Integrand size = 25, antiderivative size = 209 \[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {3 b e k n \sqrt {x}}{f}+b k n x+\frac {b e^2 k n \log \left (e+f \sqrt {x}\right )}{f^2}-b n x \log \left (d \left (e+f \sqrt {x}\right )^k\right )+\frac {2 b e^2 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {e k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {1}{2} k x \left (a+b \log \left (c x^n\right )\right )-\frac {e^2 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^2 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^2} \]
b*k*n*x-1/2*k*x*(a+b*ln(c*x^n))+b*e^2*k*n*ln(e+f*x^(1/2))/f^2-e^2*k*(a+b*l n(c*x^n))*ln(e+f*x^(1/2))/f^2+2*b*e^2*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2)) /f^2-b*n*x*ln(d*(e+f*x^(1/2))^k)+x*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)+2 *b*e^2*k*n*polylog(2,1+f*x^(1/2)/e)/f^2-3*b*e*k*n*x^(1/2)/f+e*k*(a+b*ln(c* x^n))*x^(1/2)/f
Time = 0.18 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.04 \[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {a e k \sqrt {x}}{f}-\frac {3 b e k n \sqrt {x}}{f}-\frac {a k x}{2}+b k n x+a x \log \left (d \left (e+f \sqrt {x}\right )^k\right )-b n x \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {b e^2 k n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)}{f^2}+\frac {b e k \sqrt {x} \log \left (c x^n\right )}{f}-\frac {1}{2} b k x \log \left (c x^n\right )+b x \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )-\frac {e^2 k \log \left (e+f \sqrt {x}\right ) \left (a-b n-b n \log (x)+b \log \left (c x^n\right )\right )}{f^2}-\frac {2 b e^2 k n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^2} \]
(a*e*k*Sqrt[x])/f - (3*b*e*k*n*Sqrt[x])/f - (a*k*x)/2 + b*k*n*x + a*x*Log[ d*(e + f*Sqrt[x])^k] - b*n*x*Log[d*(e + f*Sqrt[x])^k] - (b*e^2*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x])/f^2 + (b*e*k*Sqrt[x]*Log[c*x^n])/f - (b*k*x*Log[c* x^n])/2 + b*x*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] - (e^2*k*Log[e + f*Sqrt[ x]]*(a - b*n - b*n*Log[x] + b*Log[c*x^n]))/f^2 - (2*b*e^2*k*n*PolyLog[2, - ((f*Sqrt[x])/e)])/f^2
Time = 0.39 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2817, 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.
| \(\displaystyle \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \, dx\) |
| \(\Big \downarrow \) 2817 |
| \(\displaystyle -b n \int \left (-\frac {k \log \left (e+f \sqrt {x}\right ) e^2}{f^2 x}+\frac {k e}{f \sqrt {x}}-\frac {k}{2}+\log \left (d \left (e+f \sqrt {x}\right )^k\right )\right )dx+x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^2 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {e k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {1}{2} k x \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {e^2 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {e k \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {1}{2} k x \left (a+b \log \left (c x^n\right )\right )-b n \left (x \log \left (d \left (e+f \sqrt {x}\right )^k\right )-\frac {2 e^2 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{f^2}-\frac {e^2 k \log \left (e+f \sqrt {x}\right )}{f^2}-\frac {2 e^2 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {3 e k \sqrt {x}}{f}-k x\right )\) |
(e*k*Sqrt[x]*(a + b*Log[c*x^n]))/f - (k*x*(a + b*Log[c*x^n]))/2 - (e^2*k*L og[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + x*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]) - b*n*((3*e*k*Sqrt[x])/f - k*x - (e^2*k*Log[e + f*Sqrt[x]] )/f^2 + x*Log[d*(e + f*Sqrt[x])^k] - (2*e^2*k*Log[e + f*Sqrt[x]]*Log[-((f* Sqrt[x])/e)])/f^2 - (2*e^2*k*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2)
3.2.17.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], 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, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))
\[\int \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )d x\]
\[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \]
\[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
1/9*(9*b*e*x*log(d)*log(x^n) + 9*(a*e*log(d) - (e*n*log(d) - e*log(c)*log( d))*b)*x + 9*(b*e*x*log(x^n) - ((e*n - e*log(c))*b - a*e)*x)*log((f*sqrt(x ) + e)^k) - (3*b*f*k*x^2*log(x^n) + (3*a*f*k - (5*f*k*n - 3*f*k*log(c))*b) *x^2)/sqrt(x))/e + integrate(1/2*(b*f^2*k*x*log(x^n) + (a*f^2*k - (f^2*k*n - f^2*k*log(c))*b)*x)/(e*f*sqrt(x) + e^2), x)
\[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int \ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \]