Integrand size = 30, antiderivative size = 849 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=-\frac {175 b^3 d f n^3}{216 x^{3/2}}+\frac {45 b^3 d^2 f^2 n^3}{16 x}-\frac {255 b^3 d^3 f^3 n^3}{8 \sqrt {x}}+\frac {3}{8} b^3 d^4 f^4 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}-\frac {3}{16} b^3 d^4 f^4 n^3 \log (x)+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {1}{2} d^4 f^4 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right ) \]
-1/6*d*f*(a+b*ln(c*x^n))^3/x^(3/2)+1/4*d^2*f^2*(a+b*ln(c*x^n))^3/x-3/8*b^3 *n^3*ln(1+d*f*x^(1/2))/x^2+1/2*d^4*f^4*(a+b*ln(c*x^n))^3*ln(1+d*f*x^(1/2)) -1/2*d^3*f^3*(a+b*ln(c*x^n))^3/x^(1/2)-37/36*b^2*d*f*n^2*(a+b*ln(c*x^n))/x ^(3/2)+21/8*b^2*d^2*f^2*n^2*(a+b*ln(c*x^n))/x-3/8*b^2*d^4*f^4*n^2*ln(x)*(a +b*ln(c*x^n))-7/12*b*d*f*n*(a+b*ln(c*x^n))^2/x^(3/2)+9/8*b*d^2*f^2*n*(a+b* ln(c*x^n))^2/x+3/4*b^2*d^4*f^4*n^2*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))+3/4*b *d^4*f^4*n*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))+3*b^2*d^4*f^4*n^2*(a+b*ln(c *x^n))*polylog(2,-d*f*x^(1/2))+3*b*d^4*f^4*n*(a+b*ln(c*x^n))^2*polylog(2,- d*f*x^(1/2))-12*b^2*d^4*f^4*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))-63 /4*b^2*d^3*f^3*n^2*(a+b*ln(c*x^n))/x^(1/2)-15/4*b*d^3*f^3*n*(a+b*ln(c*x^n) )^2/x^(1/2)-1/8*d^4*f^4*(a+b*ln(c*x^n))^3-1/2*(a+b*ln(c*x^n))^3*ln(1+d*f*x ^(1/2))/x^2-175/216*b^3*d*f*n^3/x^(3/2)+45/16*b^3*d^2*f^2*n^3/x-3/16*b^3*d ^4*f^4*n^3*ln(x)+3/16*b^3*d^4*f^4*n^3*ln(x)^2-1/16*d^4*f^4*(a+b*ln(c*x^n)) ^4/b/n+3/8*b^3*d^4*f^4*n^3*ln(1+d*f*x^(1/2))-3/4*b^2*n^2*(a+b*ln(c*x^n))*l n(1+d*f*x^(1/2))/x^2-3/4*b*n*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))/x^2+3/2*b ^3*d^4*f^4*n^3*polylog(2,-d*f*x^(1/2))-6*b^3*d^4*f^4*n^3*polylog(3,-d*f*x^ (1/2))+24*b^3*d^4*f^4*n^3*polylog(4,-d*f*x^(1/2))-255/8*b^3*d^3*f^3*n^3/x^ (1/2)
Leaf count is larger than twice the leaf count of optimal. \(2009\) vs. \(2(849)=1698\).
Time = 0.65 (sec) , antiderivative size = 2009, normalized size of antiderivative = 2.37 \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\text {Result too large to show} \]
-1/6*(a^3*d*f)/x^(3/2) - (7*a^2*b*d*f*n)/(12*x^(3/2)) - (37*a*b^2*d*f*n^2) /(36*x^(3/2)) - (175*b^3*d*f*n^3)/(216*x^(3/2)) + (a^3*d^2*f^2)/(4*x) + (9 *a^2*b*d^2*f^2*n)/(8*x) + (21*a*b^2*d^2*f^2*n^2)/(8*x) + (45*b^3*d^2*f^2*n ^3)/(16*x) - (a^3*d^3*f^3)/(2*Sqrt[x]) - (15*a^2*b*d^3*f^3*n)/(4*Sqrt[x]) - (63*a*b^2*d^3*f^3*n^2)/(4*Sqrt[x]) - (255*b^3*d^3*f^3*n^3)/(8*Sqrt[x]) + (a^3*d^4*f^4*Log[1 + d*f*Sqrt[x]])/2 + (3*a^2*b*d^4*f^4*n*Log[1 + d*f*Sqr t[x]])/4 + (3*a*b^2*d^4*f^4*n^2*Log[1 + d*f*Sqrt[x]])/4 + (3*b^3*d^4*f^4*n ^3*Log[1 + d*f*Sqrt[x]])/8 - (a^3*Log[1 + d*f*Sqrt[x]])/(2*x^2) - (3*a^2*b *n*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (3*a*b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*x ^2) - (3*b^3*n^3*Log[1 + d*f*Sqrt[x]])/(8*x^2) - (a^3*d^4*f^4*Log[x])/4 - (3*a^2*b*d^4*f^4*n*Log[x])/8 - (3*a*b^2*d^4*f^4*n^2*Log[x])/8 - (3*b^3*d^4 *f^4*n^3*Log[x])/16 + (3*a^2*b*d^4*f^4*n*Log[x]^2)/8 + (3*a*b^2*d^4*f^4*n^ 2*Log[x]^2)/8 + (3*b^3*d^4*f^4*n^3*Log[x]^2)/16 - (a*b^2*d^4*f^4*n^2*Log[x ]^3)/4 - (b^3*d^4*f^4*n^3*Log[x]^3)/8 + (b^3*d^4*f^4*n^3*Log[1 + 1/(d*f*Sq rt[x])]*Log[x]^3)/2 - (b^3*d^4*f^4*n^3*Log[1 + d*f*Sqrt[x]]*Log[x]^3)/2 + (b^3*d^4*f^4*n^3*Log[x]^4)/8 - (a^2*b*d*f*Log[c*x^n])/(2*x^(3/2)) - (7*a*b ^2*d*f*n*Log[c*x^n])/(6*x^(3/2)) - (37*b^3*d*f*n^2*Log[c*x^n])/(36*x^(3/2) ) + (3*a^2*b*d^2*f^2*Log[c*x^n])/(4*x) + (9*a*b^2*d^2*f^2*n*Log[c*x^n])/(4 *x) + (21*b^3*d^2*f^2*n^2*Log[c*x^n])/(8*x) - (3*a^2*b*d^3*f^3*Log[c*x^n]) /(2*Sqrt[x]) - (15*a*b^2*d^3*f^3*n*Log[c*x^n])/(2*Sqrt[x]) - (63*b^3*d^...
