Integrand size = 25, antiderivative size = 405 \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {14 b^2 e n^2 \sqrt {x}}{f}+a b n x-3 b^2 n^2 x-\frac {2 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right )}{f^2}+2 b^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {4 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}+b^2 n x \log \left (c x^n\right )-\frac {6 b e n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}+b n x \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-2 b n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac {4 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {4 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {8 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^2} \]
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*e^2*n^2*ln(e+f*x^(1/2))/f^2+2*b*e^2*n*(a+b*ln(c*x^n))*ln(e +f*x^(1/2))/f^2-4*b^2*e^2*n^2*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f^2+2*b^2*n ^2*x*ln(d*(e+f*x^(1/2)))-2*b*n*x*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2)))+x*(a+ b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))-e^2*(a+b*ln(c*x^n))^2*ln(1+f*x^(1/2)/e) /f^2-4*b*e^2*n*(a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/f^2-4*b^2*e^2*n^2*p olylog(2,1+f*x^(1/2)/e)/f^2+8*b^2*e^2*n^2*polylog(3,-f*x^(1/2)/e)/f^2+14*b ^2*e*n^2*x^(1/2)/f-6*b*e*n*(a+b*ln(c*x^n))*x^(1/2)/f+e*(a+b*ln(c*x^n))^2*x ^(1/2)/f
Time = 0.26 (sec) , antiderivative size = 718, normalized size of antiderivative = 1.77 \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=-\frac {-2 a^2 e f \sqrt {x}+12 a b e f n \sqrt {x}-28 b^2 e f n^2 \sqrt {x}+a^2 f^2 x-4 a b f^2 n x+6 b^2 f^2 n^2 x+2 a^2 e^2 \log \left (e+f \sqrt {x}\right )-4 a b e^2 n \log \left (e+f \sqrt {x}\right )+4 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right )-2 a^2 f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right )+4 a b f^2 n x \log \left (d \left (e+f \sqrt {x}\right )\right )-4 b^2 f^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right )-4 a b e^2 n \log \left (e+f \sqrt {x}\right ) \log (x)+4 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log (x)+4 a b e^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-4 b^2 e^2 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+2 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x)-2 b^2 e^2 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)-4 a b e f \sqrt {x} \log \left (c x^n\right )+12 b^2 e f n \sqrt {x} \log \left (c x^n\right )+2 a b f^2 x \log \left (c x^n\right )-4 b^2 f^2 n x \log \left (c x^n\right )+4 a b e^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-4 b^2 e^2 n \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-4 a b f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+4 b^2 f^2 n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )-4 b^2 e^2 n \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )+4 b^2 e^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )-2 b^2 e f \sqrt {x} \log ^2\left (c x^n\right )+b^2 f^2 x \log ^2\left (c x^n\right )+2 b^2 e^2 \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )-2 b^2 f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+8 b e^2 n \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )-16 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{2 f^2} \]
-1/2*(-2*a^2*e*f*Sqrt[x] + 12*a*b*e*f*n*Sqrt[x] - 28*b^2*e*f*n^2*Sqrt[x] + a^2*f^2*x - 4*a*b*f^2*n*x + 6*b^2*f^2*n^2*x + 2*a^2*e^2*Log[e + f*Sqrt[x] ] - 4*a*b*e^2*n*Log[e + f*Sqrt[x]] + 4*b^2*e^2*n^2*Log[e + f*Sqrt[x]] - 2* a^2*f^2*x*Log[d*(e + f*Sqrt[x])] + 4*a*b*f^2*n*x*Log[d*(e + f*Sqrt[x])] - 4*b^2*f^2*n^2*x*Log[d*(e + f*Sqrt[x])] - 4*a*b*e^2*n*Log[e + f*Sqrt[x]]*Lo g[x] + 4*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x] + 4*a*b*e^2*n*Log[1 + (f*Sq rt[x])/e]*Log[x] - 4*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] + 2*b^2*e^2 *n^2*Log[e + f*Sqrt[x]]*Log[x]^2 - 2*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Lo g[x]^2 - 4*a*b*e*f*Sqrt[x]*Log[c*x^n] + 12*b^2*e*f*n*Sqrt[x]*Log[c*x^n] + 2*a*b*f^2*x*Log[c*x^n] - 4*b^2*f^2*n*x*Log[c*x^n] + 4*a*b*e^2*Log[e + f*Sq rt[x]]*Log[c*x^n] - 4*b^2*e^2*n*Log[e + f*Sqrt[x]]*Log[c*x^n] - 4*a*b*f^2* x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 4*b^2*f^2*n*x*Log[d*(e + f*Sqrt[x])] *Log[c*x^n] - 4*b^2*e^2*n*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] + 4*b^2*e^2 *n*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] - 2*b^2*e*f*Sqrt[x]*Log[c*x^n] ^2 + b^2*f^2*x*Log[c*x^n]^2 + 2*b^2*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n]^2 - 2*b^2*f^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + 8*b*e^2*n*(a - b*n + b*L og[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)] - 16*b^2*e^2*n^2*PolyLog[3, -((f*S qrt[x])/e)])/f^2
Time = 0.73 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.12, 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.
