\(\int \frac {\log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3}{x} \, dx\) [136]

Optimal result

Integrand size = 28, antiderivative size = 178 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )+96 b^3 n^3 \operatorname {PolyLog}\left (5,-\frac {f \sqrt {x}}{e}\right ) \] Output:

1/4*ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))^4/b/n-1/4*ln(1+f*x^(1/2)/e)*(a+b*l 
n(c*x^n))^4/b/n-2*(a+b*ln(c*x^n))^3*polylog(2,-f*x^(1/2)/e)+12*b*n*(a+b*ln 
(c*x^n))^2*polylog(3,-f*x^(1/2)/e)-48*b^2*n^2*(a+b*ln(c*x^n))*polylog(4,-f 
*x^(1/2)/e)+96*b^3*n^3*polylog(5,-f*x^(1/2)/e)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(403\) vs. \(2(178)=356\).

Time = 0.51 (sec) , antiderivative size = 403, normalized size of antiderivative = 2.26 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx=\frac {1}{8} \left (-2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log (x) \left (b^3 n^3 \log ^3(x)-4 b^2 n^2 \log ^2(x) \left (a+b \log \left (c x^n\right )\right )+6 b n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-4 \left (a+b \log \left (c x^n\right )\right )^3\right )-8 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )\right )-12 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+4 \log (x) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )-8 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )\right )-8 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^3(x)+6 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )-24 \log (x) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )+48 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )\right )-2 b^3 n^3 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^4(x)+8 \log ^3(x) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )-48 \log ^2(x) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )+192 \log (x) \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )-384 \operatorname {PolyLog}\left (5,-\frac {f \sqrt {x}}{e}\right )\right )\right ) \] Input:

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

Output:

(-2*Log[d*(e + f*Sqrt[x])]*Log[x]*(b^3*n^3*Log[x]^3 - 4*b^2*n^2*Log[x]^2*( 
a + b*Log[c*x^n]) + 6*b*n*Log[x]*(a + b*Log[c*x^n])^2 - 4*(a + b*Log[c*x^n 
])^3) - 8*(a - b*n*Log[x] + b*Log[c*x^n])^3*(Log[1 + (f*Sqrt[x])/e]*Log[x] 
 + 2*PolyLog[2, -((f*Sqrt[x])/e)]) - 12*b*n*(a - b*n*Log[x] + b*Log[c*x^n] 
)^2*(Log[1 + (f*Sqrt[x])/e]*Log[x]^2 + 4*Log[x]*PolyLog[2, -((f*Sqrt[x])/e 
)] - 8*PolyLog[3, -((f*Sqrt[x])/e)]) - 8*b^2*n^2*(a - b*n*Log[x] + b*Log[c 
*x^n])*(Log[1 + (f*Sqrt[x])/e]*Log[x]^3 + 6*Log[x]^2*PolyLog[2, -((f*Sqrt[ 
x])/e)] - 24*Log[x]*PolyLog[3, -((f*Sqrt[x])/e)] + 48*PolyLog[4, -((f*Sqrt 
[x])/e)]) - 2*b^3*n^3*(Log[1 + (f*Sqrt[x])/e]*Log[x]^4 + 8*Log[x]^3*PolyLo 
g[2, -((f*Sqrt[x])/e)] - 48*Log[x]^2*PolyLog[3, -((f*Sqrt[x])/e)] + 192*Lo 
g[x]*PolyLog[4, -((f*Sqrt[x])/e)] - 384*PolyLog[5, -((f*Sqrt[x])/e)]))/8
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2822, 2775, 2821, 2830, 2830, 7143}

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.

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

\(\Big \downarrow \) 2822

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

\(\Big \downarrow \) 2775

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

\(\Big \downarrow \) 2821

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 2830

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2775

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

rule 2821

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b 
_.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c 
*x^n])^p/m), x] + Simp[b*n*(p/m)   Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c 
*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 
0] && EqQ[d*e, 1]
 

rule 2822

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

rule 2830

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ 
.)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) 
, x] - Simp[b*n*(p/q)   Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 
1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(log(sqrt(x)*d*f + d*e)/(e**2*x - f**2*x**2),x)*a**3*e**2 + int((log(sq 
rt(x)*d*f + d*e)*log(x**n*c)**3)/x,x)*b**3 + 3*int((log(sqrt(x)*d*f + d*e) 
*log(x**n*c)**2)/x,x)*a*b**2 + 3*int((log(sqrt(x)*d*f + d*e)*log(x**n*c))/ 
x,x)*a**2*b - int((sqrt(x)*log(sqrt(x)*d*f + d*e))/(e**2*x - f**2*x**2),x) 
*a**3*e*f + log(sqrt(x)*d*f + d*e)**2*a**3