Integrand size = 30, antiderivative size = 555 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {37 b^2 d f n^2}{108 x^{3/2}}+\frac {7 b^2 d^2 f^2 n^2}{8 x}-\frac {21 b^2 d^3 f^3 n^2}{4 \sqrt {x}}+\frac {1}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right )-\frac {b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b^2 d^4 f^4 n^2 \log (x)+\frac {1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac {3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-4 b^2 d^4 f^4 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \] Output:
-37/108*b^2*d*f*n^2/x^(3/2)+7/8*b^2*d^2*f^2*n^2/x-21/4*b^2*d^3*f^3*n^2/x^( 1/2)+1/4*b^2*d^4*f^4*n^2*ln(1+d*f*x^(1/2))-1/4*b^2*n^2*ln(1+d*f*x^(1/2))/x ^2-1/8*b^2*d^4*f^4*n^2*ln(x)+1/8*b^2*d^4*f^4*n^2*ln(x)^2-7/18*b*d*f*n*(a+b *ln(c*x^n))/x^(3/2)+3/4*b*d^2*f^2*n*(a+b*ln(c*x^n))/x-5/2*b*d^3*f^3*n*(a+b *ln(c*x^n))/x^(1/2)+1/2*b*d^4*f^4*n*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))-1/2* b*n*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/x^2-1/4*b*d^4*f^4*n*ln(x)*(a+b*ln(c* x^n))-1/6*d*f*(a+b*ln(c*x^n))^2/x^(3/2)+1/4*d^2*f^2*(a+b*ln(c*x^n))^2/x-1/ 2*d^3*f^3*(a+b*ln(c*x^n))^2/x^(1/2)+1/2*d^4*f^4*ln(1+d*f*x^(1/2))*(a+b*ln( c*x^n))^2-1/2*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))^2/x^2-1/12*d^4*f^4*(a+b*ln (c*x^n))^3/b/n+b^2*d^4*f^4*n^2*polylog(2,-d*f*x^(1/2))+2*b*d^4*f^4*n*(a+b* ln(c*x^n))*polylog(2,-d*f*x^(1/2))-4*b^2*d^4*f^4*n^2*polylog(3,-d*f*x^(1/2 ))
Time = 0.67 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.59 \[ \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^3} \, dx=-\frac {36 a^2 d f \sqrt {x}+84 a b d f n \sqrt {x}+74 b^2 d f n^2 \sqrt {x}-54 a^2 d^2 f^2 x-162 a b d^2 f^2 n x-189 b^2 d^2 f^2 n^2 x+108 a^2 d^3 f^3 x^{3/2}+540 a b d^3 f^3 n x^{3/2}+1134 b^2 d^3 f^3 n^2 x^{3/2}+108 a^2 \log \left (1+d f \sqrt {x}\right )+108 a b n \log \left (1+d f \sqrt {x}\right )+54 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-108 a^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-108 a b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )-54 b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )+54 a^2 d^4 f^4 x^2 \log (x)+54 a b d^4 f^4 n x^2 \log (x)+27 b^2 d^4 f^4 n^2 x^2 \log (x)-54 a b d^4 f^4 n x^2 \log ^2(x)-27 b^2 d^4 f^4 n^2 x^2 \log ^2(x)+18 b^2 d^4 f^4 n^2 x^2 \log ^3(x)+72 a b d f \sqrt {x} \log \left (c x^n\right )+84 b^2 d f n \sqrt {x} \log \left (c x^n\right )-108 a b d^2 f^2 x \log \left (c x^n\right )-162 b^2 d^2 f^2 n x \log \left (c x^n\right )+216 a b d^3 f^3 x^{3/2} \log \left (c x^n\right )+540 b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )+216 