Integrand size = 25, antiderivative size = 639 \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {90 b^3 e n^3 \sqrt {x}}{f}-6 a b^2 n^2 x+12 b^3 n^3 x+\frac {6 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right )}{f^2}-6 b^3 n^3 x \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {12 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 e n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \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^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac {12 b^3 e^2 n^3 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {12 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {24 b^3 e^2 n^3 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {48 b^3 e^2 n^3 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{f^2} \]
-6*a*b^2*n^2*x-6*b^3*n^2*x*ln(c*x^n)-3*b^2*n^2*x*(a+b*ln(c*x^n))+3*b*n*x*( a+b*ln(c*x^n))^2-48*b^3*e^2*n^3*polylog(4,-f*x^(1/2)/e)/f^2-90*b^3*e*n^3*x ^(1/2)/f+6*b^3*e^2*n^3*ln(e+f*x^(1/2))/f^2+6*b^2*n^2*x*(a+b*ln(c*x^n))*ln( d*(e+f*x^(1/2)))-3*b*n*x*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))+12*b^3*e^2* n^3*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f^2+3*b*e^2*n*(a+b*ln(c*x^n))^2*ln(1+ f*x^(1/2)/e)/f^2+12*b^2*e^2*n^2*(a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/f^ 2-6*b*e^2*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x^(1/2)/e)/f^2+24*b^2*e^2*n^2*( a+b*ln(c*x^n))*polylog(3,-f*x^(1/2)/e)/f^2+42*b^2*e*n^2*(a+b*ln(c*x^n))*x^ (1/2)/f-9*b*e*n*(a+b*ln(c*x^n))^2*x^(1/2)/f-6*b^2*e^2*n^2*(a+b*ln(c*x^n))* ln(e+f*x^(1/2))/f^2+12*b^3*e^2*n^3*polylog(2,1+f*x^(1/2)/e)/f^2-24*b^3*e^2 *n^3*polylog(3,-f*x^(1/2)/e)/f^2+x*(a+b*ln(c*x^n))^3*ln(d*(e+f*x^(1/2)))-e ^2*(a+b*ln(c*x^n))^3*ln(1+f*x^(1/2)/e)/f^2+e*(a+b*ln(c*x^n))^3*x^(1/2)/f-1 /2*x*(a+b*ln(c*x^n))^3-6*b^3*n^3*x*ln(d*(e+f*x^(1/2)))+12*b^3*n^3*x
Leaf count is larger than twice the leaf count of optimal. \(1522\) vs. \(2(639)=1278\).
Time = 0.49 (sec) , antiderivative size = 1522, normalized size of antiderivative = 2.38 \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx =\text {Too large to display} \]
-1/2*(-2*a^3*e*f*Sqrt[x] + 18*a^2*b*e*f*n*Sqrt[x] - 84*a*b^2*e*f*n^2*Sqrt[ x] + 180*b^3*e*f*n^3*Sqrt[x] + a^3*f^2*x - 6*a^2*b*f^2*n*x + 18*a*b^2*f^2* n^2*x - 24*b^3*f^2*n^3*x + 2*a^3*e^2*Log[e + f*Sqrt[x]] - 6*a^2*b*e^2*n*Lo g[e + f*Sqrt[x]] + 12*a*b^2*e^2*n^2*Log[e + f*Sqrt[x]] - 12*b^3*e^2*n^3*Lo g[e + f*Sqrt[x]] - 2*a^3*f^2*x*Log[d*(e + f*Sqrt[x])] + 6*a^2*b*f^2*n*x*Lo g[d*(e + f*Sqrt[x])] - 12*a*b^2*f^2*n^2*x*Log[d*(e + f*Sqrt[x])] + 12*b^3* f^2*n^3*x*Log[d*(e + f*Sqrt[x])] - 6*a^2*b*e^2*n*Log[e + f*Sqrt[x]]*Log[x] + 12*a*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x] - 12*b^3*e^2*n^3*Log[e + f*S qrt[x]]*Log[x] + 6*a^2*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 12*a*b^2*e^ 2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] + 12*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e ]*Log[x] + 6*a*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x]^2 - 6*b^3*e^2*n^3*Log [e + f*Sqrt[x]]*Log[x]^2 - 6*a*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 + 6*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 - 2*b^3*e^2*n^3*Log[e + f *Sqrt[x]]*Log[x]^3 + 2*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^3 - 6*a^2 *b*e*f*Sqrt[x]*Log[c*x^n] + 36*a*b^2*e*f*n*Sqrt[x]*Log[c*x^n] - 84*b^3*e*f *n^2*Sqrt[x]*Log[c*x^n] + 3*a^2*b*f^2*x*Log[c*x^n] - 12*a*b^2*f^2*n*x*Log[ c*x^n] + 18*b^3*f^2*n^2*x*Log[c*x^n] + 6*a^2*b*e^2*Log[e + f*Sqrt[x]]*Log[ c*x^n] - 12*a*b^2*e^2*n*Log[e + f*Sqrt[x]]*Log[c*x^n] + 12*b^3*e^2*n^2*Log [e + f*Sqrt[x]]*Log[c*x^n] - 6*a^2*b*f^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^ n] + 12*a*b^2*f^2*n*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] - 12*b^3*f^2*n^...