Time = 1.40 (sec) , antiderivative size = 870, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -3 b n \int \left (\frac {d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{2 x}-\frac {d^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{4 x}-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^2 f^3}{2 x^{3/2}}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^2 f^2}{4 x^2}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 f}{6 x^{5/2}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^3}\right )dx+\frac {1}{2} d^4 f^4 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 \sqrt {x}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4-\frac {1}{4} d^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3 f^4-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 \sqrt {x}}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3 f}{6 x^{3/2}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-3 b n \left (\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^4 f^4}{48 b^2 n^2}-\frac {d^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{12 b n}+\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^3 f^4}{24 b n}-\frac {1}{16} b^2 d^4 n^2 \log ^2(x) f^4-\frac {1}{4} d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4-\frac {1}{8} b^2 d^4 n^2 \log \left (d \sqrt {x} f+1\right ) f^4+\frac {1}{16} b^2 d^4 n^2 \log (x) f^4-\frac {1}{4} b d^4 n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) f^4+\frac {1}{8} b d^4 n \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4-\frac {1}{2} b^2 d^4 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4-d^4 \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4-b d^4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4+2 b^2 d^4 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) f^4+4 b d^4 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) f^4-8 b^2 d^4 n^2 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right ) f^4+\frac {5 d^3 \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 \sqrt {x}}+\frac {21 b d^3 n \left (a+b \log \left (c x^n\right )\right ) f^3}{4 \sqrt {x}}+\frac {85 b^2 d^3 n^2 f^3}{8 \sqrt {x}}-\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 x}-\frac {7 b d^2 n \left (a+b \log \left (c x^n\right )\right ) f^2}{8 x}-\frac {15 b^2 d^2 n^2 f^2}{16 x}+\frac {7 d \left (a+b \log \left (c x^n\right )\right )^2 f}{36 x^{3/2}}+\frac {37 b d n \left (a+b \log \left (c x^n\right )\right ) f}{108 x^{3/2}}+\frac {175 b^2 d n^2 f}{648 x^{3/2}}+\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac {b^2 n^2 \log \left (d \sqrt {x} f+1\right )}{8 x^2}+\frac {b n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}\right )\) |
-1/6*(d*f*(a + b*Log[c*x^n])^3)/x^(3/2) + (d^2*f^2*(a + b*Log[c*x^n])^3)/( 4*x) - (d^3*f^3*(a + b*Log[c*x^n])^3)/(2*Sqrt[x]) + (d^4*f^4*Log[1 + d*f*S qrt[x]]*(a + b*Log[c*x^n])^3)/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) ^3)/(2*x^2) - (d^4*f^4*Log[x]*(a + b*Log[c*x^n])^3)/4 - 3*b*n*((175*b^2*d* f*n^2)/(648*x^(3/2)) - (15*b^2*d^2*f^2*n^2)/(16*x) + (85*b^2*d^3*f^3*n^2)/ (8*Sqrt[x]) - (b^2*d^4*f^4*n^2*Log[1 + d*f*Sqrt[x]])/8 + (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(8*x^2) + (b^2*d^4*f^4*n^2*Log[x])/16 - (b^2*d^4*f^4*n^2*Log [x]^2)/16 + (37*b*d*f*n*(a + b*Log[c*x^n]))/(108*x^(3/2)) - (7*b*d^2*f^2*n *(a + b*Log[c*x^n]))/(8*x) + (21*b*d^3*f^3*n*(a + b*Log[c*x^n]))/(4*Sqrt[x ]) - (b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/4 + (b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*x^2) + (b*d^4*f^4*n*Log[x]*(a + b*Lo g[c*x^n]))/8 + (7*d*f*(a + b*Log[c*x^n])^2)/(36*x^(3/2)) - (3*d^2*f^2*(a + b*Log[c*x^n])^2)/(8*x) + (5*d^3*f^3*(a + b*Log[c*x^n])^2)/(4*Sqrt[x]) - ( d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/4 + (Log[1 + d*f*Sqrt[x ]]*(a + b*Log[c*x^n])^2)/(4*x^2) + (d^4*f^4*(a + b*Log[c*x^n])^3)/(24*b*n) - (d^4*f^4*Log[x]*(a + b*Log[c*x^n])^3)/(12*b*n) + (d^4*f^4*(a + b*Log[c* x^n])^4)/(48*b^2*n^2) - (b^2*d^4*f^4*n^2*PolyLog[2, -(d*f*Sqrt[x])])/2 - b *d^4*f^4*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - d^4*f^4*(a + b* Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])] + 2*b^2*d^4*f^4*n^2*PolyLog[3, -( d*f*Sqrt[x])] + 4*b*d^4*f^4*n*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[...
3.1.63.3.1 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]))
\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{3}}d x\]
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
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) + 1)/x^3, x)
Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
\[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx=\int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^3} \,d x \]