Integrand size = 27, antiderivative size = 604 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {90 b^3 n^3 \sqrt {x}}{d f}-6 a b^2 n^2 x+12 b^3 n^3 x-6 b^3 n^3 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )+\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{d^2 f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )+6 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {9 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\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}{d^2 f^2}-\frac {12 b^3 n^3 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {6 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {24 b^3 n^3 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {24 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {48 b^3 n^3 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right )}{d^2 f^2} \]
-6*a*b^2*n^2*x+12*b^3*n^3*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))^2-1/2*x*(a+b*ln(c*x^n))^3-6*b^3*n^3*x*ln(d*(1/ d+f*x^(1/2)))+6*b^2*n^2*x*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^(1/2)))-3*b*n*x*(a +b*ln(c*x^n))^2*ln(d*(1/d+f*x^(1/2)))+x*(a+b*ln(c*x^n))^3*ln(d*(1/d+f*x^(1 /2)))+6*b^3*n^3*ln(1+d*f*x^(1/2))/d^2/f^2-6*b^2*n^2*(a+b*ln(c*x^n))*ln(1+d *f*x^(1/2))/d^2/f^2+3*b*n*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))/d^2/f^2-(a+b *ln(c*x^n))^3*ln(1+d*f*x^(1/2))/d^2/f^2-12*b^3*n^3*polylog(2,-d*f*x^(1/2)) /d^2/f^2+12*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))/d^2/f^2-6*b*n* (a+b*ln(c*x^n))^2*polylog(2,-d*f*x^(1/2))/d^2/f^2-24*b^3*n^3*polylog(3,-d* f*x^(1/2))/d^2/f^2+24*b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))/d^2/ f^2-48*b^3*n^3*polylog(4,-d*f*x^(1/2))/d^2/f^2-90*b^3*n^3*x^(1/2)/d/f+42*b ^2*n^2*(a+b*ln(c*x^n))*x^(1/2)/d/f-9*b*n*(a+b*ln(c*x^n))^2*x^(1/2)/d/f+(a+ b*ln(c*x^n))^3*x^(1/2)/d/f
Time = 0.33 (sec) , antiderivative size = 986, normalized size of antiderivative = 1.63 \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=-\frac {-2 a^3 d f \sqrt {x}+18 a^2 b d f n \sqrt {x}-84 a b^2 d f n^2 \sqrt {x}+180 b^3 d f n^3 \sqrt {x}+a^3 d^2 f^2 x-6 a^2 b d^2 f^2 n x+18 a b^2 d^2 f^2 n^2 x-24 b^3 d^2 f^2 n^3 x+2 a^3 \log \left (1+d f \sqrt {x}\right )-6 a^2 b n \log \left (1+d f \sqrt {x}\right )+12 a b^2 n^2 \log \left (1+d f \sqrt {x}\right )-12 b^3 n^3 \log \left (1+d f \sqrt {x}\right )-2 a^3 d^2 f^2 x \log \left (1+d f \sqrt {x}\right )+6 a^2 b d^2 f^2 n x \log \left (1+d f \sqrt {x}\right )-12 a b^2 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right )+12 b^3 d^2 f^2 n^3 x \log \left (1+d f \sqrt {x}\right )-6 a^2 b d f \sqrt {x} \log \left (c x^n\right )+36 a b^2 d f n \sqrt {x} \log \left (c x^n\right )-84 b^3 d f n^2 \sqrt {x} \log \left (c x^n\right )+3 a^2 b d^2 f^2 x \log \left (c x^n\right )-12 a b^2 d^2 f^2 n x \log \left (c x^n\right )+18 b^3 d^2 f^2 n^2 x \log \left (c x^n\right )+6 a^2 b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 a b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+12 b^3 n^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-6 a^2 b d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+12 a b^2 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-12 b^3 d^2 f^2 n^2 x \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-6 a b^2 d f \sqrt {x} \log ^2\left (c x^n\right )+18 b^3 d f n \sqrt {x} \log ^2\left (c x^n\right )+3 a b^2 d^2 f^2 x \log ^2\left (c x^n\right )-6 b^3 d^2 f^2 n x \log ^2\left (c x^n\right )+6 a b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 b^3 n \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 a b^2 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+6 b^3 d^2 f^2 n x \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-2 b^3 d f \sqrt {x} \log ^3\left (c x^n\right )+b^3 d^2 f^2 x \log ^3\left (c x^n\right )+2 b^3 \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )-2 b^3 d^2 f^2 x \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )+12 b n \left (a^2-2 a b n+2 b^2 n^2+2 b (a-b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )-48 b^2 n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )+96 b^3 n^3 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right )}{2 d^2 f^2} \]
-1/2*(-2*a^3*d*f*Sqrt[x] + 18*a^2*b*d*f*n*Sqrt[x] - 84*a*b^2*d*f*n^2*Sqrt[ x] + 180*b^3*d*f*n^3*Sqrt[x] + a^3*d^2*f^2*x - 6*a^2*b*d^2*f^2*n*x + 18*a* b^2*d^2*f^2*n^2*x - 24*b^3*d^2*f^2*n^3*x + 2*a^3*Log[1 + d*f*Sqrt[x]] - 6* a^2*b*n*Log[1 + d*f*Sqrt[x]] + 12*a*b^2*n^2*Log[1 + d*f*Sqrt[x]] - 12*b^3* n^3*Log[1 + d*f*Sqrt[x]] - 2*a^3*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] + 6*a^2*b* d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]] - 12*a*b^2*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt [x]] + 12*b^3*d^2*f^2*n^3*x*Log[1 + d*f*Sqrt[x]] - 6*a^2*b*d*f*Sqrt[x]*Log [c*x^n] + 36*a*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 84*b^3*d*f*n^2*Sqrt[x]*Log[c *x^n] + 3*a^2*b*d^2*f^2*x*Log[c*x^n] - 12*a*b^2*d^2*f^2*n*x*Log[c*x^n] + 1 8*b^3*d^2*f^2*n^2*x*Log[c*x^n] + 6*a^2*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 12*a*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 12*b^3*n^2*Log[1 + d*f*Sqrt[ x]]*Log[c*x^n] - 6*a^2*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 12*a* b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 12*b^3*d^2*f^2*n^2*x*Log [1 + d*f*Sqrt[x]]*Log[c*x^n] - 6*a*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 + 18*b^3*d *f*n*Sqrt[x]*Log[c*x^n]^2 + 3*a*b^2*d^2*f^2*x*Log[c*x^n]^2 - 6*b^3*d^2*f^2 *n*x*Log[c*x^n]^2 + 6*a*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 6*b^3*n*Lo g[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 6*a*b^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*L og[c*x^n]^2 + 6*b^3*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 2*b^3* d*f*Sqrt[x]*Log[c*x^n]^3 + b^3*d^2*f^2*x*Log[c*x^n]^3 + 2*b^3*Log[1 + d*f* Sqrt[x]]*Log[c*x^n]^3 - 2*b^3*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]...
