Integrand size = 28, antiderivative size = 608 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {37 b^2 f n^2}{108 e x^{3/2}}+\frac {7 b^2 f^2 n^2}{8 e^2 x}-\frac {21 b^2 f^3 n^2}{4 e^3 \sqrt {x}}+\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right )}{4 e^4}-\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{4 x^2}-\frac {b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b^2 f^4 n^2 \log (x)}{8 e^4}+\frac {b^2 f^4 n^2 \log ^2(x)}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )}{18 e x^{3/2}}+\frac {3 b f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 e^2 x}-\frac {5 b f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt {x}}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac {b f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b e^4 n}-\frac {b^2 f^4 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {2 b f^4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {4 b^2 f^4 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^4} \]
-37/108*b^2*f*n^2/e/x^(3/2)+7/8*b^2*f^2*n^2/e^2/x-1/8*b^2*f^4*n^2*ln(x)/e^ 4+1/8*b^2*f^4*n^2*ln(x)^2/e^4-7/18*b*f*n*(a+b*ln(c*x^n))/e/x^(3/2)+3/4*b*f ^2*n*(a+b*ln(c*x^n))/e^2/x-1/4*b*f^4*n*ln(x)*(a+b*ln(c*x^n))/e^4-1/6*f*(a+ b*ln(c*x^n))^2/e/x^(3/2)+1/4*f^2*(a+b*ln(c*x^n))^2/e^2/x-1/12*f^4*(a+b*ln( c*x^n))^3/b/e^4/n+1/4*b^2*f^4*n^2*ln(e+f*x^(1/2))/e^4+1/2*b*f^4*n*(a+b*ln( c*x^n))*ln(e+f*x^(1/2))/e^4-b^2*f^4*n^2*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/e ^4-1/4*b^2*n^2*ln(d*(e+f*x^(1/2)))/x^2-1/2*b*n*(a+b*ln(c*x^n))*ln(d*(e+f*x ^(1/2)))/x^2-1/2*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))/x^2+1/2*f^4*(a+b*ln (c*x^n))^2*ln(1+f*x^(1/2)/e)/e^4+2*b*f^4*n*(a+b*ln(c*x^n))*polylog(2,-f*x^ (1/2)/e)/e^4-b^2*f^4*n^2*polylog(2,1+f*x^(1/2)/e)/e^4-4*b^2*f^4*n^2*polylo g(3,-f*x^(1/2)/e)/e^4-21/4*b^2*f^3*n^2/e^3/x^(1/2)-5/2*b*f^3*n*(a+b*ln(c*x ^n))/e^3/x^(1/2)-1/2*f^3*(a+b*ln(c*x^n))^2/e^3/x^(1/2)
Time = 0.42 (sec) , antiderivative size = 1078, normalized size of antiderivative = 1.77 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=-\frac {36 a^2 e^3 f \sqrt {x}+84 a b e^3 f n \sqrt {x}+74 b^2 e^3 f n^2 \sqrt {x}-54 a^2 e^2 f^2 x-162 a b e^2 f^2 n x-189 b^2 e^2 f^2 n^2 x+108 a^2 e f^3 x^{3/2}+540 a b e f^3 n x^{3/2}+1134 b^2 e f^3 n^2 x^{3/2}-108 a^2 f^4 x^2 \log \left (e+f \sqrt {x}\right )-108 a b f^4 n x^2 \log \left (e+f \sqrt {x}\right )-54 b^2 f^4 n^2 x^2 \log \left (e+f \sqrt {x}\right )+108 a^2 e^4 \log \left (d \left (e+f \sqrt {x}\right )\right )+108 a b e^4 n \log \left (d \left (e+f \sqrt {x}\right )\right )+54 b^2 e^4 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+54 a^2 f^4 x^2 \log (x)+54 a b f^4 n x^2 \log (x)+27 b^2 f^4 n^2 x^2 \log (x)+216 a b f^4 n x^2 \log \left (e+f \sqrt {x}\right ) \log (x)+108 b^2 f^4 n^2 x^2 \log \left (e+f \sqrt {x}\right ) \log (x)-216 a b f^4 n x^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-108 b^2 f^4 n^2 x^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-54 a b f^4 n x^2 \log ^2(x)-27 b^2 f^4 n^2 x^2 \log ^2(x)-108 b^2 f^4 n^2 