| \(\displaystyle \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2817 |
| \(\displaystyle -2 b n \int \left (-\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) e^2}{f^2 x}+\frac {\left (a+b \log \left (c x^n\right )\right ) e}{f \sqrt {x}}+\frac {1}{2} \left (-a-b \log \left (c x^n\right )\right )+\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right )dx+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b f^2 n}+\frac {e^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b f^2 n}-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {3 e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )-\frac {a x}{2}-\frac {1}{2} b x \log \left (c x^n\right )-b n x \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {2 b e^2 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{f^2}-\frac {4 b e^2 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {b e^2 n \log \left (e+f \sqrt {x}\right )}{f^2}+\frac {2 b e^2 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {7 b e n \sqrt {x}}{f}+\frac {3 b n x}{2}\right )+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^2\) |
(e*Sqrt[x]*(a + b*Log[c*x^n])^2)/f - (x*(a + b*Log[c*x^n])^2)/2 - (e^2*Log [e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/f^2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - 2*b*n*((-7*b*e*n*Sqrt[x])/f - (a*x)/2 + (3*b*n*x)/2 + (b *e^2*n*Log[e + f*Sqrt[x]])/f^2 - b*n*x*Log[d*(e + f*Sqrt[x])] + (2*b*e^2*n *Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 - (b*x*Log[c*x^n])/2 + (3*e *Sqrt[x]*(a + b*Log[c*x^n]))/f - (x*(a + b*Log[c*x^n]))/2 - (e^2*Log[e + f *Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c* x^n]) - (e^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*b*f^2*n) + (e^2*L og[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(2*b*f^2*n) + (2*b*e^2*n*PolyL og[2, 1 + (f*Sqrt[x])/e])/f^2 + (2*e^2*(a + b*Log[c*x^n])*PolyLog[2, -((f* Sqrt[x])/e)])/f^2 - (4*b*e^2*n*PolyLog[3, -((f*Sqrt[x])/e)])/f^2)
3.2.24.3.1 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]))
\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (e +f \sqrt {x}\right )\right )d x\]
\[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
1/27*(27*b^2*e*x*log(d)*log(x^n)^2 + 54*(a*b*e*log(d) - (e*n*log(d) - e*lo g(c)*log(d))*b^2)*x*log(x^n) + 27*(a^2*e*log(d) - 2*(e*n*log(d) - e*log(c) *log(d))*a*b + (2*e*n^2*log(d) - 2*e*n*log(c)*log(d) + e*log(c)^2*log(d))* b^2)*x + 27*(b^2*e*x*log(x^n)^2 - 2*((e*n - e*log(c))*b^2 - a*b*e)*x*log(x ^n) - (2*(e*n - e*log(c))*a*b - (2*e*n^2 - 2*e*n*log(c) + e*log(c)^2)*b^2 - a^2*e)*x)*log(f*sqrt(x) + e) - (9*b^2*f*x^2*log(x^n)^2 - 6*((5*f*n - 3*f *log(c))*b^2 - 3*a*b*f)*x^2*log(x^n) - (6*(5*f*n - 3*f*log(c))*a*b - (38*f *n^2 - 30*f*n*log(c) + 9*f*log(c)^2)*b^2 - 9*a^2*f)*x^2)/sqrt(x))/e + inte grate(1/2*(b^2*f^2*x*log(x^n)^2 + 2*(a*b*f^2 - (f^2*n - f^2*log(c))*b^2)*x *log(x^n) + (a^2*f^2 - 2*(f^2*n - f^2*log(c))*a*b + (2*f^2*n^2 - 2*f^2*n*l og(c) + f^2*log(c)^2)*b^2)*x)/(e*f*sqrt(x) + e^2), x)
\[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int \ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]