\(\int x \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^3 \, dx\) [65]

Optimal result

Integrand size = 28, antiderivative size = 858 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \] Output:

12*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))/d^4/f^4+3*b^2*n^2*(a+b* 
ln(c*x^n))*polylog(2,-d*f*x^(1/2))/d^4/f^4-3*b*n*(a+b*ln(c*x^n))^2*polylog 
(2,-d*f*x^(1/2))/d^4/f^4-3/4*b^2*n^2*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/d^4 
/f^4+3/4*b*n*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^2/d^4/f^4+63/4*b^2*n^2*x^(1 
/2)*(a+b*ln(c*x^n))/d^3/f^3-15/4*b*n*x^(1/2)*(a+b*ln(c*x^n))^2/d^3/f^3-3/8 
*b^2*n^2*x*(a+b*ln(c*x^n))/d^2/f^2+37/36*b^2*n^2*x^(3/2)*(a+b*ln(c*x^n))/d 
/f+9/8*b*n*x*(a+b*ln(c*x^n))^2/d^2/f^2-7/12*b*n*x^(3/2)*(a+b*ln(c*x^n))^2/ 
d/f+1/2*x^2*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^3-9/4*a*b^2*n^2*x/d^2/f^2-9/ 
4*b^3*n^2*x*ln(c*x^n)/d^2/f^2+45/16*b^3*n^3*x/d^2/f^2-175/216*b^3*n^3*x^(3 
/2)/d/f-1/8*x^2*(a+b*ln(c*x^n))^3-9/16*b^2*n^2*x^2*(a+b*ln(c*x^n))+3/8*b*n 
*x^2*(a+b*ln(c*x^n))^2-24*b^3*n^3*polylog(4,-d*f*x^(1/2))/d^4/f^4-6*b^3*n^ 
3*polylog(3,-d*f*x^(1/2))/d^4/f^4-3/2*b^3*n^3*polylog(2,-d*f*x^(1/2))/d^4/ 
f^4+3/8*b^3*n^3*ln(1+d*f*x^(1/2))/d^4/f^4+3/4*b^2*n^2*x^2*ln(1+d*f*x^(1/2) 
)*(a+b*ln(c*x^n))-3/4*b*n*x^2*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^2-255/8*b^ 
3*n^3*x^(1/2)/d^3/f^3-1/4*x*(a+b*ln(c*x^n))^3/d^2/f^2+1/6*x^(3/2)*(a+b*ln( 
c*x^n))^3/d/f-3/8*b^3*n^3*x^2*ln(1+d*f*x^(1/2))-1/2*ln(1+d*f*x^(1/2))*(a+b 
*ln(c*x^n))^3/d^4/f^4+1/2*x^(1/2)*(a+b*ln(c*x^n))^3/d^3/f^3+3/8*b^3*n^3*x^ 
2
 

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 1432, normalized size of antiderivative = 1.67 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(216*a^3*d*f*Sqrt[x] - 1620*a^2*b*d*f*n*Sqrt[x] + 6804*a*b^2*d*f*n^2*Sqrt[ 
x] - 13770*b^3*d*f*n^3*Sqrt[x] - 108*a^3*d^2*f^2*x + 486*a^2*b*d^2*f^2*n*x 
 - 1134*a*b^2*d^2*f^2*n^2*x + 1215*b^3*d^2*f^2*n^3*x + 72*a^3*d^3*f^3*x^(3 
/2) - 252*a^2*b*d^3*f^3*n*x^(3/2) + 444*a*b^2*d^3*f^3*n^2*x^(3/2) - 350*b^ 
3*d^3*f^3*n^3*x^(3/2) - 54*a^3*d^4*f^4*x^2 + 162*a^2*b*d^4*f^4*n*x^2 - 243 
*a*b^2*d^4*f^4*n^2*x^2 + 162*b^3*d^4*f^4*n^3*x^2 - 216*a^3*Log[1 + d*f*Sqr 
t[x]] + 324*a^2*b*n*Log[1 + d*f*Sqrt[x]] - 324*a*b^2*n^2*Log[1 + d*f*Sqrt[ 
x]] + 162*b^3*n^3*Log[1 + d*f*Sqrt[x]] + 216*a^3*d^4*f^4*x^2*Log[1 + d*f*S 
qrt[x]] - 324*a^2*b*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] + 324*a*b^2*d^4*f^4 
*n^2*x^2*Log[1 + d*f*Sqrt[x]] - 162*b^3*d^4*f^4*n^3*x^2*Log[1 + d*f*Sqrt[x 
]] + 648*a^2*b*d*f*Sqrt[x]*Log[c*x^n] - 3240*a*b^2*d*f*n*Sqrt[x]*Log[c*x^n 
] + 6804*b^3*d*f*n^2*Sqrt[x]*Log[c*x^n] - 324*a^2*b*d^2*f^2*x*Log[c*x^n] + 
 972*a*b^2*d^2*f^2*n*x*Log[c*x^n] - 1134*b^3*d^2*f^2*n^2*x*Log[c*x^n] + 21 
6*a^2*b*d^3*f^3*x^(3/2)*Log[c*x^n] - 504*a*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n 
] + 444*b^3*d^3*f^3*n^2*x^(3/2)*Log[c*x^n] - 162*a^2*b*d^4*f^4*x^2*Log[c*x 
^n] + 324*a*b^2*d^4*f^4*n*x^2*Log[c*x^n] - 243*b^3*d^4*f^4*n^2*x^2*Log[c*x 
^n] - 648*a^2*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 648*a*b^2*n*Log[1 + d*f* 
Sqrt[x]]*Log[c*x^n] - 324*b^3*n^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 648*a^ 
2*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 648*a*b^2*d^4*f^4*n*x^2* 
Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 324*b^3*d^4*f^4*n^2*x^2*Log[1 + d*f*S...
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 810, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2824, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {1}{2} x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-3 b n \left (-\frac {1}{8} n^2 x^2 b^2+\frac {175 n^2 x^{3/2} b^2}{648 d f}-\frac {15 n^2 x b^2}{16 d^2 f^2}-\frac {n^2 \log \left (d \sqrt {x} f+1\right ) b^2}{8 d^4 f^4}+\frac {1}{8} n^2 x^2 \log \left (d \sqrt {x} f+1\right ) b^2+\frac {3 n x \log \left (c x^n\right ) b^2}{4 d^2 f^2}+\frac {n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) b^2}{2 d^4 f^4}+\frac {2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) b^2}{d^4 f^4}+\frac {8 n^2 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right ) b^2}{d^4 f^4}+\frac {85 n^2 \sqrt {x} b^2}{8 d^3 f^3}+\frac {3 a n x b}{4 d^2 f^2}+\frac {3}{16} n x^2 \left (a+b \log \left (c x^n\right )\right ) b-\frac {37 n x^{3/2} \left (a+b \log \left (c x^n\right )\right ) b}{108 d f}+\frac {n x \left (a+b \log \left (c x^n\right )\right ) b}{8 d^2 f^2}-\frac {1}{4} n x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b+\frac {n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b}{4 d^4 f^4}-\frac {21 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) b}{4 d^3 f^3}-\frac {n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) b}{d^4 f^4}-\frac {4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) b}{d^4 f^4}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {7 x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{36 d f}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}+\frac {1}{4} x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 d^4 f^4}+\frac {5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}\right )\)