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+108 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-216 a b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-108 b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+108 a b d^4 f^4 x^2 \log (x) \log \left (c x^n\right )+54 b^2 d^4 f^4 n x^2 \log (x) \log \left (c x^n\right )-54 b^2 d^4 f^4 n x^2 \log ^2(x) \log \left (c x^n\right )+36 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-54 b^2 d^2 f^2 x \log ^2\left (c x^n\right )+108 b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )+108 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-108 b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+54 b^2 d^4 f^4 x^2 \log (x) \log ^2\left (c x^n\right )-216 b d^4 f^4 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+864 b^2 d^4 f^4 n^2 x^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{216 x^2} \] Input:
Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^3,x]
Output:
-1/216*(36*a^2*d*f*Sqrt[x] + 84*a*b*d*f*n*Sqrt[x] + 74*b^2*d*f*n^2*Sqrt[x] - 54*a^2*d^2*f^2*x - 162*a*b*d^2*f^2*n*x - 189*b^2*d^2*f^2*n^2*x + 108*a^ 2*d^3*f^3*x^(3/2) + 540*a*b*d^3*f^3*n*x^(3/2) + 1134*b^2*d^3*f^3*n^2*x^(3/ 2) + 108*a^2*Log[1 + d*f*Sqrt[x]] + 108*a*b*n*Log[1 + d*f*Sqrt[x]] + 54*b^ 2*n^2*Log[1 + d*f*Sqrt[x]] - 108*a^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 10 8*a*b*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] - 54*b^2*d^4*f^4*n^2*x^2*Log[1 + d*f*Sqrt[x]] + 54*a^2*d^4*f^4*x^2*Log[x] + 54*a*b*d^4*f^4*n*x^2*Log[x] + 2 7*b^2*d^4*f^4*n^2*x^2*Log[x] - 54*a*b*d^4*f^4*n*x^2*Log[x]^2 - 27*b^2*d^4* f^4*n^2*x^2*Log[x]^2 + 18*b^2*d^4*f^4*n^2*x^2*Log[x]^3 + 72*a*b*d*f*Sqrt[x ]*Log[c*x^n] + 84*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 108*a*b*d^2*f^2*x*Log[c*x ^n] - 162*b^2*d^2*f^2*n*x*Log[c*x^n] + 216*a*b*d^3*f^3*x^(3/2)*Log[c*x^n] + 540*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] + 216*a*b*Log[1 + d*f*Sqrt[x]]*Log[ c*x^n] + 108*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 216*a*b*d^4*f^4*x^2*L og[1 + d*f*Sqrt[x]]*Log[c*x^n] - 108*b^2*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x] ]*Log[c*x^n] + 108*a*b*d^4*f^4*x^2*Log[x]*Log[c*x^n] + 54*b^2*d^4*f^4*n*x^ 2*Log[x]*Log[c*x^n] - 54*b^2*d^4*f^4*n*x^2*Log[x]^2*Log[c*x^n] + 36*b^2*d* f*Sqrt[x]*Log[c*x^n]^2 - 54*b^2*d^2*f^2*x*Log[c*x^n]^2 + 108*b^2*d^3*f^3*x ^(3/2)*Log[c*x^n]^2 + 108*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 108*b^2* d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 54*b^2*d^4*f^4*x^2*Log[x]* Log[c*x^n]^2 - 216*b*d^4*f^4*n*x^2*(2*a + b*n + 2*b*Log[c*x^n])*PolyLog...
