Integrand size = 26, antiderivative size = 313 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {5 b e^3 k n \sqrt {x}}{4 f^3}+\frac {3 b e^2 k n x}{8 f^2}-\frac {7 b e k n x^{3/2}}{36 f}+\frac {1}{8} b k n x^2+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right )}{4 f^4}-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {b e^4 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {b e^4 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^4} \]
3/8*b*e^2*k*n*x/f^2-7/36*b*e*k*n*x^(3/2)/f+1/8*b*k*n*x^2-1/4*e^2*k*x*(a+b* ln(c*x^n))/f^2+1/6*e*k*x^(3/2)*(a+b*ln(c*x^n))/f-1/8*k*x^2*(a+b*ln(c*x^n)) +1/4*b*e^4*k*n*ln(e+f*x^(1/2))/f^4-1/2*e^4*k*(a+b*ln(c*x^n))*ln(e+f*x^(1/2 ))/f^4+b*e^4*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f^4-1/4*b*n*x^2*ln(d*(e+ f*x^(1/2))^k)+1/2*x^2*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)+b*e^4*k*n*poly log(2,1+f*x^(1/2)/e)/f^4-5/4*b*e^3*k*n*x^(1/2)/f^3+1/2*e^3*k*(a+b*ln(c*x^n ))*x^(1/2)/f^3
Time = 0.27 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.07 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {-36 a e^3 f k \sqrt {x}+90 b e^3 f k n \sqrt {x}+18 a e^2 f^2 k x-27 b e^2 f^2 k n x-12 a e f^3 k x^{3/2}+14 b e f^3 k n x^{3/2}+9 a f^4 k x^2-9 b f^4 k n x^2-36 a f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b f^4 n x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+36 b e^4 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-36 b e^3 f k \sqrt {x} \log \left (c x^n\right )+18 b e^2 f^2 k x \log \left (c x^n\right )-12 b e f^3 k x^{3/2} \log \left (c x^n\right )+9 b f^4 k x^2 \log \left (c x^n\right )-36 b f^4 x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+18 e^4 k \log \left (e+f \sqrt {x}\right ) \left (2 a-b n-2 b n \log (x)+2 b \log \left (c x^n\right )\right )+72 b e^4 k n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{72 f^4} \]
-1/72*(-36*a*e^3*f*k*Sqrt[x] + 90*b*e^3*f*k*n*Sqrt[x] + 18*a*e^2*f^2*k*x - 27*b*e^2*f^2*k*n*x - 12*a*e*f^3*k*x^(3/2) + 14*b*e*f^3*k*n*x^(3/2) + 9*a* f^4*k*x^2 - 9*b*f^4*k*n*x^2 - 36*a*f^4*x^2*Log[d*(e + f*Sqrt[x])^k] + 18*b *f^4*n*x^2*Log[d*(e + f*Sqrt[x])^k] + 36*b*e^4*k*n*Log[1 + (f*Sqrt[x])/e]* Log[x] - 36*b*e^3*f*k*Sqrt[x]*Log[c*x^n] + 18*b*e^2*f^2*k*x*Log[c*x^n] - 1 2*b*e*f^3*k*x^(3/2)*Log[c*x^n] + 9*b*f^4*k*x^2*Log[c*x^n] - 36*b*f^4*x^2*L og[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 18*e^4*k*Log[e + f*Sqrt[x]]*(2*a - b* n - 2*b*n*Log[x] + 2*b*Log[c*x^n]) + 72*b*e^4*k*n*PolyLog[2, -((f*Sqrt[x]) /e)])/f^4
Time = 0.50 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (-\frac {k \log \left (e+f \sqrt {x}\right ) e^4}{2 f^4 x}+\frac {k e^3}{2 f^3 \sqrt {x}}-\frac {k e^2}{4 f^2}+\frac {k \sqrt {x} e}{6 f}-\frac {k x}{8}+\frac {1}{2} x \log \left (d \left (e+f \sqrt {x}\right )^k\right )\right )dx+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 f^4}+\frac {e^3 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{2 f^3}-\frac {e^2 k x \left (a+b \log \left (c x^n\right )\right )}{4 f^2}+\frac {e k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{6 f}-\frac {1}{8} k x^2 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {1}{4} x^2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^4 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{f^4}-\frac {e^4 k \log \left (e+f \sqrt {x}\right )}{4 f^4}-\frac {e^4 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {5 e^3 k \sqrt {x}}{4 f^3}-\frac {3 e^2 k x}{8 f^2}+\frac {7 e k x^{3/2}}{36 f}-\frac {k x^2}{8}\right )\) |
(e^3*k*Sqrt[x]*(a + b*Log[c*x^n]))/(2*f^3) - (e^2*k*x*(a + b*Log[c*x^n]))/ (4*f^2) + (e*k*x^(3/2)*(a + b*Log[c*x^n]))/(6*f) - (k*x^2*(a + b*Log[c*x^n ]))/8 - (e^4*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*f^4) + (x^2*Log[d *(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/2 - b*n*((5*e^3*k*Sqrt[x])/(4*f^3) - (3*e^2*k*x)/(8*f^2) + (7*e*k*x^(3/2))/(36*f) - (k*x^2)/8 - (e^4*k*Log[e + f*Sqrt[x]])/(4*f^4) + (x^2*Log[d*(e + f*Sqrt[x])^k])/4 - (e^4*k*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^4 - (e^4*k*PolyLog[2, 1 + (f*Sqrt[x]) /e])/f^4)
3.2.16.3.1 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]
\[\int x \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )d x\]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
1/100*(50*b*e*x^2*log(d)*log(x^n) + 25*(2*a*e*log(d) - (e*n*log(d) - 2*e*l og(c)*log(d))*b)*x^2 + 25*(2*b*e*x^2*log(x^n) - ((e*n - 2*e*log(c))*b - 2* a*e)*x^2)*log((f*sqrt(x) + e)^k) - (10*b*f*k*x^3*log(x^n) + (10*a*f*k - (9 *f*k*n - 10*f*k*log(c))*b)*x^3)/sqrt(x))/e + integrate(1/8*(2*b*f^2*k*x^2* log(x^n) + (2*a*f^2*k - (f^2*k*n - 2*f^2*k*log(c))*b)*x^2)/(e*f*sqrt(x) + e^2), x)
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x\,\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]