x^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x)+108 b^2 f^4 n^2 x^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+18 b^2 f^4 n^2 x^2 \log ^3(x)+72 a b e^3 f \sqrt {x} \log \left (c x^n\right )+84 b^2 e^3 f n \sqrt {x} \log \left (c x^n\right )-108 a b e^2 f^2 x \log \left (c x^n\right )-162 b^2 e^2 f^2 n x \log \left (c x^n\right )+216 a b e f^3 x^{3/2} \log \left (c x^n\right )+540 b^2 e f^3 n x^{3/2} \log \left (c x^n\right )-216 a b f^4 x^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-108 b^2 f^4 n x^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+216 a b e^4 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+108 b^2 e^4 n \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+108 a b f^4 x^2 \log (x) \log \left (c x^n\right )+54 b^2 f^4 n x^2 \log (x) \log \left (c x^n\right )+216 b^2 f^4 n x^2 \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )-216 b^2 f^4 n x^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )-54 b^2 f^4 n x^2 \log ^2(x) \log \left (c x^n\right )+36 b^2 e^3 f \sqrt {x} \log ^2\left (c x^n\right )-54 b^2 e^2 f^2 x \log ^2\left (c x^n\right )+108 b^2 e f^3 x^{3/2} \log ^2\left (c x^n\right )-108 b^2 f^4 x^2 \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )+108 b^2 e^4 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+54 b^2 f^4 x^2 \log (x) \log ^2\left (c x^n\right )-216 b f^4 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )+864 b^2 f^4 n^2 x^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{216 e^4 x^2} \]
-1/216*(36*a^2*e^3*f*Sqrt[x] + 84*a*b*e^3*f*n*Sqrt[x] + 74*b^2*e^3*f*n^2*S qrt[x] - 54*a^2*e^2*f^2*x - 162*a*b*e^2*f^2*n*x - 189*b^2*e^2*f^2*n^2*x + 108*a^2*e*f^3*x^(3/2) + 540*a*b*e*f^3*n*x^(3/2) + 1134*b^2*e*f^3*n^2*x^(3/ 2) - 108*a^2*f^4*x^2*Log[e + f*Sqrt[x]] - 108*a*b*f^4*n*x^2*Log[e + f*Sqrt [x]] - 54*b^2*f^4*n^2*x^2*Log[e + f*Sqrt[x]] + 108*a^2*e^4*Log[d*(e + f*Sq rt[x])] + 108*a*b*e^4*n*Log[d*(e + f*Sqrt[x])] + 54*b^2*e^4*n^2*Log[d*(e + f*Sqrt[x])] + 54*a^2*f^4*x^2*Log[x] + 54*a*b*f^4*n*x^2*Log[x] + 27*b^2*f^ 4*n^2*x^2*Log[x] + 216*a*b*f^4*n*x^2*Log[e + f*Sqrt[x]]*Log[x] + 108*b^2*f ^4*n^2*x^2*Log[e + f*Sqrt[x]]*Log[x] - 216*a*b*f^4*n*x^2*Log[1 + (f*Sqrt[x ])/e]*Log[x] - 108*b^2*f^4*n^2*x^2*Log[1 + (f*Sqrt[x])/e]*Log[x] - 54*a*b* f^4*n*x^2*Log[x]^2 - 27*b^2*f^4*n^2*x^2*Log[x]^2 - 108*b^2*f^4*n^2*x^2*Log [e + f*Sqrt[x]]*Log[x]^2 + 108*b^2*f^4*n^2*x^2*Log[1 + (f*Sqrt[x])/e]*Log[ x]^2 + 18*b^2*f^4*n^2*x^2*Log[x]^3 + 72*a*b*e^3*f*Sqrt[x]*Log[c*x^n] + 84* b^2*e^3*f*n*Sqrt[x]*Log[c*x^n] - 108*a*b*e^2*f^2*x*Log[c*x^n] - 162*b^2*e^ 2*f^2*n*x*Log[c*x^n] + 216*a*b*e*f^3*x^(3/2)*Log[c*x^n] + 540*b^2*e*f^3*n* x^(3/2)*Log[c*x^n] - 216*a*b*f^4*x^2*Log[e + f*Sqrt[x]]*Log[c*x^n] - 108*b ^2*f^4*n*x^2*Log[e + f*Sqrt[x]]*Log[c*x^n] + 216*a*b*e^4*Log[d*(e + f*Sqrt [x])]*Log[c*x^n] + 108*b^2*e^4*n*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 108*a *b*f^4*x^2*Log[x]*Log[c*x^n] + 54*b^2*f^4*n*x^2*Log[x]*Log[c*x^n] + 216*b^ 2*f^4*n*x^2*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] - 216*b^2*f^4*n*x^2*Lo...
