Integrand size = 26, antiderivative size = 268 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {5 b n \sqrt {x}}{4 d^3 f^3}+\frac {3 b n x}{8 d^2 f^2}-\frac {7 b n x^{3/2}}{36 d f}+\frac {1}{8} b n x^2+\frac {b n \log \left (1+d f \sqrt {x}\right )}{4 d^4 f^4}-\frac {1}{4} b n x^2 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4} \]
3/8*b*n*x/d^2/f^2-7/36*b*n*x^(3/2)/d/f+1/8*b*n*x^2-1/4*x*(a+b*ln(c*x^n))/d ^2/f^2+1/6*x^(3/2)*(a+b*ln(c*x^n))/d/f-1/8*x^2*(a+b*ln(c*x^n))+1/4*b*n*ln( 1+d*f*x^(1/2))/d^4/f^4-1/4*b*n*x^2*ln(1+d*f*x^(1/2))-1/2*(a+b*ln(c*x^n))*l n(1+d*f*x^(1/2))/d^4/f^4+1/2*x^2*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))-b*n*pol ylog(2,-d*f*x^(1/2))/d^4/f^4-5/4*b*n*x^(1/2)/d^3/f^3+1/2*(a+b*ln(c*x^n))*x ^(1/2)/d^3/f^3
Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.71 \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {18 \left (-1+d^4 f^4 x^2\right ) \log \left (1+d f \sqrt {x}\right ) \left (2 a-b n+2 b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-3 a \left (-12+6 d f \sqrt {x}-4 d^2 f^2 x+3 d^3 f^3 x^{3/2}\right )+b n \left (-90+27 d f \sqrt {x}-14 d^2 f^2 x+9 d^3 f^3 x^{3/2}\right )-3 b \left (-12+6 d f \sqrt {x}-4 d^2 f^2 x+3 d^3 f^3 x^{3/2}\right ) \log \left (c x^n\right )\right )-72 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{72 d^4 f^4} \]
(18*(-1 + d^4*f^4*x^2)*Log[1 + d*f*Sqrt[x]]*(2*a - b*n + 2*b*Log[c*x^n]) + d*f*Sqrt[x]*(-3*a*(-12 + 6*d*f*Sqrt[x] - 4*d^2*f^2*x + 3*d^3*f^3*x^(3/2)) + b*n*(-90 + 27*d*f*Sqrt[x] - 14*d^2*f^2*x + 9*d^3*f^3*x^(3/2)) - 3*b*(-1 2 + 6*d*f*Sqrt[x] - 4*d^2*f^2*x + 3*d^3*f^3*x^(3/2))*Log[c*x^n]) - 72*b*n* PolyLog[2, -(d*f*Sqrt[x])])/(72*d^4*f^4)
Time = 0.45 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
\(\displaystyle \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2823 |
\(\displaystyle -b n \int \left (\frac {1}{2} \log \left (d \sqrt {x} f+1\right ) x-\frac {x}{8}+\frac {\sqrt {x}}{6 d f}-\frac {1}{4 d^2 f^2}+\frac {1}{2 d^3 f^3 \sqrt {x}}-\frac {\log \left (d \sqrt {x} f+1\right )}{2 d^4 f^4 x}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 d f}+\frac {1}{2} x^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {\operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^4 f^4}-\frac {\log \left (d f \sqrt {x}+1\right )}{4 d^4 f^4}+\frac {5 \sqrt {x}}{4 d^3 f^3}-\frac {3 x}{8 d^2 f^2}+\frac {7 x^{3/2}}{36 d f}+\frac {1}{4} x^2 \log \left (d f \sqrt {x}+1\right )-\frac {x^2}{8}\right )\) |
(Sqrt[x]*(a + b*Log[c*x^n]))/(2*d^3*f^3) - (x*(a + b*Log[c*x^n]))/(4*d^2*f ^2) + (x^(3/2)*(a + b*Log[c*x^n]))/(6*d*f) - (x^2*(a + b*Log[c*x^n]))/8 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*d^4*f^4) + (x^2*Log[1 + d*f*S qrt[x]]*(a + b*Log[c*x^n]))/2 - b*n*((5*Sqrt[x])/(4*d^3*f^3) - (3*x)/(8*d^ 2*f^2) + (7*x^(3/2))/(36*d*f) - x^2/8 - Log[1 + d*f*Sqrt[x]]/(4*d^4*f^4) + (x^2*Log[1 + d*f*Sqrt[x]])/4 + PolyLog[2, -(d*f*Sqrt[x])]/(d^4*f^4))
3.1.47.3.1 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]
\[\int x \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )d x\]
\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
\[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]