\(\int \frac {\log (d (e+f \sqrt {x})) (a+b \log (c x^n))^2}{x^2} \, dx\) [132]

Optimal result

Integrand size = 28, antiderivative size = 441 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {14 b^2 f n^2}{e \sqrt {x}}+\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log (x)}{e^2}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {4 b^2 f^2 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {8 b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2} \] Output:

-14*b^2*f*n^2/e/x^(1/2)+2*b^2*f^2*n^2*ln(e+f*x^(1/2))/e^2-2*b^2*n^2*ln(d*( 
e+f*x^(1/2)))/x-4*b^2*f^2*n^2*ln(e+f*x^(1/2))*ln(-f*x^(1/2)/e)/e^2-b^2*f^2 
*n^2*ln(x)/e^2+1/2*b^2*f^2*n^2*ln(x)^2/e^2-6*b*f*n*(a+b*ln(c*x^n))/e/x^(1/ 
2)+2*b*f^2*n*ln(e+f*x^(1/2))*(a+b*ln(c*x^n))/e^2-2*b*n*ln(d*(e+f*x^(1/2))) 
*(a+b*ln(c*x^n))/x-b*f^2*n*ln(x)*(a+b*ln(c*x^n))/e^2-f*(a+b*ln(c*x^n))^2/e 
/x^(1/2)-ln(d*(e+f*x^(1/2)))*(a+b*ln(c*x^n))^2/x+f^2*ln(1+f*x^(1/2)/e)*(a+ 
b*ln(c*x^n))^2/e^2-1/6*f^2*(a+b*ln(c*x^n))^3/b/e^2/n-4*b^2*f^2*n^2*polylog 
(2,1+f*x^(1/2)/e)/e^2+4*b*f^2*n*(a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/e^ 
2-8*b^2*f^2*n^2*polylog(3,-f*x^(1/2)/e)/e^2
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.86 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/3*(3*a^2*e*f*Sqrt[x] + 18*a*b*e*f*n*Sqrt[x] + 42*b^2*e*f*n^2*Sqrt[x] - 
3*a^2*f^2*x*Log[e + f*Sqrt[x]] - 6*a*b*f^2*n*x*Log[e + f*Sqrt[x]] - 6*b^2* 
f^2*n^2*x*Log[e + f*Sqrt[x]] + 3*a^2*e^2*Log[d*(e + f*Sqrt[x])] + 6*a*b*e^ 
2*n*Log[d*(e + f*Sqrt[x])] + 6*b^2*e^2*n^2*Log[d*(e + f*Sqrt[x])] + (3*a^2 
*f^2*x*Log[x])/2 + 3*a*b*f^2*n*x*Log[x] + 3*b^2*f^2*n^2*x*Log[x] + 6*a*b*f 
^2*n*x*Log[e + f*Sqrt[x]]*Log[x] + 6*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]]*Log[ 
x] - 6*a*b*f^2*n*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - 6*b^2*f^2*n^2*x*Log[1 + 
 (f*Sqrt[x])/e]*Log[x] - (3*a*b*f^2*n*x*Log[x]^2)/2 - (3*b^2*f^2*n^2*x*Log 
[x]^2)/2 - 3*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]]*Log[x]^2 + 3*b^2*f^2*n^2*x*L 
og[1 + (f*Sqrt[x])/e]*Log[x]^2 + (b^2*f^2*n^2*x*Log[x]^3)/2 + 6*a*b*e*f*Sq 
rt[x]*Log[c*x^n] + 18*b^2*e*f*n*Sqrt[x]*Log[c*x^n] - 6*a*b*f^2*x*Log[e + f 
*Sqrt[x]]*Log[c*x^n] - 6*b^2*f^2*n*x*Log[e + f*Sqrt[x]]*Log[c*x^n] + 6*a*b 
*e^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 6*b^2*e^2*n*Log[d*(e + f*Sqrt[x]) 
]*Log[c*x^n] + 3*a*b*f^2*x*Log[x]*Log[c*x^n] + 3*b^2*f^2*n*x*Log[x]*Log[c* 
x^n] + 6*b^2*f^2*n*x*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] - 6*b^2*f^2*n*x* 
Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] - (3*b^2*f^2*n*x*Log[x]^2*Log[c*x 
^n])/2 + 3*b^2*e*f*Sqrt[x]*Log[c*x^n]^2 - 3*b^2*f^2*x*Log[e + f*Sqrt[x]]*L 
og[c*x^n]^2 + 3*b^2*e^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + (3*b^2*f^2*x 
*Log[x]*Log[c*x^n]^2)/2 - 12*b*f^2*n*x*(a + b*n + b*Log[c*x^n])*PolyLog[2, 
 -((f*Sqrt[x])/e)] + 24*b^2*f^2*n^2*x*PolyLog[3, -((f*Sqrt[x])/e)])/(e^...
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.22, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

\(\displaystyle -2 b n \left (\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{12 b^2 e^2 n^2}+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}-\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 b e^2 n}-\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {3 f \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}+\frac {2 b f^2 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{e^2}+\frac {4 b f^2 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b f^2 n \log ^2(x)}{4 e^2}-\frac {b f^2 n \log \left (e+f \sqrt {x}\right )}{e^2}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {b f^2 n \log (x)}{2 e^2}+\frac {7 b f n}{e \sqrt {x}}\right )-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}\)

Input:

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

Output:

