\(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^{5/2}} \, dx\) [142]

Optimal result

Integrand size = 30, antiderivative size = 310 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=-\frac {5 b f k n}{9 e x}+\frac {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right )}{9 e^3}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {2 b f^3 k n \log (x)}{9 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {4 b f^3 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{3 e^3} \] Output:

-5/9*b*f*k*n/e/x+16/9*b*f^2*k*n/e^2/x^(1/2)-4/9*b*f^3*k*n*ln(e+f*x^(1/2))/ 
e^3-4/9*b*n*ln(d*(e+f*x^(1/2))^k)/x^(3/2)+4/3*b*f^3*k*n*ln(e+f*x^(1/2))*ln 
(-f*x^(1/2)/e)/e^3+2/9*b*f^3*k*n*ln(x)/e^3-1/6*b*f^3*k*n*ln(x)^2/e^3-1/3*f 
*k*(a+b*ln(c*x^n))/e/x+2/3*f^2*k*(a+b*ln(c*x^n))/e^2/x^(1/2)-2/3*f^3*k*ln( 
e+f*x^(1/2))*(a+b*ln(c*x^n))/e^3-2/3*ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n)) 
/x^(3/2)+1/3*f^3*k*ln(x)*(a+b*ln(c*x^n))/e^3+4/3*b*f^3*k*n*polylog(2,1+f*x 
^(1/2)/e)/e^3
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.05 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\frac {-3 a e^2 f k \sqrt {x}-5 b e^2 f k n \sqrt {x}+6 a e f^2 k x+16 b e f^2 k n x-6 a e^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )-4 b e^3 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+3 a f^3 k x^{3/2} \log (x)+2 b f^3 k n x^{3/2} \log (x)-6 b f^3 k n x^{3/2} \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-\frac {3}{2} b f^3 k n x^{3/2} \log ^2(x)-3 b e^2 f k \sqrt {x} \log \left (c x^n\right )+6 b e f^2 k x \log \left (c x^n\right )-6 b e^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+3 b f^3 k x^{3/2} \log (x) \log \left (c x^n\right )-2 f^3 k x^{3/2} \log \left (e+f \sqrt {x}\right ) \left (3 a+2 b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )-12 b f^3 k n x^{3/2} \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{9 e^3 x^{3/2}} \] Input:

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(5/2),x]
 

Output:

(-3*a*e^2*f*k*Sqrt[x] - 5*b*e^2*f*k*n*Sqrt[x] + 6*a*e*f^2*k*x + 16*b*e*f^2 
*k*n*x - 6*a*e^3*Log[d*(e + f*Sqrt[x])^k] - 4*b*e^3*n*Log[d*(e + f*Sqrt[x] 
)^k] + 3*a*f^3*k*x^(3/2)*Log[x] + 2*b*f^3*k*n*x^(3/2)*Log[x] - 6*b*f^3*k*n 
*x^(3/2)*Log[1 + (f*Sqrt[x])/e]*Log[x] - (3*b*f^3*k*n*x^(3/2)*Log[x]^2)/2 
- 3*b*e^2*f*k*Sqrt[x]*Log[c*x^n] + 6*b*e*f^2*k*x*Log[c*x^n] - 6*b*e^3*Log[ 
d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 3*b*f^3*k*x^(3/2)*Log[x]*Log[c*x^n] - 2* 
f^3*k*x^(3/2)*Log[e + f*Sqrt[x]]*(3*a + 2*b*n - 3*b*n*Log[x] + 3*b*Log[c*x 
^n]) - 12*b*f^3*k*n*x^(3/2)*PolyLog[2, -((f*Sqrt[x])/e)])/(9*e^3*x^(3/2))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 299, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x^{5/2}} \, dx\)

