Integrand size = 28, antiderivative size = 441 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {14 b^2 f n^2}{e \sqrt {x}}+\frac {2 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right )}{e^2}-\frac {2 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}-\frac {4 b^2 f^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b^2 f^2 n^2 \log (x)}{e^2}+\frac {b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac {6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {2 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}-\frac {4 b^2 f^2 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {8 b^2 f^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2} \]
-b^2*f^2*n^2*ln(x)/e^2+1/2*b^2*f^2*n^2*ln(x)^2/e^2-b*f^2*n*ln(x)*(a+b*ln(c *x^n))/e^2-1/6*f^2*(a+b*ln(c*x^n))^3/b/e^2/n+2*b^2*f^2*n^2*ln(e+f*x^(1/2)) /e^2+2*b*f^2*n*(a+b*ln(c*x^n))*ln(e+f*x^(1/2))/e^2-4*b^2*f^2*n^2*ln(-f*x^( 1/2)/e)*ln(e+f*x^(1/2))/e^2-2*b^2*n^2*ln(d*(e+f*x^(1/2)))/x-2*b*n*(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)))/x+f^2* (a+b*ln(c*x^n))^2*ln(1+f*x^(1/2)/e)/e^2+4*b*f^2*n*(a+b*ln(c*x^n))*polylog( 2,-f*x^(1/2)/e)/e^2-4*b^2*f^2*n^2*polylog(2,1+f*x^(1/2)/e)/e^2-8*b^2*f^2*n ^2*polylog(3,-f*x^(1/2)/e)/e^2-14*b^2*f*n^2/e/x^(1/2)-6*b*f*n*(a+b*ln(c*x^ n))/e/x^(1/2)-f*(a+b*ln(c*x^n))^2/e/x^(1/2)
Time = 0.34 (sec) , antiderivative size = 821, normalized size of antiderivative = 1.86 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=-\frac {3 a^2 e f \sqrt {x}+18 a b e f n \sqrt {x}+42 b^2 e f n^2 \sqrt {x}-3 a^2 f^2 x \log \left (e+f \sqrt {x}\right )-6 a b f^2 n x \log \left (e+f \sqrt {x}\right )-6 b^2 f^2 n^2 x \log \left (e+f \sqrt {x}\right )+3 a^2 e^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+6 a b e^2 n \log \left (d \left (e+f \sqrt {x}\right )\right )+6 b^2 e^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3}{2} a^2 f^2 x \log (x)+3 a b f^2 n x \log (x)+3 b^2 f^2 n^2 x \log (x)+6 a b f^2 n x \log \left (e+f \sqrt {x}\right ) \log (x)+6 b^2 f^2 n^2 x \log \left (e+f \sqrt {x}\right ) \log (x)-6 a b f^2 n x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-6 b^2 f^2 n^2 x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-\frac {3}{2} a b f^2 n x \log ^2(x)-\frac {3}{2} b^2 f^2 n^2 x \log ^2(x)-3 b^2 f^2 n^2 x \log \left (e+f \sqrt {x}\right ) \log ^2(x)+3 b^2 f^2 n^2 x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+\frac {1}{2} b^2 f^2 n^2 x \log ^3(x)+6 a b e f \sqrt {x} \log \left (c x^n\right )+18 b^2 e f n \sqrt {x} \log \left (c x^n\right )-6 a b f^2 x \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-6 b^2 f^2 n x \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+6 a b e^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+6 b^2 e^2 n \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+3 a b f^2 x \log (x) \log \left (c x^n\right )+3 b^2 f^2 n x \log (x) \log \left (c x^n\right )+6 b^2 f^2 n x \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-6 b^2 f^2 n x \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )-\frac {3}{2} b^2 f^2 n x \log ^2(x) \log \left (c x^n\right )+3 b^2 e f \sqrt {x} \log ^2\left (c x^n\right )-3 b^2 f^2 x \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+3 b^2 e^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+\frac {3}{2} b^2 f^2 x \log (x) \log ^2\left (c x^n\right )-12 b f^2 n x \left (a+b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )+24 b^2 f^2 n^2 x \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{3 e^2 x} \]
-1/3*(3*a^2*e*f*Sqrt[x] + 18*a*b*e*f*n*Sqrt[x] + 42*b^2*e*f*n^2*Sqrt[x] - 3*a^2*f^2*x*Log[e + f*Sqrt[x]] - 6*a*b*f^2*n*x*Log[e + f*Sqrt[x]] - 6*b^2* f^2*n^2*x*Log[e + f*Sqrt[x]] + 3*a^2*e^2*Log[d*(e + f*Sqrt[x])] + 6*a*b*e^ 2*n*Log[d*(e + f*Sqrt[x])] + 6*b^2*e^2*n^2*Log[d*(e + f*Sqrt[x])] + (3*a^2 *f^2*x*Log[x])/2 + 3*a*b*f^2*n*x*Log[x] + 3*b^2*f^2*n^2*x*Log[x] + 6*a*b*f ^2*n*x*Log[e + f*Sqrt[x]]*Log[x] + 6*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]]*Log[ x] - 6*a*b*f^2*n*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - 6*b^2*f^2*n^2*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - (3*a*b*f^2*n*x*Log[x]^2)/2 - (3*b^2*f^2*n^2*x*Log [x]^2)/2 - 3*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]]*Log[x]^2 + 3*b^2*f^2*n^2*x*L og[1 + (f*Sqrt[x])/e]*Log[x]^2 + (b^2*f^2*n^2*x*Log[x]^3)/2 + 6*a*b*e*f*Sq rt[x]*Log[c*x^n] + 18*b^2*e*f*n*Sqrt[x]*Log[c*x^n] - 6*a*b*f^2*x*Log[e + f *Sqrt[x]]*Log[c*x^n] - 6*b^2*f^2*n*x*Log[e + f*Sqrt[x]]*Log[c*x^n] + 6*a*b *e^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 6*b^2*e^2*n*Log[d*(e + f*Sqrt[x]) ]*Log[c*x^n] + 3*a*b*f^2*x*Log[x]*Log[c*x^n] + 3*b^2*f^2*n*x*Log[x]*Log[c* x^n] + 6*b^2*f^2*n*x*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] - 6*b^2*f^2*n*x* Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] - (3*b^2*f^2*n*x*Log[x]^2*Log[c*x ^n])/2 + 3*b^2*e*f*Sqrt[x]*Log[c*x^n]^2 - 3*b^2*f^2*x*Log[e + f*Sqrt[x]]*L og[c*x^n]^2 + 3*b^2*e^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + (3*b^2*f^2*x *Log[x]*Log[c*x^n]^2)/2 - 12*b*f^2*n*x*(a + b*n + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)] + 24*b^2*f^2*n^2*x*PolyLog[3, -((f*Sqrt[x])/e)])/(e^...
Time = 0.90 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.22, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2824, 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.
| \(\displaystyle \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -2 b n \int \left (\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) f^2}{e^2 x}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right ) f^2}{2 e^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right ) f}{e x^{3/2}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}\right )dx-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{12 b^2 e^2 n^2}+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 f^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}-\frac {f^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b e^2 n}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 b e^2 n}-\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2}+\frac {3 f \left (a+b \log \left (c x^n\right )\right )}{e \sqrt {x}}+\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right )}{x}+\frac {2 b f^2 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{e^2}+\frac {4 b f^2 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^2}-\frac {b f^2 n \log ^2(x)}{4 e^2}-\frac {b f^2 n \log \left (e+f \sqrt {x}\right )}{e^2}+\frac {2 b f^2 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^2}+\frac {b f^2 n \log (x)}{2 e^2}+\frac {7 b f n}{e \sqrt {x}}\right )-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {f^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac {f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt {x}}\) |
-((f*(a + b*Log[c*x^n])^2)/(e*Sqrt[x])) + (f^2*Log[e + f*Sqrt[x]]*(a + b*L og[c*x^n])^2)/e^2 - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x - (f^2 *Log[x]*(a + b*Log[c*x^n])^2)/(2*e^2) - 2*b*n*((7*b*f*n)/(e*Sqrt[x]) - (b* f^2*n*Log[e + f*Sqrt[x]])/e^2 + (b*n*Log[d*(e + f*Sqrt[x])])/x + (2*b*f^2* n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 + (b*f^2*n*Log[x])/(2*e^2) - (b*f^2*n*Log[x]^2)/(4*e^2) + (3*f*(a + b*Log[c*x^n]))/(e*Sqrt[x]) - (f^ 2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2 + (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x + (f^2*Log[x]*(a + b*Log[c*x^n]))/(2*e^2) + (f^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*b*e^2*n) - (f^2*Log[1 + (f*Sqrt[x])/ e]*(a + b*Log[c*x^n])^2)/(2*b*e^2*n) - (f^2*Log[x]*(a + b*Log[c*x^n])^2)/( 4*b*e^2*n) + (f^2*(a + b*Log[c*x^n])^3)/(12*b^2*e^2*n^2) + (2*b*f^2*n*Poly Log[2, 1 + (f*Sqrt[x])/e])/e^2 - (2*f^2*(a + b*Log[c*x^n])*PolyLog[2, -((f *Sqrt[x])/e)])/e^2 + (4*b*f^2*n*PolyLog[3, -((f*Sqrt[x])/e)])/e^2)
3.2.26.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_ .))^(p_.)*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)], 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, g, m, n, q}, x] && IGtQ[p, 0] && RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ [p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q, 0] && IntegerQ[ (q + 1)/m] && EqQ[d*e, 1]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x^{2}}d x\]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}} \,d x } \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx=\int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^2} \,d x \]