Integrand size = 28, antiderivative size = 914 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {175 b^3 f n^3}{216 e x^{3/2}}+\frac {45 b^3 f^2 n^3}{16 e^2 x}-\frac {255 b^3 f^3 n^3}{8 e^3 \sqrt {x}}+\frac {3 b^3 f^4 n^3 \log \left (e+f \sqrt {x}\right )}{8 e^4}-\frac {3 b^3 n^3 \log \left (d \left (e+f \sqrt {x}\right )\right )}{8 x^2}-\frac {3 b^3 f^4 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 e^4}-\frac {3 b^3 f^4 n^3 \log (x)}{16 e^4}+\frac {3 b^3 f^4 n^3 \log ^2(x)}{16 e^4}-\frac {37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac {21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac {63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt {x}}+\frac {3 b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {3 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac {9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac {15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt {x}}-\frac {3 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac {3 b f^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3}{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 )^3}{2 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}-\frac {3 b^3 f^4 n^3 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{2 e^4}+\frac {3 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {6 b^3 f^4 n^3 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {24 b^3 f^4 n^3 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{e^4} \]
-37/36*b^2*f*n^2*(a+b*ln(c*x^n))/e/x^(3/2)+21/8*b^2*f^2*n^2*(a+b*ln(c*x^n) )/e^2/x-3/8*b^2*f^4*n^2*ln(x)*(a+b*ln(c*x^n))/e^4-7/12*b*f*n*(a+b*ln(c*x^n ))^2/e/x^(3/2)+9/8*b*f^2*n*(a+b*ln(c*x^n))^2/e^2/x+3/4*b^2*f^4*n^2*(a+b*ln (c*x^n))*ln(e+f*x^(1/2))/e^4-3/2*b^3*f^4*n^3*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/ 2))/e^4+3/4*b*f^4*n*(a+b*ln(c*x^n))^2*ln(1+f*x^(1/2)/e)/e^4+3*b^2*f^4*n^2* (a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/e^4+3*b*f^4*n*(a+b*ln(c*x^n))^2*po lylog(2,-f*x^(1/2)/e)/e^4-12*b^2*f^4*n^2*(a+b*ln(c*x^n))*polylog(3,-f*x^(1 /2)/e)/e^4-63/4*b^2*f^3*n^2*(a+b*ln(c*x^n))/e^3/x^(1/2)-15/4*b*f^3*n*(a+b* ln(c*x^n))^2/e^3/x^(1/2)-1/2*(a+b*ln(c*x^n))^3*ln(d*(e+f*x^(1/2)))/x^2-1/8 *f^4*(a+b*ln(c*x^n))^3/e^4-1/6*f*(a+b*ln(c*x^n))^3/e/x^(3/2)+1/4*f^2*(a+b* ln(c*x^n))^3/e^2/x-3/8*b^3*n^3*ln(d*(e+f*x^(1/2)))/x^2+1/2*f^4*(a+b*ln(c*x ^n))^3*ln(1+f*x^(1/2)/e)/e^4-1/2*f^3*(a+b*ln(c*x^n))^3/e^3/x^(1/2)-3/16*b^ 3*f^4*n^3*ln(x)/e^4+3/16*b^3*f^4*n^3*ln(x)^2/e^4-1/16*f^4*(a+b*ln(c*x^n))^ 4/b/e^4/n+3/8*b^3*f^4*n^3*ln(e+f*x^(1/2))/e^4-3/4*b^2*n^2*(a+b*ln(c*x^n))* ln(d*(e+f*x^(1/2)))/x^2-3/4*b*n*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))/x^2- 3/2*b^3*f^4*n^3*polylog(2,1+f*x^(1/2)/e)/e^4-6*b^3*f^4*n^3*polylog(3,-f*x^ (1/2)/e)/e^4+24*b^3*f^4*n^3*polylog(4,-f*x^(1/2)/e)/e^4-255/8*b^3*f^3*n^3/ e^3/x^(1/2)-175/216*b^3*f*n^3/e/x^(3/2)+45/16*b^3*f^2*n^3/e^2/x
Time = 1.02 (sec) , antiderivative size = 1549, normalized size of antiderivative = 1.69 \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \]
-1/432*(54*e^4*Log[d*(e + f*Sqrt[x])]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3 *b^3*n^3 + 6*b*(2*a^2 + 2*a*b*n + b^2*n^2)*Log[c*x^n] + 6*b^2*(2*a + b*n)* Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3) + 18*e^3*f*Sqrt[x]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2* n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a *b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 27*e^2*f^2*x*(4*a^3 + 6*a^2*b*n + 6 *a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n* (-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b ^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4 *b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 54*e*f^3*x^(3/2)*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2* n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a *b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 54*f^4*x^2*Log[e + f*Sqrt[x]]*(4*a^ 3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^ n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log [c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 27*f^4*x^2*Log[x]*(4 *a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Lo...