Time = 0.95 (sec) , antiderivative size = 567, normalized size of antiderivative = 1.02, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -2 b n \int \left (\frac {d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) f^4}{2 x}-\frac {d^4 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4}{4 x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right ) f^3}{2 x^{3/2}}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{4 x^2}-\frac {d \left (a+b \log \left (c x^n\right )\right ) f}{6 x^{5/2}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}\right )dx+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{24 b^2 n^2}-d^4 f^4 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{8 b n}-\frac {1}{4} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {1}{8} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+\frac {5 d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}-\frac {3 d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}+\frac {7 d f \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {1}{2} b d^4 f^4 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+2 b d^4 f^4 n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )-\frac {1}{16} b d^4 f^4 n \log ^2(x)-\frac {1}{8} b d^4 f^4 n \log \left (d f \sqrt {x}+1\right )+\frac {1}{16} b d^4 f^4 n \log (x)+\frac {21 b d^3 f^3 n}{8 \sqrt {x}}-\frac {7 b d^2 f^2 n}{16 x}+\frac {37 b d f n}{216 x^{3/2}}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{8 x^2}\right )+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}\) |
Input:
Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^3,x]
Output:
-1/6*(d*f*(a + b*Log[c*x^n])^2)/x^(3/2) + (d^2*f^2*(a + b*Log[c*x^n])^2)/( 4*x) - (d^3*f^3*(a + b*Log[c*x^n])^2)/(2*Sqrt[x]) + (d^4*f^4*Log[1 + d*f*S qrt[x]]*(a + b*Log[c*x^n])^2)/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) ^2)/(2*x^2) - (d^4*f^4*Log[x]*(a + b*Log[c*x^n])^2)/4 - 2*b*n*((37*b*d*f*n )/(216*x^(3/2)) - (7*b*d^2*f^2*n)/(16*x) + (21*b*d^3*f^3*n)/(8*Sqrt[x]) - (b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]])/8 + (b*n*Log[1 + d*f*Sqrt[x]])/(8*x^2) + (b*d^4*f^4*n*Log[x])/16 - (b*d^4*f^4*n*Log[x]^2)/16 + (7*d*f*(a + b*Log[ c*x^n]))/(36*x^(3/2)) - (3*d^2*f^2*(a + b*Log[c*x^n]))/(8*x) + (5*d^3*f^3* (a + b*Log[c*x^n]))/(4*Sqrt[x]) - (d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log [c*x^n]))/4 + (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*x^2) + (d^4*f^4 *Log[x]*(a + b*Log[c*x^n]))/8 - (d^4*f^4*Log[x]*(a + b*Log[c*x^n])^2)/(8*b *n) + (d^4*f^4*(a + b*Log[c*x^n])^3)/(24*b^2*n^2) - (b*d^4*f^4*n*PolyLog[2 , -(d*f*Sqrt[x])])/2 - d^4*f^4*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x] )] + 2*b*d^4*f^4*n*PolyLog[3, -(d*f*Sqrt[x])])
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]))
\[\int \frac {\ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{x^{3}}d x\]
Input:
int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^2/x^3,x)
Output:
int(ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n))^2/x^3,x)
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:
integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2/x^3,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^3, x)
Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \] Input:
integrate(ln(d*(1/d+f*x**(1/2)))*(a+b*ln(c*x**n))**2/x**3,x)
Output:
Timed out
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:
integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2/x^3,x, algorithm="maxi ma")
Output:
integrate((b*log(c*x^n) + a)^2*log((f*sqrt(x) + 1/d)*d)/x^3, x)
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \] Input:
integrate(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2/x^3,x, algorithm="giac ")
Output:
