Integrand size = 26, antiderivative size = 907 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {255 b^3 e^3 n^3 \sqrt {x}}{8 f^3}-\frac {9 a b^2 e^2 n^2 x}{4 f^2}+\frac {45 b^3 e^2 n^3 x}{16 f^2}-\frac {175 b^3 e n^3 x^{3/2}}{216 f}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right )}{8 f^4}-\frac {3}{8} b^3 n^3 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3 b^3 e^4 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 f^4}-\frac {9 b^3 e^2 n^2 x \log \left (c x^n\right )}{4 f^2}+\frac {63 b^2 e^3 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 f^3}-\frac {3 b^2 e^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 f^2}+\frac {37 b^2 e n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 f}-\frac {9}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 e^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b e^3 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 f^3}+\frac {9 b e^2 n x \left (a+b \log \left (c x^n\right )\right )^2}{8 f^2}-\frac {7 b e n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 f}+\frac {3}{8} b n x^2 \left (a+b \log \left (c x^n\right )\right )^2-\frac {3}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {3 b^3 e^4 n^3 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{2 f^4}+\frac {3 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {3 b e^4 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {6 b^3 e^4 n^3 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^4}+\frac {12 b^2 e^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^4}-\frac {24 b^3 e^4 n^3 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{f^4} \]
-9/16*b^2*n^2*x^2*(a+b*ln(c*x^n))+3/8*b*n*x^2*(a+b*ln(c*x^n))^2+3/8*b^3*e^ 4*n^3*ln(e+f*x^(1/2))/f^4+3/4*b^2*n^2*x^2*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2 )))-3/4*b*n*x^2*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))+3/2*b^3*e^4*n^3*poly log(2,1+f*x^(1/2)/e)/f^4-6*b^3*e^4*n^3*polylog(3,-f*x^(1/2)/e)/f^4-24*b^3* e^4*n^3*polylog(4,-f*x^(1/2)/e)/f^4-255/8*b^3*e^3*n^3*x^(1/2)/f^3+45/16*b^ 3*e^2*n^3*x/f^2-175/216*b^3*e*n^3*x^(3/2)/f+3/8*b^3*n^3*x^2-9/4*a*b^2*e^2* n^2*x/f^2-9/4*b^3*e^2*n^2*x*ln(c*x^n)/f^2-3/8*b^2*e^2*n^2*x*(a+b*ln(c*x^n) )/f^2+37/36*b^2*e*n^2*x^(3/2)*(a+b*ln(c*x^n))/f+9/8*b*e^2*n*x*(a+b*ln(c*x^ n))^2/f^2-7/12*b*e*n*x^(3/2)*(a+b*ln(c*x^n))^2/f-3/4*b^2*e^4*n^2*(a+b*ln(c *x^n))*ln(e+f*x^(1/2))/f^4+3/2*b^3*e^4*n^3*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2) )/f^4+3/4*b*e^4*n*(a+b*ln(c*x^n))^2*ln(1+f*x^(1/2)/e)/f^4+3*b^2*e^4*n^2*(a +b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/f^4-3*b*e^4*n*(a+b*ln(c*x^n))^2*poly log(2,-f*x^(1/2)/e)/f^4+12*b^2*e^4*n^2*(a+b*ln(c*x^n))*polylog(3,-f*x^(1/2 )/e)/f^4+63/4*b^2*e^3*n^2*(a+b*ln(c*x^n))*x^(1/2)/f^3-15/4*b*e^3*n*(a+b*ln (c*x^n))^2*x^(1/2)/f^3+1/2*x^2*(a+b*ln(c*x^n))^3*ln(d*(e+f*x^(1/2)))-1/8*x ^2*(a+b*ln(c*x^n))^3-1/4*e^2*x*(a+b*ln(c*x^n))^3/f^2+1/6*e*x^(3/2)*(a+b*ln (c*x^n))^3/f-3/8*b^3*n^3*x^2*ln(d*(e+f*x^(1/2)))-1/2*e^4*(a+b*ln(c*x^n))^3 *ln(1+f*x^(1/2)/e)/f^4+1/2*e^3*(a+b*ln(c*x^n))^3*x^(1/2)/f^3
Leaf count is larger than twice the leaf count of optimal. \(1968\) vs. \(2(907)=1814\).