Time = 1.07 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.14, 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.
| \(\displaystyle \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -2 b n \int \left (\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) f^4}{2 e^4 x}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right ) f^4}{4 e^4 x}-\frac {\left (a+b \log \left (c x^n\right )\right ) f^3}{2 e^3 x^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right ) f^2}{4 e^2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) f}{6 e x^{5/2}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}\right )dx-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{24 b^2 e^4 n^2}+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {f^4 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^4}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 b e^4 n}-\frac {f^4 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 b e^4 n}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{8 b e^4 n}-\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}+\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}+\frac {5 f^3 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt {x}}-\frac {3 f^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}+\frac {7 f \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right )}{8 x^2}+\frac {b f^4 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{2 e^4}+\frac {2 b f^4 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {b f^4 n \log ^2(x)}{16 e^4}-\frac {b f^4 n \log \left (e+f \sqrt {x}\right )}{8 e^4}+\frac {b f^4 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 e^4}+\frac {b f^4 n \log (x)}{16 e^4}+\frac {21 b f^3 n}{8 e^3 \sqrt {x}}-\frac {7 b f^2 n}{16 e^2 x}+\frac {37 b f n}{216 e x^{3/2}}\right )-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^4}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 e^3 \sqrt {x}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 e^2 x}-\frac {f \left (a+b \log \left (c x^n\right )\right )^2}{6 e x^{3/2}}\) |
-1/6*(f*(a + b*Log[c*x^n])^2)/(e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^2)/(4* e^2*x) - (f^3*(a + b*Log[c*x^n])^2)/(2*e^3*Sqrt[x]) + (f^4*Log[e + f*Sqrt[ x]]*(a + b*Log[c*x^n])^2)/(2*e^4) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x ^n])^2)/(2*x^2) - (f^4*Log[x]*(a + b*Log[c*x^n])^2)/(4*e^4) - 2*b*n*((37*b *f*n)/(216*e*x^(3/2)) - (7*b*f^2*n)/(16*e^2*x) + (21*b*f^3*n)/(8*e^3*Sqrt[ x]) - (b*f^4*n*Log[e + f*Sqrt[x]])/(8*e^4) + (b*n*Log[d*(e + f*Sqrt[x])])/ (8*x^2) + (b*f^4*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(2*e^4) + (b* f^4*n*Log[x])/(16*e^4) - (b*f^4*n*Log[x]^2)/(16*e^4) + (7*f*(a + b*Log[c*x ^n]))/(36*e*x^(3/2)) - (3*f^2*(a + b*Log[c*x^n]))/(8*e^2*x) + (5*f^3*(a + b*Log[c*x^n]))/(4*e^3*Sqrt[x]) - (f^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n] ))/(4*e^4) + (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) + (f^4*Lo g[x]*(a + b*Log[c*x^n]))/(8*e^4) + (f^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^ n])^2)/(4*b*e^4*n) - (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4* b*e^4*n) - (f^4*Log[x]*(a + b*Log[c*x^n])^2)/(8*b*e^4*n) + (f^4*(a + b*Log [c*x^n])^3)/(24*b^2*e^4*n^2) + (b*f^4*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2* e^4) - (f^4*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 + (2*b*f^ 4*n*PolyLog[3, -((f*Sqrt[x])/e)])/e^4)
3.2.27.3.1 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]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x^{3}}d x\]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx=\int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3} \,d x \]