Input:

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

Output:

(Sqrt[x]*(a + b*Log[c*x^n])^3)/(2*d^3*f^3) - (x*(a + b*Log[c*x^n])^3)/(4*d 
^2*f^2) + (x^(3/2)*(a + b*Log[c*x^n])^3)/(6*d*f) - (x^2*(a + b*Log[c*x^n]) 
^3)/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(2*d^4*f^4) + (x^2*Log 
[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/2 - 3*b*n*((85*b^2*n^2*Sqrt[x])/(8 
*d^3*f^3) + (3*a*b*n*x)/(4*d^2*f^2) - (15*b^2*n^2*x)/(16*d^2*f^2) + (175*b 
^2*n^2*x^(3/2))/(648*d*f) - (b^2*n^2*x^2)/8 - (b^2*n^2*Log[1 + d*f*Sqrt[x] 
])/(8*d^4*f^4) + (b^2*n^2*x^2*Log[1 + d*f*Sqrt[x]])/8 + (3*b^2*n*x*Log[c*x 
^n])/(4*d^2*f^2) - (21*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(4*d^3*f^3) + (b*n* 
x*(a + b*Log[c*x^n]))/(8*d^2*f^2) - (37*b*n*x^(3/2)*(a + b*Log[c*x^n]))/(1 
08*d*f) + (3*b*n*x^2*(a + b*Log[c*x^n]))/16 + (b*n*Log[1 + d*f*Sqrt[x]]*(a 
 + b*Log[c*x^n]))/(4*d^4*f^4) - (b*n*x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c 
*x^n]))/4 + (5*Sqrt[x]*(a + b*Log[c*x^n])^2)/(4*d^3*f^3) - (3*x*(a + b*Log 
[c*x^n])^2)/(8*d^2*f^2) + (7*x^(3/2)*(a + b*Log[c*x^n])^2)/(36*d*f) - (x^2 
*(a + b*Log[c*x^n])^2)/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(4* 
d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/4 + (b^2*n^2*Po 
lyLog[2, -(d*f*Sqrt[x])])/(2*d^4*f^4) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, 
 -(d*f*Sqrt[x])])/(d^4*f^4) + ((a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[ 
x])])/(d^4*f^4) + (2*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(d^4*f^4) - (4*b* 
n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])])/(d^4*f^4) + (8*b^2*n^2*Po 
lyLog[4, -(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 2824

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ 
.))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* 
(e + f*x^m)], 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, g, m, n, 
 q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ 
[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ 
(q + 1)/m] && EqQ[d*e, 1]))
 

Maple [F]

\[\int x \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integral((b^3*x*log(c*x^n)^3 + 3*a*b^2*x*log(c*x^n)^2 + 3*a^2*b*x*log(c*x^ 
n) + a^3*x)*log(d*f*sqrt(x) + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)^3*x*log((f*sqrt(x) + 1/d)*d), x)
 

Giac [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:

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

Output:

integrate((b*log(c*x^n) + a)^3*x*log((f*sqrt(x) + 1/d)*d), x)
 

Mupad [F(-1)]

Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int x\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int x \,\mathrm {log}\left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3}d x \] Input:

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

Output:

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