\(\Big \downarrow \) 2823

\(\displaystyle -b n \int \left (-\frac {2 k \log \left (e+f \sqrt {x}\right ) f^3}{3 e^3 x}+\frac {k \log (x) f^3}{3 e^3 x}+\frac {2 k f^2}{3 e^2 x^{3/2}}-\frac {k f}{3 e x^2}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{5/2}}\right )dx-\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}-b n \left (\frac {4 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}-\frac {4 f^3 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{3 e^3}+\frac {f^3 k \log ^2(x)}{6 e^3}+\frac {4 f^3 k \log \left (e+f \sqrt {x}\right )}{9 e^3}-\frac {4 f^3 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}-\frac {2 f^3 k \log (x)}{9 e^3}-\frac {16 f^2 k}{9 e^2 \sqrt {x}}+\frac {5 f k}{9 e x}\right )\)

Input:

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(5/2),x]
 

Output:

-1/3*(f*k*(a + b*Log[c*x^n]))/(e*x) + (2*f^2*k*(a + b*Log[c*x^n]))/(3*e^2* 
Sqrt[x]) - (2*f^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^3) - (2*Lo 
g[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/(3*x^(3/2)) + (f^3*k*Log[x]*(a 
+ b*Log[c*x^n]))/(3*e^3) - b*n*((5*f*k)/(9*e*x) - (16*f^2*k)/(9*e^2*Sqrt[x 
]) + (4*f^3*k*Log[e + f*Sqrt[x]])/(9*e^3) + (4*Log[d*(e + f*Sqrt[x])^k])/( 
9*x^(3/2)) - (4*f^3*k*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^3) - 
(2*f^3*k*Log[x])/(9*e^3) + (f^3*k*Log[x]^2)/(6*e^3) - (4*f^3*k*PolyLog[2, 
1 + (f*Sqrt[x])/e])/(3*e^3))
 

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]

\[\int \frac {\ln \left (d \left (e +f \sqrt {x}\right )^{k}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{\frac {5}{2}}}d x\]

Input:

int(ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n))/x^(5/2),x)
 

Output:

int(ln(d*(e+f*x^(1/2))^k)*(a+b*ln(c*x^n))/x^(5/2),x)
 

Fricas [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}} \,d x } \] Input:

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(5/2),x, algorithm="fr 
icas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\text {Timed out} \] Input:

integrate(ln(d*(e+f*x**(1/2))**k)*(a+b*ln(c*x**n))/x**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}} \,d x } \] Input:

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(5/2),x, algorithm="ma 
xima")
 

Output:

1/9*integrate((3*b*f*k*x*log(x^n) + (3*a*f*k + (2*f*k*n + 3*f*k*log(c))*b) 
*x)/x^3, x)/e + 1/9*integrate((3*b*f^3*k*x*log(x^n) + (3*a*f^3*k + (2*f^3* 
k*n + 3*f^3*k*log(c))*b)*x)/x^2, x)/e^3 - 1/9*(2*(b*f^6*k*x^2*log(x^n) + ( 
b*f^6*k*log(c) + a*f^6*k)*x^2)/sqrt(x) - (3*b*e*f^5*k*x^2*log(x^n) + (3*a* 
e*f^5*k - (e*f^5*k*n - 3*e*f^5*k*log(c))*b)*x^2)/x + 2*(3*b*e^2*f^4*k*x^2* 
log(x^n) + (3*a*e^2*f^4*k - (4*e^2*f^4*k*n - 3*e^2*f^4*k*log(c))*b)*x^2)/x 
^(3/2) + 2*(3*b*e^6*x*log(x^n) + (3*a*e^6 + (2*e^6*n + 3*e^6*log(c))*b)*x) 
*log((f*sqrt(x) + e)^k)/x^(5/2) - 2*((3*a*e^4*f^2*k + (8*e^4*f^2*k*n + 3*e 
^4*f^2*k*log(c))*b)*x^2 - (3*a*e^6*log(d) + (2*e^6*n*log(d) + 3*e^6*log(c) 
*log(d))*b)*x + 3*(b*e^4*f^2*k*x^2 - b*e^6*x*log(d))*log(x^n))/x^(5/2))/e^ 
6 + integrate(1/9*(3*b*f^7*k*x*log(x^n) + (3*a*f^7*k + (2*f^7*k*n + 3*f^7* 
k*log(c))*b)*x)/(e^6*f*sqrt(x) + e^7), x)
 

