\(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^2}{x^2} \, dx\) [63]

Optimal result

Integrand size = 30, antiderivative size = 389 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {14 b^2 d f n^2}{\sqrt {x}}+2 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{x}-b^2 d^2 f^2 n^2 \log (x)+\frac {1}{2} b^2 d^2 f^2 n^2 \log ^2(x)-\frac {6 b d f n \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+2 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-b d^2 f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b n}+4 b^2 d^2 f^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+4 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-8 b^2 d^2 f^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.61 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {6 a^2 d f \sqrt {x}+36 a b d f n \sqrt {x}+84 b^2 d f n^2 \sqrt {x}+6 a^2 \log \left (1+d f \sqrt {x}\right )+12 a b n \log \left (1+d f \sqrt {x}\right )+12 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-6 a^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right )-12 a b d^2 f^2 n x \log \left (1+d f \sqrt {x}\right )-12 b^2 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right )+3 a^2 d^2 f^2 x \log (x)+6 a b d^2 f^2 n x \log (x)+6 b^2 d^2 f^2 n^2 x \log (x)-3 a b d^2 f^2 n x \log ^2(x)-3 b^2 d^2 f^2 n^2 x \log ^2(x)+b^2 d^2 f^2 n^2 x \log ^3(x)+12 a b d f \sqrt {x} \log \left (c x^n\right )+36 b^2 d f n \sqrt {x} \log \left (c x^n\right )+12 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+12 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 a b d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 b^2 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+6 a b d^2 f^2 x \log (x) \log \left (c x^n\right )+6 b^2 d^2 f^2 n x \log (x) \log \left (c x^n\right )-3 b^2 d^2 f^2 n x \log ^2(x) \log \left (c x^n\right )+6 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+6 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 b^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+3 b^2 d^2 f^2 x \log (x) \log ^2\left (c x^n\right )-24 b d^2 f^2 n x \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+48 b^2 d^2 f^2 n^2 x \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{6 x} \] Input:

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^2,x]
 

Output:

-1/6*(6*a^2*d*f*Sqrt[x] + 36*a*b*d*f*n*Sqrt[x] + 84*b^2*d*f*n^2*Sqrt[x] + 
6*a^2*Log[1 + d*f*Sqrt[x]] + 12*a*b*n*Log[1 + d*f*Sqrt[x]] + 12*b^2*n^2*Lo 
g[1 + d*f*Sqrt[x]] - 6*a^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] - 12*a*b*d^2*f^2 
*n*x*Log[1 + d*f*Sqrt[x]] - 12*b^2*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt[x]] + 3* 
a^2*d^2*f^2*x*Log[x] + 6*a*b*d^2*f^2*n*x*Log[x] + 6*b^2*d^2*f^2*n^2*x*Log[ 
x] - 3*a*b*d^2*f^2*n*x*Log[x]^2 - 3*b^2*d^2*f^2*n^2*x*Log[x]^2 + b^2*d^2*f 
^2*n^2*x*Log[x]^3 + 12*a*b*d*f*Sqrt[x]*Log[c*x^n] + 36*b^2*d*f*n*Sqrt[x]*L 
og[c*x^n] + 12*a*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 12*b^2*n*Log[1 + d*f* 
Sqrt[x]]*Log[c*x^n] - 12*a*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 1 
2*b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 6*a*b*d^2*f^2*x*Log[x] 
*Log[c*x^n] + 6*b^2*d^2*f^2*n*x*Log[x]*Log[c*x^n] - 3*b^2*d^2*f^2*n*x*Log[ 
x]^2*Log[c*x^n] + 6*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 + 6*b^2*Log[1 + d*f*Sqrt[ 
x]]*Log[c*x^n]^2 - 6*b^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 3*b 
^2*d^2*f^2*x*Log[x]*Log[c*x^n]^2 - 24*b*d^2*f^2*n*x*(a + b*n + b*Log[c*x^n 
])*PolyLog[2, -(d*f*Sqrt[x])] + 48*b^2*d^2*f^2*n^2*x*PolyLog[3, -(d*f*Sqrt 
[x])])/x
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.

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

\(\Big \downarrow \) 2824

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

\(\Big \downarrow \) 2009

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

Input:

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^2,x]
 

Output:

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

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]

\[\int \frac {\ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{x^{2}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{2}} \,d x } \] Input:

integrate(log(d*(1/d+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) + 1/d)*d)/x^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(x)*a**2*d*f - 4*sqrt(x)*a*b*d*f*n - 4*sqrt(x)*b**2*d*f*n**2 + i 
nt(log(x**n*c)**2/(d**2*f**2*x**3 - x**2),x)*b**2*x + 2*int(log(x**n*c)/(d 
**2*f**2*x**3 - x**2),x)*a*b*x + 2*int(log(x**n*c)/(d**2*f**2*x**3 - x**2) 
,x)*b**2*n*x - int((sqrt(x)*log(x**n*c)**2)/(d**2*f**2*x**3 - x**2),x)*b** 
2*d*f*x - 2*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**3 - x**2),x)*a*b*d*f*x 
 - 2*int((sqrt(x)*log(x**n*c))/(d**2*f**2*x**3 - x**2),x)*b**2*d*f*n*x - 2 
*log(sqrt(x)*d*f + 1)*log(x**n*c)**2*b**2 - 4*log(sqrt(x)*d*f + 1)*log(x** 
n*c)*a*b - 4*log(sqrt(x)*d*f + 1)*log(x**n*c)*b**2*n + 2*log(sqrt(x)*d*f + 
 1)*a**2*d**2*f**2*x - 2*log(sqrt(x)*d*f + 1)*a**2 + 4*log(sqrt(x)*d*f + 1 
)*a*b*d**2*f**2*n*x - 4*log(sqrt(x)*d*f + 1)*a*b*n + 4*log(sqrt(x)*d*f + 1 
)*b**2*d**2*f**2*n**2*x - 4*log(sqrt(x)*d*f + 1)*b**2*n**2 - 2*log(sqrt(x) 
)*a**2*d**2*f**2*x - 4*log(sqrt(x))*a*b*d**2*f**2*n*x - 4*log(sqrt(x))*b** 
2*d**2*f**2*n**2*x - log(x**n*c)**2*b**2 - 2*log(x**n*c)*a*b - 4*log(x**n* 
c)*b**2*n - 2*a*b*n - 4*b**2*n**2)/(2*x)