integrate((b*log(c*x^n) + a)^2*log((f*sqrt(x) + 1/d)*d)/x^3, x)
Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \] Input:
int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^2)/x^3,x)
Output:
int((log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n))^2)/x^3, x)
\[ \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^3} \, dx=\frac {6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )^{2}}{d^{2} f^{2} x^{4}-x^{3}}d x \right ) b^{2} x^{2}-4 \sqrt {x}\, a^{2} d f -24 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) a b -12 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b^{2} n -12 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a b n +12 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d^{2} f^{2} x^{4}-x^{3}}d x \right ) a b \,x^{2}+6 \left (\int \frac {\mathrm {log}\left (x^{n} c \right )}{d^{2} f^{2} x^{4}-x^{3}}d x \right ) b^{2} n \,x^{2}-12 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right )^{2} b^{2}-6 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b^{2} n^{2}-4 \sqrt {x}\, a b d f n -6 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )^{2}}{d^{2} f^{2} x^{4}-x^{3}}d x \right ) b^{2} d f \,x^{2}-12 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )}{d^{2} f^{2} x^{4}-x^{3}}d x \right ) a b d f \,x^{2}-6 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (x^{n} c \right )}{d^{2} f^{2} x^{4}-x^{3}}d x \right ) b^{2} d f n \,x^{2}-12 \,\mathrm {log}\left (\sqrt {x}\right ) a b \,d^{4} f^{4} n \,x^{2}-2 \sqrt {x}\, b^{2} d f \,n^{2}-3 b^{2} n^{2}-3 \mathrm {log}\left (x^{n} c \right )^{2} b^{2}-6 \,\mathrm {log}\left (x^{n} c \right ) a b -6 \,\mathrm {log}\left (x^{n} c \right ) b^{2} n -3 a b n -6 \sqrt {x}\, b^{2} d^{3} f^{3} n^{2} x +6 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b^{2} d^{4} f^{4} n^{2} x^{2}-6 \,\mathrm {log}\left (\sqrt {x}\right ) b^{2} d^{4} f^{4} n^{2} x^{2}+6 a b \,d^{2} f^{2} n x -12 \sqrt {x}\, a b \,d^{3} f^{3} n x +12 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a b \,d^{4} f^{4} n \,x^{2}-12 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a^{2}-12 \sqrt {x}\, a^{2} d^{3} f^{3} x +12 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a^{2} d^{4} f^{4} x^{2}-12 \,\mathrm {log}\left (\sqrt {x}\right ) a^{2} d^{4} f^{4} x^{2}+3 b^{2} d^{2} f^{2} n^{2} x +6 a^{2} d^{2} f^{2} x}{24 x^{2}} \] Input:
int(log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n))^2/x^3,x)
Output:
( - 12*sqrt(x)*a**2*d**3*f**3*x - 4*sqrt(x)*a**2*d*f - 12*sqrt(x)*a*b*d**3 *f**3*n*x - 4*sqrt(x)*a*b*d*f*n - 6*sqrt(x)*b**2*d**3*f**3*n**2*x - 2*sqrt (x)*b**2*d*f*n**2 + 6*int(log(x**n*c)**2/(d**2*f**2*x**4 - x**3),x)*b**2*x **2 + 12*int(log(x**n*c)/(d**2*f**2*x**4 - x**3),x)*a*b*x**2 + 6*int(log(x **n*c)/(d**2*f**2*x**4 - x**3),x)*b**2*n*x**2 - 6*int((sqrt(x)*log(x**n*c) **2)/(d**2*f**2*x**4 - x**3),x)*b**2*d*f*x**2 - 12*int((sqrt(x)*log(x**n*c ))/(d**2*f**2*x**4 - x**3),x)*a*b*d*f*x**2 - 6*int((sqrt(x)*log(x**n*c))/( d**2*f**2*x**4 - x**3),x)*b**2*d*f*n*x**2 - 12*log(sqrt(x)*d*f + 1)*log(x* *n*c)**2*b**2 - 24*log(sqrt(x)*d*f + 1)*log(x**n*c)*a*b - 12*log(sqrt(x)*d *f + 1)*log(x**n*c)*b**2*n + 12*log(sqrt(x)*d*f + 1)*a**2*d**4*f**4*x**2 - 12*log(sqrt(x)*d*f + 1)*a**2 + 12*log(sqrt(x)*d*f + 1)*a*b*d**4*f**4*n*x* *2 - 12*log(sqrt(x)*d*f + 1)*a*b*n + 6*log(sqrt(x)*d*f + 1)*b**2*d**4*f**4 *n**2*x**2 - 6*log(sqrt(x)*d*f + 1)*b**2*n**2 - 12*log(sqrt(x))*a**2*d**4* f**4*x**2 - 12*log(sqrt(x))*a*b*d**4*f**4*n*x**2 - 6*log(sqrt(x))*b**2*d** 4*f**4*n**2*x**2 - 3*log(x**n*c)**2*b**2 - 6*log(x**n*c)*a*b - 6*log(x**n* c)*b**2*n + 6*a**2*d**2*f**2*x + 6*a*b*d**2*f**2*n*x - 3*a*b*n + 3*b**2*d* *2*f**2*n**2*x - 3*b**2*n**2)/(24*x**2)