Integrand size = 30, antiderivative size = 283 \[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {16 b e^2 k n \sqrt {x}}{9 f^2}-\frac {5 b e k n x}{9 f}+\frac {8}{27} b k n x^{3/2}-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right )}{9 f^3}-\frac {4}{9} b n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {4 b e^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}+\frac {2}{3} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b e^3 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{3 f^3} \]
-5/9*b*e*k*n*x/f+8/27*b*k*n*x^(3/2)+1/3*e*k*x*(a+b*ln(c*x^n))/f-2/9*k*x^(3 /2)*(a+b*ln(c*x^n))-4/9*b*e^3*k*n*ln(e+f*x^(1/2))/f^3+2/3*e^3*k*(a+b*ln(c* x^n))*ln(e+f*x^(1/2))/f^3-4/3*b*e^3*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f ^3-4/9*b*n*x^(3/2)*ln(d*(e+f*x^(1/2))^k)+2/3*x^(3/2)*(a+b*ln(c*x^n))*ln(d* (e+f*x^(1/2))^k)-4/3*b*e^3*k*n*polylog(2,1+f*x^(1/2)/e)/f^3+16/9*b*e^2*k*n *x^(1/2)/f^2-2/3*e^2*k*(a+b*ln(c*x^n))*x^(1/2)/f^2
Time = 0.22 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.05 \[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {-18 a e^2 f k \sqrt {x}+48 b e^2 f k n \sqrt {x}+9 a e f^2 k x-15 b e f^2 k n x-6 a f^3 k x^{3/2}+8 b f^3 k n x^{3/2}+18 a f^3 x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-12 b f^3 n x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )+18 b e^3 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-18 b e^2 f k \sqrt {x} \log \left (c x^n\right )+9 b e f^2 k x \log \left (c x^n\right )-6 b f^3 k x^{3/2} \log \left (c x^n\right )+18 b f^3 x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+6 e^3 k \log \left (e+f \sqrt {x}\right ) \left (3 a-2 b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )+36 b e^3 k n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{27 f^3} \]
(-18*a*e^2*f*k*Sqrt[x] + 48*b*e^2*f*k*n*Sqrt[x] + 9*a*e*f^2*k*x - 15*b*e*f ^2*k*n*x - 6*a*f^3*k*x^(3/2) + 8*b*f^3*k*n*x^(3/2) + 18*a*f^3*x^(3/2)*Log[ d*(e + f*Sqrt[x])^k] - 12*b*f^3*n*x^(3/2)*Log[d*(e + f*Sqrt[x])^k] + 18*b* e^3*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 18*b*e^2*f*k*Sqrt[x]*Log[c*x^n] + 9*b*e*f^2*k*x*Log[c*x^n] - 6*b*f^3*k*x^(3/2)*Log[c*x^n] + 18*b*f^3*x^(3/2) *Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 6*e^3*k*Log[e + f*Sqrt[x]]*(3*a - 2 *b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + 36*b*e^3*k*n*PolyLog[2, -((f*Sqrt[ x])/e)])/(27*f^3)
Time = 0.47 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {2 k \log \left (e+f \sqrt {x}\right ) e^3}{3 f^3 x}-\frac {2 k e^2}{3 f^2 \sqrt {x}}+\frac {k e}{3 f}+\frac {2}{3} \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {2 k \sqrt {x}}{9}\right )dx+\frac {2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} x^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 e^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^3}-\frac {2 e^2 k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 f^2}+\frac {e k x \left (a+b \log \left (c x^n\right )\right )}{3 f}-\frac {2}{9} k x^{3/2} \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {4}{9} x^{3/2} \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {4 e^3 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{3 f^3}+\frac {4 e^3 k \log \left (e+f \sqrt {x}\right )}{9 f^3}+\frac {4 e^3 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 f^3}-\frac {16 e^2 k \sqrt {x}}{9 f^2}+\frac {5 e k x}{9 f}-\frac {8}{27} k x^{3/2}\right )\) |
(-2*e^2*k*Sqrt[x]*(a + b*Log[c*x^n]))/(3*f^2) + (e*k*x*(a + b*Log[c*x^n])) /(3*f) - (2*k*x^(3/2)*(a + b*Log[c*x^n]))/9 + (2*e^3*k*Log[e + f*Sqrt[x]]* (a + b*Log[c*x^n]))/(3*f^3) + (2*x^(3/2)*Log[d*(e + f*Sqrt[x])^k]*(a + b*L og[c*x^n]))/3 - b*n*((-16*e^2*k*Sqrt[x])/(9*f^2) + (5*e*k*x)/(9*f) - (8*k* x^(3/2))/27 + (4*e^3*k*Log[e + f*Sqrt[x]])/(9*f^3) + (4*x^(3/2)*Log[d*(e + f*Sqrt[x])^k])/9 + (4*e^3*k*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3* f^3) + (4*e^3*k*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^3))
3.2.34.3.1 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]
\[\int \sqrt {x}\, \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )d x\]
\[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \sqrt {x} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
\[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \sqrt {x} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
2/9*(3*b*x*log(x^n) - (b*(2*n - 3*log(c)) - 3*a)*x)*sqrt(x)*log((f*sqrt(x) + e)^k) + 2/9*(3*b*x*log(d)*log(x^n) - ((2*n*log(d) - 3*log(c)*log(d))*b - 3*a*log(d))*x)*sqrt(x) - integrate(1/9*(3*b*f*k*x*log(x^n) + (3*a*f*k - (2*f*k*n - 3*f*k*log(c))*b)*x)/(f*sqrt(x) + e), x)
\[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \sqrt {x} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int \sqrt {x} \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \sqrt {x}\,\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]