Giac [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}} \,d x } \] Input:

integrate(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(5/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{5/2}} \,d x \] Input:

int((log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)))/x^(5/2),x)
 

Output:

int((log(d*(e + f*x^(1/2))^k)*(a + b*log(c*x^n)))/x^(5/2), x)
 

Reduce [F]

\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\frac {12 \sqrt {x}\, \left (\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b \,e^{2} f^{3} k \,n^{2} x -12 \sqrt {x}\, \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )}{-f^{2} x^{2}+e^{2} x}d x \right ) b e \,f^{4} k \,n^{2} x -12 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, f +e \right ) a \,f^{3} k^{2} n x -52 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, f +e \right ) b \,f^{3} k^{2} n^{2} x -44 \sqrt {x}\, \mathrm {log}\left (\sqrt {x}\right ) b \,f^{3} k^{2} n^{2} x +12 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right )^{2} b \,f^{3} n^{2} x -12 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,f^{3} k n x +44 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,f^{3} k \,n^{2} x +3 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right )^{2} b \,f^{3} k^{2} x +6 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) a \,f^{3} k^{2} x -6 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,e^{2} f \,k^{2} n +26 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,f^{3} k^{2} n x -6 \sqrt {x}\, a \,e^{2} f \,k^{2} n -10 \sqrt {x}\, b \,e^{2} f \,k^{2} n^{2}-12 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) \mathrm {log}\left (x^{n} c \right ) b \,e^{3} k n -12 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) a \,e^{3} k n -8 \,\mathrm {log}\left (\left (\sqrt {x}\, f +e \right )^{k} d \right ) b \,e^{3} k \,n^{2}+12 \,\mathrm {log}\left (x^{n} c \right ) b e \,f^{2} k^{2} n x +12 a e \,f^{2} k^{2} n x +32 b e \,f^{2} k^{2} n^{2} x}{18 \sqrt {x}\, e^{3} k n x} \] Input:

int(log(d*(e+f*x^(1/2))^k)*(a+b*log(c*x^n))/x^(5/2),x)
 

Output:

(12*sqrt(x)*int(log((sqrt(x)*f + e)**k*d)/(e**2*x - f**2*x**2),x)*b*e**2*f 
**3*k*n**2*x - 12*sqrt(x)*int((sqrt(x)*log((sqrt(x)*f + e)**k*d))/(e**2*x 
- f**2*x**2),x)*b*e*f**4*k*n**2*x - 12*sqrt(x)*log(sqrt(x)*f + e)*a*f**3*k 
**2*n*x - 52*sqrt(x)*log(sqrt(x)*f + e)*b*f**3*k**2*n**2*x - 44*sqrt(x)*lo 
g(sqrt(x))*b*f**3*k**2*n**2*x + 12*sqrt(x)*log((sqrt(x)*f + e)**k*d)**2*b* 
f**3*n**2*x - 12*sqrt(x)*log((sqrt(x)*f + e)**k*d)*log(x**n*c)*b*f**3*k*n* 
x + 44*sqrt(x)*log((sqrt(x)*f + e)**k*d)*b*f**3*k*n**2*x + 3*sqrt(x)*log(x 
**n*c)**2*b*f**3*k**2*x + 6*sqrt(x)*log(x**n*c)*a*f**3*k**2*x - 6*sqrt(x)* 
log(x**n*c)*b*e**2*f*k**2*n + 26*sqrt(x)*log(x**n*c)*b*f**3*k**2*n*x - 6*s 
qrt(x)*a*e**2*f*k**2*n - 10*sqrt(x)*b*e**2*f*k**2*n**2 - 12*log((sqrt(x)*f 
 + e)**k*d)*log(x**n*c)*b*e**3*k*n - 12*log((sqrt(x)*f + e)**k*d)*a*e**3*k 
*n - 8*log((sqrt(x)*f + e)**k*d)*b*e**3*k*n**2 + 12*log(x**n*c)*b*e*f**2*k 
**2*n*x + 12*a*e*f**2*k**2*n*x + 32*b*e*f**2*k**2*n**2*x)/(18*sqrt(x)*e**3 
*k*n*x)