\(\int x \log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3 \, dx\) [134]

Optimal result

Integrand size = 26, antiderivative size = 907 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1968\) vs. \(2(907)=1814\).

Time = 1.08 (sec) , antiderivative size = 1968, normalized size of antiderivative = 2.17 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(216*a^3*e^3*f*Sqrt[x] - 1620*a^2*b*e^3*f*n*Sqrt[x] + 6804*a*b^2*e^3*f*n^2 
*Sqrt[x] - 13770*b^3*e^3*f*n^3*Sqrt[x] - 108*a^3*e^2*f^2*x + 486*a^2*b*e^2 
*f^2*n*x - 1134*a*b^2*e^2*f^2*n^2*x + 1215*b^3*e^2*f^2*n^3*x + 72*a^3*e*f^ 
3*x^(3/2) - 252*a^2*b*e*f^3*n*x^(3/2) + 444*a*b^2*e*f^3*n^2*x^(3/2) - 350* 
b^3*e*f^3*n^3*x^(3/2) - 54*a^3*f^4*x^2 + 162*a^2*b*f^4*n*x^2 - 243*a*b^2*f 
^4*n^2*x^2 + 162*b^3*f^4*n^3*x^2 - 216*a^3*e^4*Log[e + f*Sqrt[x]] + 324*a^ 
2*b*e^4*n*Log[e + f*Sqrt[x]] - 324*a*b^2*e^4*n^2*Log[e + f*Sqrt[x]] + 162* 
b^3*e^4*n^3*Log[e + f*Sqrt[x]] + 216*a^3*f^4*x^2*Log[d*(e + f*Sqrt[x])] - 
324*a^2*b*f^4*n*x^2*Log[d*(e + f*Sqrt[x])] + 324*a*b^2*f^4*n^2*x^2*Log[d*( 
e + f*Sqrt[x])] - 162*b^3*f^4*n^3*x^2*Log[d*(e + f*Sqrt[x])] + 648*a^2*b*e 
^4*n*Log[e + f*Sqrt[x]]*Log[x] - 648*a*b^2*e^4*n^2*Log[e + f*Sqrt[x]]*Log[ 
x] + 324*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[x] - 648*a^2*b*e^4*n*Log[1 + ( 
f*Sqrt[x])/e]*Log[x] + 648*a*b^2*e^4*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] - 3 
24*b^3*e^4*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x] - 648*a*b^2*e^4*n^2*Log[e + f 
*Sqrt[x]]*Log[x]^2 + 324*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[x]^2 + 648*a*b 
^2*e^4*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 - 324*b^3*e^4*n^3*Log[1 + (f*Sq 
rt[x])/e]*Log[x]^2 + 216*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[x]^3 - 216*b^3 
*e^4*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^3 + 648*a^2*b*e^3*f*Sqrt[x]*Log[c*x 
^n] - 3240*a*b^2*e^3*f*n*Sqrt[x]*Log[c*x^n] + 6804*b^3*e^3*f*n^2*Sqrt[x]*L 
og[c*x^n] - 324*a^2*b*e^2*f^2*x*Log[c*x^n] + 972*a*b^2*e^2*f^2*n*x*Log[...
 

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 935, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(e^3*Sqrt[x]*(a + b*Log[c*x^n])^3)/(2*f^3) - (e^2*x*(a + b*Log[c*x^n])^3)/ 
(4*f^2) + (e*x^(3/2)*(a + b*Log[c*x^n])^3)/(6*f) - (x^2*(a + b*Log[c*x^n]) 
^3)/8 - (e^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(2*f^4) + (x^2*Log[d 
*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/2 - 3*b*n*((85*b^2*e^3*n^2*Sqrt[x] 
)/(8*f^3) + (3*a*b*e^2*n*x)/(4*f^2) - (15*b^2*e^2*n^2*x)/(16*f^2) + (175*b 
^2*e*n^2*x^(3/2))/(648*f) - (b^2*n^2*x^2)/8 - (b^2*e^4*n^2*Log[e + f*Sqrt[ 
x]])/(8*f^4) + (b^2*n^2*x^2*Log[d*(e + f*Sqrt[x])])/8 - (b^2*e^4*n^2*Log[e 
 + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(2*f^4) + (3*b^2*e^2*n*x*Log[c*x^n])/ 
(4*f^2) - (21*b*e^3*n*Sqrt[x]*(a + b*Log[c*x^n]))/(4*f^3) + (b*e^2*n*x*(a 
+ b*Log[c*x^n]))/(8*f^2) - (37*b*e*n*x^(3/2)*(a + b*Log[c*x^n]))/(108*f) + 
 (3*b*n*x^2*(a + b*Log[c*x^n]))/16 + (b*e^4*n*Log[e + f*Sqrt[x]]*(a + b*Lo 
g[c*x^n]))/(4*f^4) - (b*n*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/4 
 + (5*e^3*Sqrt[x]*(a + b*Log[c*x^n])^2)/(4*f^3) - (3*e^2*x*(a + b*Log[c*x^ 
n])^2)/(8*f^2) + (7*e*x^(3/2)*(a + b*Log[c*x^n])^2)/(36*f) - (x^2*(a + b*L 
og[c*x^n])^2)/8 + (x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/4 - (e 
^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*f^4) - (e^4*Log[e + f*S 
qrt[x]]*(a + b*Log[c*x^n])^3)/(6*b*f^4*n) + (e^4*Log[1 + (f*Sqrt[x])/e]*(a 
 + b*Log[c*x^n])^3)/(6*b*f^4*n) - (b^2*e^4*n^2*PolyLog[2, 1 + (f*Sqrt[x])/ 
e])/(2*f^4) - (b*e^4*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/f^ 
4 + (e^4*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^4 + (2*b^...
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x*log(d*(e+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) + d*e), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate(x*log(d*(e+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) + e)*d), x)
 

Giac [F]

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

integrate(x*log(d*(e+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) + e)*d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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