f +e \right ) a f \,k^{2} n -16 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, f +e \right ) b f \,k^{2} n^{2}+4 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )^{2} b f \,n^{2}-4 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b f k n +\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )^{2} b f \,k^{2}+2 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) a f \,k^{2}+8 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b f \,k^{2} n -4 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b e k n -4 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) a e k n -16 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b e k \,n^{2}}{2 \sqrt {x}\, e k n} \] Input:

int(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(3/2),x)
 

Output:

(4*sqrt(x)*int(log((sqrt(x)*f + e)**k*d)/(e**2*x - f**2*x**2),x)*b*e**2*f* 
k*n**2 - 4*sqrt(x)*int((sqrt(x)*log((sqrt(x)*f + e)**k*d))/(e**2*x**2 - f* 
*2*x**3),x)*b*e**3*k*n**2 - 4*sqrt(x)*log(sqrt(x)*f + e)*a*f*k**2*n - 16*s 
qrt(x)*log(sqrt(x)*f + e)*b*f*k**2*n**2 + 4*sqrt(x)*log((sqrt(x)*f + e)**k 
*d)**2*b*f*n**2 - 4*sqrt(x)*log((sqrt(x)*f + e)**k*d)*log(x**n*c)*b*f*k*n 
+ sqrt(x)*log(x**n*c)**2*b*f*k**2 + 2*sqrt(x)*log(x**n*c)*a*f*k**2 + 8*sqr 
t(x)*log(x**n*c)*b*f*k**2*n - 4*log((sqrt(x)*f + e)**k*d)*log(x**n*c)*b*e* 
k*n - 4*log((sqrt(x)*f + e)**k*d)*a*e*k*n - 16*log((sqrt(x)*f + e)**k*d)*b 
*e*k*n**2)/(2*sqrt(x)*e*k*n)