Integrand size = 30, antiderivative size = 310 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=-\frac {5 b f k n}{9 e x}+\frac {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right )}{9 e^3}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {2 b f^3 k n \log (x)}{9 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {4 b f^3 k n \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{3 e^3} \]
-5/9*b*f*k*n/e/x+2/9*b*f^3*k*n*ln(x)/e^3-1/6*b*f^3*k*n*ln(x)^2/e^3-1/3*f*k *(a+b*ln(c*x^n))/e/x+1/3*f^3*k*ln(x)*(a+b*ln(c*x^n))/e^3-4/9*b*f^3*k*n*ln( e+f*x^(1/2))/e^3-2/3*f^3*k*(a+b*ln(c*x^n))*ln(e+f*x^(1/2))/e^3+4/3*b*f^3*k *n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/e^3-4/9*b*n*ln(d*(e+f*x^(1/2))^k)/x^(3 /2)-2/3*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^(3/2)+4/3*b*f^3*k*n*polylo g(2,1+f*x^(1/2)/e)/e^3+16/9*b*f^2*k*n/e^2/x^(1/2)+2/3*f^2*k*(a+b*ln(c*x^n) )/e^2/x^(1/2)
Time = 0.27 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.05 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\frac {-3 a e^2 f k \sqrt {x}-5 b e^2 f k n \sqrt {x}+6 a e f^2 k x+16 b e f^2 k n x-6 a e^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right )-4 b e^3 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+3 a f^3 k x^{3/2} \log (x)+2 b f^3 k n x^{3/2} \log (x)-6 b f^3 k n x^{3/2} \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-\frac {3}{2} b f^3 k n x^{3/2} \log ^2(x)-3 b e^2 f k \sqrt {x} \log \left (c x^n\right )+6 b e f^2 k x \log \left (c x^n\right )-6 b e^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+3 b f^3 k x^{3/2} \log (x) \log \left (c x^n\right )-2 f^3 k x^{3/2} \log \left (e+f \sqrt {x}\right ) \left (3 a+2 b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )-12 b f^3 k n x^{3/2} \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{9 e^3 x^{3/2}} \]
(-3*a*e^2*f*k*Sqrt[x] - 5*b*e^2*f*k*n*Sqrt[x] + 6*a*e*f^2*k*x + 16*b*e*f^2 *k*n*x - 6*a*e^3*Log[d*(e + f*Sqrt[x])^k] - 4*b*e^3*n*Log[d*(e + f*Sqrt[x] )^k] + 3*a*f^3*k*x^(3/2)*Log[x] + 2*b*f^3*k*n*x^(3/2)*Log[x] - 6*b*f^3*k*n *x^(3/2)*Log[1 + (f*Sqrt[x])/e]*Log[x] - (3*b*f^3*k*n*x^(3/2)*Log[x]^2)/2 - 3*b*e^2*f*k*Sqrt[x]*Log[c*x^n] + 6*b*e*f^2*k*x*Log[c*x^n] - 6*b*e^3*Log[ d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 3*b*f^3*k*x^(3/2)*Log[x]*Log[c*x^n] - 2* f^3*k*x^(3/2)*Log[e + f*Sqrt[x]]*(3*a + 2*b*n - 3*b*n*Log[x] + 3*b*Log[c*x ^n]) - 12*b*f^3*k*n*x^(3/2)*PolyLog[2, -((f*Sqrt[x])/e)])/(9*e^3*x^(3/2))
Time = 0.50 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{x^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (-\frac {2 k \log \left (e+f \sqrt {x}\right ) f^3}{3 e^3 x}+\frac {k \log (x) f^3}{3 e^3 x}+\frac {2 k f^2}{3 e^2 x^{3/2}}-\frac {k f}{3 e x^2}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{5/2}}\right )dx-\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{3 x^{3/2}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}-b n \left (\frac {4 \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}-\frac {4 f^3 k \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{3 e^3}+\frac {f^3 k \log ^2(x)}{6 e^3}+\frac {4 f^3 k \log \left (e+f \sqrt {x}\right )}{9 e^3}-\frac {4 f^3 k \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}-\frac {2 f^3 k \log (x)}{9 e^3}-\frac {16 f^2 k}{9 e^2 \sqrt {x}}+\frac {5 f k}{9 e x}\right )\) |
-1/3*(f*k*(a + b*Log[c*x^n]))/(e*x) + (2*f^2*k*(a + b*Log[c*x^n]))/(3*e^2* Sqrt[x]) - (2*f^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^3) - (2*Lo g[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/(3*x^(3/2)) + (f^3*k*Log[x]*(a + b*Log[c*x^n]))/(3*e^3) - b*n*((5*f*k)/(9*e*x) - (16*f^2*k)/(9*e^2*Sqrt[x ]) + (4*f^3*k*Log[e + f*Sqrt[x]])/(9*e^3) + (4*Log[d*(e + f*Sqrt[x])^k])/( 9*x^(3/2)) - (4*f^3*k*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^3) - (2*f^3*k*Log[x])/(9*e^3) + (f^3*k*Log[x]^2)/(6*e^3) - (4*f^3*k*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*e^3))
3.2.36.3.1 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]
\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )}{x^{\frac {5}{2}}}d x\]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}} \,d x } \]
1/9*integrate((3*b*f*k*x*log(x^n) + (3*a*f*k + (2*f*k*n + 3*f*k*log(c))*b) *x)/x^3, x)/e + 1/9*integrate((3*b*f^3*k*x*log(x^n) + (3*a*f^3*k + (2*f^3* k*n + 3*f^3*k*log(c))*b)*x)/x^2, x)/e^3 - 1/9*(2*(b*f^6*k*x^2*log(x^n) + ( b*f^6*k*log(c) + a*f^6*k)*x^2)/sqrt(x) - (3*b*e*f^5*k*x^2*log(x^n) + (3*a* e*f^5*k - (e*f^5*k*n - 3*e*f^5*k*log(c))*b)*x^2)/x + 2*(3*b*e^2*f^4*k*x^2* log(x^n) + (3*a*e^2*f^4*k - (4*e^2*f^4*k*n - 3*e^2*f^4*k*log(c))*b)*x^2)/x ^(3/2) + 2*(3*b*e^6*x*log(x^n) + (3*a*e^6 + (2*e^6*n + 3*e^6*log(c))*b)*x) *log((f*sqrt(x) + e)^k)/x^(5/2) - 2*((3*a*e^4*f^2*k + (8*e^4*f^2*k*n + 3*e ^4*f^2*k*log(c))*b)*x^2 - (3*a*e^6*log(d) + (2*e^6*n*log(d) + 3*e^6*log(c) *log(d))*b)*x + 3*(b*e^4*f^2*k*x^2 - b*e^6*x*log(d))*log(x^n))/x^(5/2))/e^ 6 + integrate(1/9*(3*b*f^7*k*x*log(x^n) + (3*a*f^7*k + (2*f^7*k*n + 3*f^7* k*log(c))*b)*x)/(e^6*f*sqrt(x) + e^7), x)
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + e\right )}^{k} d\right )}{x^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{x^{5/2}} \, dx=\int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{5/2}} \,d x \]