Time = 1.71 (sec) , antiderivative size = 1011, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -3 b n \int \left (\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{2 e^4 x}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{4 e^4 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 f^3}{2 e^3 x^{3/2}}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 f^2}{4 e^2 x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 f}{6 e x^{5/2}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^3}\right )dx-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {f^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac {f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{4 e^4}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt {x}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{2 e^4}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{4 e^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 e^3 \sqrt {x}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 e^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f}{6 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 )^3}{2 x^2}-3 b n \left (\frac {\left (a+b \log \left (c x^n\right )\right )^4 f^4}{48 b^2 e^4 n^2}+\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{6 b e^4 n}-\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{6 b e^4 n}-\frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{12 b e^4 n}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^4}{24 b e^4 n}-\frac {b^2 n^2 \log ^2(x) f^4}{16 e^4}-\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{4 e^4}-\frac {b^2 n^2 \log \left (e+f \sqrt {x}\right ) f^4}{8 e^4}+\frac {b^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right ) f^4}{2 e^4}+\frac {b^2 n^2 \log (x) f^4}{16 e^4}-\frac {b n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) f^4}{4 e^4}+\frac {b n \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4}{8 e^4}+\frac {b^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right ) f^4}{2 e^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {4 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {8 b^2 n^2 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {5 \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 e^3 \sqrt {x}}+\frac {21 b n \left (a+b \log \left (c x^n\right )\right ) f^3}{4 e^3 \sqrt {x}}+\frac {85 b^2 n^2 f^3}{8 e^3 \sqrt {x}}-\frac {3 \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 e^2 x}-\frac {7 b n \left (a+b \log \left (c x^n\right )\right ) f^2}{8 e^2 x}-\frac {15 b^2 n^2 f^2}{16 e^2 x}+\frac {7 \left (a+b \log \left (c x^n\right )\right )^2 f}{36 e x^{3/2}}+\frac {37 b n \left (a+b \log \left (c x^n\right )\right ) f}{108 e x^{3/2}}+\frac {175 b^2 n^2 f}{648 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 )^2}{4 x^2}+\frac {b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right )}{8 x^2}+\frac {b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}\right )\) |
-1/6*(f*(a + b*Log[c*x^n])^3)/(e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^3)/(4* e^2*x) - (f^3*(a + b*Log[c*x^n])^3)/(2*e^3*Sqrt[x]) + (f^4*Log[e + f*Sqrt[ x]]*(a + b*Log[c*x^n])^3)/(2*e^4) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x ^n])^3)/(2*x^2) - (f^4*Log[x]*(a + b*Log[c*x^n])^3)/(4*e^4) - 3*b*n*((175* b^2*f*n^2)/(648*e*x^(3/2)) - (15*b^2*f^2*n^2)/(16*e^2*x) + (85*b^2*f^3*n^2 )/(8*e^3*Sqrt[x]) - (b^2*f^4*n^2*Log[e + f*Sqrt[x]])/(8*e^4) + (b^2*n^2*Lo g[d*(e + f*Sqrt[x])])/(8*x^2) + (b^2*f^4*n^2*Log[e + f*Sqrt[x]]*Log[-((f*S qrt[x])/e)])/(2*e^4) + (b^2*f^4*n^2*Log[x])/(16*e^4) - (b^2*f^4*n^2*Log[x] ^2)/(16*e^4) + (37*b*f*n*(a + b*Log[c*x^n]))/(108*e*x^(3/2)) - (7*b*f^2*n* (a + b*Log[c*x^n]))/(8*e^2*x) + (21*b*f^3*n*(a + b*Log[c*x^n]))/(4*e^3*Sqr t[x]) - (b*f^4*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*e^4) + (b*n*Log [d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) + (b*f^4*n*Log[x]*(a + b*L og[c*x^n]))/(8*e^4) + (7*f*(a + b*Log[c*x^n])^2)/(36*e*x^(3/2)) - (3*f^2*( a + b*Log[c*x^n])^2)/(8*e^2*x) + (5*f^3*(a + b*Log[c*x^n])^2)/(4*e^3*Sqrt[ x]) + (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(4*x^2) - (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*e^4) + (f^4*(a + b*Log[c*x^n])^3) /(24*b*e^4*n) + (f^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(6*b*e^4*n) - (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(6*b*e^4*n) - (f^4*Log [x]*(a + b*Log[c*x^n])^3)/(12*b*e^4*n) + (f^4*(a + b*Log[c*x^n])^4)/(48*b^ 2*e^4*n^2) + (b^2*f^4*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*e^4) - (b*f...
3.2.32.3.1 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]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x^{3}}d x\]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}} \,d x } \]
integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a ^3)*log(d*f*sqrt(x) + d*e)/x^3, x)
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]