Time = 0.50 (sec) , antiderivative size = 1968, normalized size of antiderivative = 2.17 \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \]
(216*a^3*e^3*f*Sqrt[x] - 1620*a^2*b*e^3*f*n*Sqrt[x] + 6804*a*b^2*e^3*f*n^2 *Sqrt[x] - 13770*b^3*e^3*f*n^3*Sqrt[x] - 108*a^3*e^2*f^2*x + 486*a^2*b*e^2 *f^2*n*x - 1134*a*b^2*e^2*f^2*n^2*x + 1215*b^3*e^2*f^2*n^3*x + 72*a^3*e*f^ 3*x^(3/2) - 252*a^2*b*e*f^3*n*x^(3/2) + 444*a*b^2*e*f^3*n^2*x^(3/2) - 350* b^3*e*f^3*n^3*x^(3/2) - 54*a^3*f^4*x^2 + 162*a^2*b*f^4*n*x^2 - 243*a*b^2*f ^4*n^2*x^2 + 162*b^3*f^4*n^3*x^2 - 216*a^3*e^4*Log[e + f*Sqrt[x]] + 324*a^ 2*b*e^4*n*Log[e + f*Sqrt[x]] - 324*a*b^2*e^4*n^2*Log[e + f*Sqrt[x]] + 162* b^3*e^4*n^3*Log[e + f*Sqrt[x]] + 216*a^3*f^4*x^2*Log[d*(e + f*Sqrt[x])] - 324*a^2*b*f^4*n*x^2*Log[d*(e + f*Sqrt[x])] + 324*a*b^2*f^4*n^2*x^2*Log[d*( e + f*Sqrt[x])] - 162*b^3*f^4*n^3*x^2*Log[d*(e + f*Sqrt[x])] + 648*a^2*b*e ^4*n*Log[e + f*Sqrt[x]]*Log[x] - 648*a*b^2*e^4*n^2*Log[e + f*Sqrt[x]]*Log[ x] + 324*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[x] - 648*a^2*b*e^4*n*Log[1 + ( f*Sqrt[x])/e]*Log[x] + 648*a*b^2*e^4*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] - 3 24*b^3*e^4*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x] - 648*a*b^2*e^4*n^2*Log[e + f *Sqrt[x]]*Log[x]^2 + 324*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[x]^2 + 648*a*b ^2*e^4*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 - 324*b^3*e^4*n^3*Log[1 + (f*Sq rt[x])/e]*Log[x]^2 + 216*b^3*e^4*n^3*Log[e + f*Sqrt[x]]*Log[x]^3 - 216*b^3 *e^4*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^3 + 648*a^2*b*e^3*f*Sqrt[x]*Log[c*x ^n] - 3240*a*b^2*e^3*f*n*Sqrt[x]*Log[c*x^n] + 6804*b^3*e^3*f*n^2*Sqrt[x]*L og[c*x^n] - 324*a^2*b*e^2*f^2*x*Log[c*x^n] + 972*a*b^2*e^2*f^2*n*x*Log[...
Time = 1.47 (sec) , antiderivative size = 935, normalized size of antiderivative = 1.03, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, 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.
