Integrand size = 28, antiderivative size = 289 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {7 b d f n}{36 x^{3/2}}+\frac {3 b d^2 f^2 n}{8 x}-\frac {5 b d^3 f^3 n}{4 \sqrt {x}}+\frac {1}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right )-\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 x^2}-\frac {1}{8} b d^4 f^4 n \log (x)+\frac {1}{8} b d^4 f^4 n \log ^2(x)-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )+b d^4 f^4 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \]
-7/36*b*d*f*n/x^(3/2)+3/8*b*d^2*f^2*n/x-1/8*b*d^4*f^4*n*ln(x)+1/8*b*d^4*f^ 4*n*ln(x)^2-1/6*d*f*(a+b*ln(c*x^n))/x^(3/2)+1/4*d^2*f^2*(a+b*ln(c*x^n))/x- 1/4*d^4*f^4*ln(x)*(a+b*ln(c*x^n))+1/4*b*d^4*f^4*n*ln(1+d*f*x^(1/2))-1/4*b* n*ln(1+d*f*x^(1/2))/x^2+1/2*d^4*f^4*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))-1/2* (a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/x^2+b*d^4*f^4*n*polylog(2,-d*f*x^(1/2))- 5/4*b*d^3*f^3*n/x^(1/2)-1/2*d^3*f^3*(a+b*ln(c*x^n))/x^(1/2)
Time = 0.20 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.72 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {\left (-1+d^4 f^4 x^2\right ) \log \left (1+d f \sqrt {x}\right ) \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{4 x^2}-\frac {d f \left (12 a+14 b n-18 a d f \sqrt {x}-27 b d f n \sqrt {x}+36 a d^2 f^2 x+90 b d^2 f^2 n x-9 b d^3 f^3 n x^{3/2} \log ^2(x)+6 b \left (2-3 d f \sqrt {x}+6 d^2 f^2 x\right ) \log \left (c x^n\right )+9 d^3 f^3 x^{3/2} \log (x) \left (2 a+b n+2 b \log \left (c x^n\right )\right )\right )}{72 x^{3/2}}+b d^4 f^4 n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \]
((-1 + d^4*f^4*x^2)*Log[1 + d*f*Sqrt[x]]*(2*a + b*n + 2*b*Log[c*x^n]))/(4* x^2) - (d*f*(12*a + 14*b*n - 18*a*d*f*Sqrt[x] - 27*b*d*f*n*Sqrt[x] + 36*a* d^2*f^2*x + 90*b*d^2*f^2*n*x - 9*b*d^3*f^3*n*x^(3/2)*Log[x]^2 + 6*b*(2 - 3 *d*f*Sqrt[x] + 6*d^2*f^2*x)*Log[c*x^n] + 9*d^3*f^3*x^(3/2)*Log[x]*(2*a + b *n + 2*b*Log[c*x^n])))/(72*x^(3/2)) + b*d^4*f^4*n*PolyLog[2, -(d*f*Sqrt[x] )]
Time = 0.47 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2823, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {d^4 \log \left (d \sqrt {x} f+1\right ) f^4}{2 x}-\frac {d^4 \log (x) f^4}{4 x}-\frac {d^3 f^3}{2 x^{3/2}}+\frac {d^2 f^2}{4 x^2}-\frac {d f}{6 x^{5/2}}-\frac {\log \left (d \sqrt {x} f+1\right )}{2 x^3}\right )dx+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-b n \left (-d^4 f^4 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-\frac {1}{8} d^4 f^4 \log ^2(x)-\frac {1}{4} d^4 f^4 \log \left (d f \sqrt {x}+1\right )+\frac {1}{8} d^4 f^4 \log (x)+\frac {5 d^3 f^3}{4 \sqrt {x}}-\frac {3 d^2 f^2}{8 x}+\frac {7 d f}{36 x^{3/2}}+\frac {\log \left (d f \sqrt {x}+1\right )}{4 x^2}\right )\) |
-1/6*(d*f*(a + b*Log[c*x^n]))/x^(3/2) + (d^2*f^2*(a + b*Log[c*x^n]))/(4*x) - (d^3*f^3*(a + b*Log[c*x^n]))/(2*Sqrt[x]) + (d^4*f^4*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^2 ) - (d^4*f^4*Log[x]*(a + b*Log[c*x^n]))/4 - b*n*((7*d*f)/(36*x^(3/2)) - (3 *d^2*f^2)/(8*x) + (5*d^3*f^3)/(4*Sqrt[x]) - (d^4*f^4*Log[1 + d*f*Sqrt[x]]) /4 + Log[1 + d*f*Sqrt[x]]/(4*x^2) + (d^4*f^4*Log[x])/8 - (d^4*f^4*Log[x]^2 )/8 - d^4*f^4*PolyLog[2, -(d*f*Sqrt[x])])
3.1.51.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{3}}d x\]
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^3} \,d x \]