Integrand size = 27, antiderivative size = 374 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {14 b^2 n^2 \sqrt {x}}{d f}+a b n x-3 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-\frac {2 b^2 n^2 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}+b^2 n x \log \left (c x^n\right )-\frac {6 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2} \]
a*b*n*x-3*b^2*n^2*x+b^2*n*x*ln(c*x^n)+b*n*x*(a+b*ln(c*x^n))-1/2*x*(a+b*ln( c*x^n))^2+2*b^2*n^2*x*ln(d*(1/d+f*x^(1/2)))-2*b*n*x*(a+b*ln(c*x^n))*ln(d*( 1/d+f*x^(1/2)))+x*(a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^(1/2)))-2*b^2*n^2*ln(1+d *f*x^(1/2))/d^2/f^2+2*b*n*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/d^2/f^2-(a+b*l n(c*x^n))^2*ln(1+d*f*x^(1/2))/d^2/f^2+4*b^2*n^2*polylog(2,-d*f*x^(1/2))/d^ 2/f^2-4*b*n*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))/d^2/f^2+8*b^2*n^2*poly log(3,-d*f*x^(1/2))/d^2/f^2+14*b^2*n^2*x^(1/2)/d/f-6*b*n*(a+b*ln(c*x^n))*x ^(1/2)/d/f+(a+b*ln(c*x^n))^2*x^(1/2)/d/f
Time = 0.22 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.41 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {-2 a^2 d f \sqrt {x}+12 a b d f n \sqrt {x}-28 b^2 d f n^2 \sqrt {x}+a^2 d^2 f^2 x-4 a b d^2 f^2 n x+6 b^2 d^2 f^2 n^2 x+2 a^2 \log \left (1+d f \sqrt {x}\right )-4 a b n \log \left (1+d f \sqrt {x}\right )+4 b^2 n^2 \log \left (1+d f \sqrt {x}\right )-2 a^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right )+4 a b d^2 f^2 n x \log \left (1+d f \sqrt {x}\right )-4 b^2 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right )-4 a b d f \sqrt {x} \log \left (c x^n\right )+12 b^2 d f n \sqrt {x} \log \left (c x^n\right )+2 a b d^2 f^2 x \log \left (c x^n\right )-4 b^2 d^2 f^2 n x \log \left (c x^n\right )+4 a b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-4 b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-4 a b d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+4 b^2 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-2 b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+b^2 d^2 f^2 x \log ^2\left (c x^n\right )+2 b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-2 b^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+8 b n \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-16 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{2 d^2 f^2} \]
-1/2*(-2*a^2*d*f*Sqrt[x] + 12*a*b*d*f*n*Sqrt[x] - 28*b^2*d*f*n^2*Sqrt[x] + a^2*d^2*f^2*x - 4*a*b*d^2*f^2*n*x + 6*b^2*d^2*f^2*n^2*x + 2*a^2*Log[1 + d *f*Sqrt[x]] - 4*a*b*n*Log[1 + d*f*Sqrt[x]] + 4*b^2*n^2*Log[1 + d*f*Sqrt[x] ] - 2*a^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] + 4*a*b*d^2*f^2*n*x*Log[1 + d*f*S qrt[x]] - 4*b^2*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt[x]] - 4*a*b*d*f*Sqrt[x]*Log [c*x^n] + 12*b^2*d*f*n*Sqrt[x]*Log[c*x^n] + 2*a*b*d^2*f^2*x*Log[c*x^n] - 4 *b^2*d^2*f^2*n*x*Log[c*x^n] + 4*a*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 4*b^ 2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 4*a*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] *Log[c*x^n] + 4*b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 2*b^2*d* f*Sqrt[x]*Log[c*x^n]^2 + b^2*d^2*f^2*x*Log[c*x^n]^2 + 2*b^2*Log[1 + d*f*Sq rt[x]]*Log[c*x^n]^2 - 2*b^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 8*b*n*(a - b*n + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 16*b^2*n^2*Pol yLog[3, -(d*f*Sqrt[x])])/(d^2*f^2)
Time = 0.55 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2817 |
| \(\displaystyle -2 b n \int \left (\frac {1}{2} \left (-a-b \log \left (c x^n\right )\right )+\log \left (d \left (\sqrt {x} f+\frac {1}{d}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2 x}+\frac {a+b \log \left (c x^n\right )}{d f \sqrt {x}}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )-\frac {a x}{2}-\frac {1}{2} b x \log \left (c x^n\right )-\frac {2 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {4 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {b n \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}-\frac {7 b n \sqrt {x}}{d f}-b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {3 b n x}{2}\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\) |
(Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - (x*(a + b*Log[c*x^n])^2)/2 + x*Log[ d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) - 2*b*n*((-7*b*n*Sqrt[x])/(d*f) - (a*x)/2 + (3* b*n*x)/2 - b*n*x*Log[d*(d^(-1) + f*Sqrt[x])] + (b*n*Log[1 + d*f*Sqrt[x]])/ (d^2*f^2) - (b*x*Log[c*x^n])/2 + (3*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) - (x *(a + b*Log[c*x^n]))/2 + x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(d^2*f^2) - (2*b*n*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x ])])/(d^2*f^2) - (4*b*n*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2))
3.1.55.3.1 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]))
\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )d x\]
\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
(b^2*x*log(x^n)^2 - 2*(b^2*(n - log(c)) - a*b)*x*log(x^n) + ((2*n^2 - 2*n* log(c) + log(c)^2)*b^2 - 2*a*b*(n - log(c)) + a^2)*x)*log(d*f*sqrt(x) + 1) - 1/27*(9*b^2*d*f*x^2*log(x^n)^2 + 6*(3*a*b*d*f - (5*d*f*n - 3*d*f*log(c) )*b^2)*x^2*log(x^n) + (9*a^2*d*f - 6*(5*d*f*n - 3*d*f*log(c))*a*b + (38*d* f*n^2 - 30*d*f*n*log(c) + 9*d*f*log(c)^2)*b^2)*x^2)/sqrt(x) + integrate(1/ 2*(b^2*d^2*f^2*x*log(x^n)^2 + 2*(a*b*d^2*f^2 - (d^2*f^2*n - d^2*f^2*log(c) )*b^2)*x*log(x^n) + (a^2*d^2*f^2 - 2*(d^2*f^2*n - d^2*f^2*log(c))*a*b + (2 *d^2*f^2*n^2 - 2*d^2*f^2*n*log(c) + d^2*f^2*log(c)^2)*b^2)*x)/(d*f*sqrt(x) + 1), x)
\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]