Time = 1.13 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.06, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2817, 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.
| \(\displaystyle \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2817 |
| \(\displaystyle -3 b n \int \left (-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2 x}+\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{f \sqrt {x}}-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2\right )dx+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 b n \left (x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^2 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac {4 b e^2 n \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac {8 b e^2 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b f^2 n}+\frac {e^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b f^2 n}-\frac {e^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {2 b e^2 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {3 e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}-\frac {14 b e n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-x \left (a+b \log \left (c x^n\right )\right )^2+b n x \left (a+b \log \left (c x^n\right )\right )+2 a b n x+2 b^2 n x \log \left (c x^n\right )+2 b^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {4 b^2 e^2 n^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{f^2}+\frac {8 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {16 b^2 e^2 n^2 \operatorname {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {2 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right )}{f^2}-\frac {4 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {30 b^2 e n^2 \sqrt {x}}{f}-4 b^2 n^2 x\right )+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3\) |
(e*Sqrt[x]*(a + b*Log[c*x^n])^3)/f - (x*(a + b*Log[c*x^n])^3)/2 - (e^2*Log [e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/f^2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - 3*b*n*((30*b^2*e*n^2*Sqrt[x])/f + 2*a*b*n*x - 4*b^2*n^2* x - (2*b^2*e^2*n^2*Log[e + f*Sqrt[x]])/f^2 + 2*b^2*n^2*x*Log[d*(e + f*Sqrt [x])] - (4*b^2*e^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 + 2*b ^2*n*x*Log[c*x^n] - (14*b*e*n*Sqrt[x]*(a + b*Log[c*x^n]))/f + b*n*x*(a + b *Log[c*x^n]) + (2*b*e^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 - 2*b *n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (3*e*Sqrt[x]*(a + b*Log[c *x^n])^2)/f - x*(a + b*Log[c*x^n])^2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log [c*x^n])^2 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/f^2 - (e^2* Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(3*b*f^2*n) + (e^2*Log[1 + (f*Sqr t[x])/e]*(a + b*Log[c*x^n])^3)/(3*b*f^2*n) - (4*b^2*e^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 - (4*b*e^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[ x])/e)])/f^2 + (2*e^2*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f ^2 + (8*b^2*e^2*n^2*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 - (8*b*e^2*n*(a + b* Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 + (16*b^2*e^2*n^2*PolyLog[4, -((f*Sqrt[x])/e)])/f^2)
3.2.29.3.1 Defintions of rubi rules used
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))^(p_.), x_Symbol] :> With[{u = IntHide[Log[d*(e + f*x^m)^r], 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, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))
\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )d x\]
\[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
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)
Timed out. \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Timed out} \]
\[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
1/27*(27*b^3*e*x*log(d)*log(x^n)^3 + 81*(a*b^2*e*log(d) - (e*n*log(d) - e* log(c)*log(d))*b^3)*x*log(x^n)^2 + 81*(a^2*b*e*log(d) - 2*(e*n*log(d) - e* log(c)*log(d))*a*b^2 + (2*e*n^2*log(d) - 2*e*n*log(c)*log(d) + e*log(c)^2* log(d))*b^3)*x*log(x^n) + 27*(a^3*e*log(d) - 3*(e*n*log(d) - e*log(c)*log( d))*a^2*b + 3*(2*e*n^2*log(d) - 2*e*n*log(c)*log(d) + e*log(c)^2*log(d))*a *b^2 - (6*e*n^3*log(d) - 6*e*n^2*log(c)*log(d) + 3*e*n*log(c)^2*log(d) - e *log(c)^3*log(d))*b^3)*x + 27*(b^3*e*x*log(x^n)^3 - 3*((e*n - e*log(c))*b^ 3 - a*b^2*e)*x*log(x^n)^2 - 3*(2*(e*n - e*log(c))*a*b^2 - (2*e*n^2 - 2*e*n *log(c) + e*log(c)^2)*b^3 - a^2*b*e)*x*log(x^n) - (3*(e*n - e*log(c))*a^2* b - 3*(2*e*n^2 - 2*e*n*log(c) + e*log(c)^2)*a*b^2 + (6*e*n^3 - 6*e*n^2*log (c) + 3*e*n*log(c)^2 - e*log(c)^3)*b^3 - a^3*e)*x)*log(f*sqrt(x) + e) - (9 *b^3*f*x^2*log(x^n)^3 - 9*((5*f*n - 3*f*log(c))*b^3 - 3*a*b^2*f)*x^2*log(x ^n)^2 - 3*(6*(5*f*n - 3*f*log(c))*a*b^2 - (38*f*n^2 - 30*f*n*log(c) + 9*f* log(c)^2)*b^3 - 9*a^2*b*f)*x^2*log(x^n) - (9*(5*f*n - 3*f*log(c))*a^2*b - 3*(38*f*n^2 - 30*f*n*log(c) + 9*f*log(c)^2)*a*b^2 + (130*f*n^3 - 114*f*n^2 *log(c) + 45*f*n*log(c)^2 - 9*f*log(c)^3)*b^3 - 9*a^3*f)*x^2)/sqrt(x))/e + integrate(1/2*(b^3*f^2*x*log(x^n)^3 + 3*(a*b^2*f^2 - (f^2*n - f^2*log(c)) *b^3)*x*log(x^n)^2 + 3*(a^2*b*f^2 - 2*(f^2*n - f^2*log(c))*a*b^2 + (2*f^2* n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*b^3)*x*log(x^n) + (a^3*f^2 - 3*(f^2*n - f^2*log(c))*a^2*b + 3*(2*f^2*n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*a*...
\[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int \ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]