-((f*(a + b*Log[c*x^n])^2)/(e*Sqrt[x])) + (f^2*Log[e + f*Sqrt[x]]*(a + b*L 
og[c*x^n])^2)/e^2 - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x - (f^2 
*Log[x]*(a + b*Log[c*x^n])^2)/(2*e^2) - 2*b*n*((7*b*f*n)/(e*Sqrt[x]) - (b* 
f^2*n*Log[e + f*Sqrt[x]])/e^2 + (b*n*Log[d*(e + f*Sqrt[x])])/x + (2*b*f^2* 
n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 + (b*f^2*n*Log[x])/(2*e^2) 
 - (b*f^2*n*Log[x]^2)/(4*e^2) + (3*f*(a + b*Log[c*x^n]))/(e*Sqrt[x]) - (f^ 
2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 + (Log[d*(e + f*Sqrt[x])]*(a 
+ b*Log[c*x^n]))/x + (f^2*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) + (f^2*Log[e 
+ f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*b*e^2*n) - (f^2*Log[1 + (f*Sqrt[x])/ 
e]*(a + b*Log[c*x^n])^2)/(2*b*e^2*n) - (f^2*Log[x]*(a + b*Log[c*x^n])^2)/( 
4*b*e^2*n) + (f^2*(a + b*Log[c*x^n])^3)/(12*b^2*e^2*n^2) + (2*b*f^2*n*Poly 
Log[2, 1 + (f*Sqrt[x])/e])/e^2 - (2*f^2*(a + b*Log[c*x^n])*PolyLog[2, -((f 
*Sqrt[x])/e)])/e^2 + (4*b*f^2*n*PolyLog[3, -((f*Sqrt[x])/e)])/e^2)
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

int((log(d*(e + f*x^(1/2)))*(a + b*log(c*x^n))^2)/x^2, 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 )^2}{x^2} \, dx=\frac {-\left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) b^{2} e^{4} x -2 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) a^{2} e^{2}-4 b^{2} e^{2} n^{2}-4 \sqrt {x}\, b^{2} e f \,n^{2}-2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) a b \,e^{4} x -2 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) b^{2} e^{4} n x +\left (\int \frac {\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )^{2}}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) b^{2} e^{3} f x -\mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2}-2 \sqrt {x}\, a^{2} e f -2 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2} e^{2}+2 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) a^{2} f^{2} x -4 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) b^{2} e^{2} n^{2}-2 \,\mathrm {log}\left (\sqrt {x}\right ) a^{2} f^{2} x -2 \,\mathrm {log}\left (x^{n} c \right ) a b \,e^{2}-4 \,\mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n -2 a b \,e^{2} n -4 \sqrt {x}\, a b e f n +4 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) a b \,f^{2} n x -4 \,\mathrm {log}\left (\sqrt {x}\right ) a b \,f^{2} n x +2 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) a b \,e^{3} f x +2 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )}{-f^{2} x^{3}+e^{2} x^{2}}d x \right ) b^{2} e^{3} f n x -4 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) \mathrm {log}\left (x^{n} c \right ) a b \,e^{2}-4 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) \mathrm {log}\left (x^{n} c \right ) b^{2} e^{2} n -4 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) a b \,e^{2} n +4 \,\mathrm {log}\left (\sqrt {x}\, d f +d e \right ) b^{2} f^{2} n^{2} x -4 \,\mathrm {log}\left (\sqrt {x}\right ) b^{2} f^{2} n^{2} x}{2 e^{2} x} \] Input:

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

Output:

( - 2*sqrt(x)*a**2*e*f - 4*sqrt(x)*a*b*e*f*n - 4*sqrt(x)*b**2*e*f*n**2 - i 
nt(log(x**n*c)**2/(e**2*x**2 - f**2*x**3),x)*b**2*e**4*x - 2*int(log(x**n* 
c)/(e**2*x**2 - f**2*x**3),x)*a*b*e**4*x - 2*int(log(x**n*c)/(e**2*x**2 - 
f**2*x**3),x)*b**2*e**4*n*x + int((sqrt(x)*log(x**n*c)**2)/(e**2*x**2 - f* 
*2*x**3),x)*b**2*e**3*f*x + 2*int((sqrt(x)*log(x**n*c))/(e**2*x**2 - f**2* 
x**3),x)*a*b*e**3*f*x + 2*int((sqrt(x)*log(x**n*c))/(e**2*x**2 - f**2*x**3 
),x)*b**2*e**3*f*n*x - 2*log(sqrt(x)*d*f + d*e)*log(x**n*c)**2*b**2*e**2 - 
 4*log(sqrt(x)*d*f + d*e)*log(x**n*c)*a*b*e**2 - 4*log(sqrt(x)*d*f + d*e)* 
log(x**n*c)*b**2*e**2*n - 2*log(sqrt(x)*d*f + d*e)*a**2*e**2 + 2*log(sqrt( 
x)*d*f + d*e)*a**2*f**2*x - 4*log(sqrt(x)*d*f + d*e)*a*b*e**2*n + 4*log(sq 
rt(x)*d*f + d*e)*a*b*f**2*n*x - 4*log(sqrt(x)*d*f + d*e)*b**2*e**2*n**2 + 
4*log(sqrt(x)*d*f + d*e)*b**2*f**2*n**2*x - 2*log(sqrt(x))*a**2*f**2*x - 4 
*log(sqrt(x))*a*b*f**2*n*x - 4*log(sqrt(x))*b**2*f**2*n**2*x - log(x**n*c) 
**2*b**2*e**2 - 2*log(x**n*c)*a*b*e**2 - 4*log(x**n*c)*b**2*e**2*n - 2*a*b 
*e**2*n - 4*b**2*e**2*n**2)/(2*e**2*x)