| \(\displaystyle \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -3 b n \int \left (-\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2 e^4}{2 f^4 x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 e^3}{2 f^3 \sqrt {x}}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 e^2}{4 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 e}{6 f}-\frac {1}{8} x \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{2} x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2\right )dx+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^4 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 f^4}+\frac {e^3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 f^3}-\frac {e^2 x \left (a+b \log \left (c x^n\right )\right )^3}{4 f^2}+\frac {e x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 e^4}{2 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3 e^3}{2 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3 e^2}{4 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3 e}{6 f}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {1}{2} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-3 b n \left (-\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3 e^4}{6 b f^4 n}+\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 e^4}{6 b f^4 n}-\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 e^4}{4 f^4}-\frac {b^2 n^2 \log \left (e+f \sqrt {x}\right ) e^4}{8 f^4}-\frac {b^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right ) e^4}{2 f^4}+\frac {b n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) e^4}{4 f^4}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right ) e^4}{2 f^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^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 ) e^4}{f^4}+\frac {8 b^2 n^2 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) e^4}{f^4}+\frac {5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 e^3}{4 f^3}-\frac {21 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) e^3}{4 f^3}+\frac {85 b^2 n^2 \sqrt {x} e^3}{8 f^3}-\frac {3 x \left (a+b \log \left (c x^n\right )\right )^2 e^2}{8 f^2}-\frac {15 b^2 n^2 x e^2}{16 f^2}+\frac {3 a b n x e^2}{4 f^2}+\frac {3 b^2 n x \log \left (c x^n\right ) e^2}{4 f^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right ) e^2}{8 f^2}+\frac {7 x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 e}{36 f}+\frac {175 b^2 n^2 x^{3/2} e}{648 f}-\frac {37 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right ) e}{108 f}-\frac {1}{8} b^2 n^2 x^2-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{4} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{8} b^2 n^2 x^2 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {3}{16} b n x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {1}{4} b n x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right )\) |
(e^3*Sqrt[x]*(a + b*Log[c*x^n])^3)/(2*f^3) - (e^2*x*(a + b*Log[c*x^n])^3)/ (4*f^2) + (e*x^(3/2)*(a + b*Log[c*x^n])^3)/(6*f) - (x^2*(a + b*Log[c*x^n]) ^3)/8 - (e^4*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(2*f^4) + (x^2*Log[d *(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/2 - 3*b*n*((85*b^2*e^3*n^2*Sqrt[x] )/(8*f^3) + (3*a*b*e^2*n*x)/(4*f^2) - (15*b^2*e^2*n^2*x)/(16*f^2) + (175*b ^2*e*n^2*x^(3/2))/(648*f) - (b^2*n^2*x^2)/8 - (b^2*e^4*n^2*Log[e + f*Sqrt[ x]])/(8*f^4) + (b^2*n^2*x^2*Log[d*(e + f*Sqrt[x])])/8 - (b^2*e^4*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(2*f^4) + (3*b^2*e^2*n*x*Log[c*x^n])/ (4*f^2) - (21*b*e^3*n*Sqrt[x]*(a + b*Log[c*x^n]))/(4*f^3) + (b*e^2*n*x*(a + b*Log[c*x^n]))/(8*f^2) - (37*b*e*n*x^(3/2)*(a + b*Log[c*x^n]))/(108*f) + (3*b*n*x^2*(a + b*Log[c*x^n]))/16 + (b*e^4*n*Log[e + f*Sqrt[x]]*(a + b*Lo g[c*x^n]))/(4*f^4) - (b*n*x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/4 + (5*e^3*Sqrt[x]*(a + b*Log[c*x^n])^2)/(4*f^3) - (3*e^2*x*(a + b*Log[c*x^ n])^2)/(8*f^2) + (7*e*x^(3/2)*(a + b*Log[c*x^n])^2)/(36*f) - (x^2*(a + b*L og[c*x^n])^2)/8 + (x^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/4 - (e ^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*f^4) - (e^4*Log[e + f*S qrt[x]]*(a + b*Log[c*x^n])^3)/(6*b*f^4*n) + (e^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(6*b*f^4*n) - (b^2*e^4*n^2*PolyLog[2, 1 + (f*Sqrt[x])/ e])/(2*f^4) - (b*e^4*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/f^ 4 + (e^4*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^4 + (2*b^...
3.2.28.3.1 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]))
\[\int x {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )d x\]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
integral((b^3*x*log(c*x^n)^3 + 3*a*b^2*x*log(c*x^n)^2 + 3*a^2*b*x*log(c*x^ n) + a^3*x)*log(d*f*sqrt(x) + d*e), x)
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Timed out} \]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
\[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} x \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int x\,\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]