Time = 0.84 (sec) , antiderivative size = 567, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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.
| \(\displaystyle \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2817 |
| \(\displaystyle -3 b n \int \left (\log \left (d \left (\sqrt {x} f+\frac {1}{d}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2 x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d f \sqrt {x}}-\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )^2\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 b n \left (-\frac {4 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {8 b n \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}+\frac {2 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-2 b n x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {14 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d f}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-x \left (a+b \log \left (c x^n\right )\right )^2+2 a b n x+2 b^2 n x \log \left (c x^n\right )+\frac {4 b^2 n^2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {8 b^2 n^2 \operatorname {PolyLog}\left (3,-d f \sqrt {x}\right )}{d^2 f^2}+\frac {16 b^2 n^2 \operatorname {PolyLog}\left (4,-d f \sqrt {x}\right )}{d^2 f^2}-\frac {2 b^2 n^2 \log \left (d f \sqrt {x}+1\right )}{d^2 f^2}+\frac {30 b^2 n^2 \sqrt {x}}{d f}+2 b^2 n^2 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )-4 b^2 n^2 x\right )-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}+x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3\) |
(Sqrt[x]*(a + b*Log[c*x^n])^3)/(d*f) - (x*(a + b*Log[c*x^n])^3)/2 + x*Log[ d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(d^2*f^2) - 3*b*n*((30*b^2*n^2*Sqrt[x])/(d*f) + 2*a*b*n*x - 4*b^2*n^2*x + 2*b^2*n^2*x*Log[d*(d^(-1) + f*Sqrt[x])] - (2*b^2*n^2*Log[ 1 + d*f*Sqrt[x]])/(d^2*f^2) + 2*b^2*n*x*Log[c*x^n] - (14*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) + b*n*x*(a + b*Log[c*x^n]) - 2*b*n*x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c* x^n]))/(d^2*f^2) + (3*Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - x*(a + b*Log[c *x^n])^2 + x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d *f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) + (4*b^2*n^2*PolyLog[2, -(d*f* Sqrt[x])])/(d^2*f^2) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x]) ])/(d^2*f^2) + (2*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^ 2) + (8*b^2*n^2*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) - (8*b*n*(a + b*Log[ c*x^n])*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) + (16*b^2*n^2*PolyLog[4, -(d *f*Sqrt[x])])/(d^2*f^2))
3.1.60.3.1 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]))
\[\int {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )d x\]
\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
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) + 1), x)
Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\text {Timed out} \]
\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
(b^3*x*log(x^n)^3 - 3*(b^3*(n - log(c)) - a*b^2)*x*log(x^n)^2 + 3*((2*n^2 - 2*n*log(c) + log(c)^2)*b^3 - 2*a*b^2*(n - log(c)) + a^2*b)*x*log(x^n) + (3*(2*n^2 - 2*n*log(c) + log(c)^2)*a*b^2 - (6*n^3 - 6*n^2*log(c) + 3*n*log (c)^2 - log(c)^3)*b^3 - 3*a^2*b*(n - log(c)) + a^3)*x)*log(d*f*sqrt(x) + 1 ) - 1/27*(9*b^3*d*f*x^2*log(x^n)^3 + 9*(3*a*b^2*d*f - (5*d*f*n - 3*d*f*log (c))*b^3)*x^2*log(x^n)^2 + 3*(9*a^2*b*d*f - 6*(5*d*f*n - 3*d*f*log(c))*a*b ^2 + (38*d*f*n^2 - 30*d*f*n*log(c) + 9*d*f*log(c)^2)*b^3)*x^2*log(x^n) + ( 9*a^3*d*f - 9*(5*d*f*n - 3*d*f*log(c))*a^2*b + 3*(38*d*f*n^2 - 30*d*f*n*lo g(c) + 9*d*f*log(c)^2)*a*b^2 - (130*d*f*n^3 - 114*d*f*n^2*log(c) + 45*d*f* n*log(c)^2 - 9*d*f*log(c)^3)*b^3)*x^2)/sqrt(x) + integrate(1/2*(b^3*d^2*f^ 2*x*log(x^n)^3 + 3*(a*b^2*d^2*f^2 - (d^2*f^2*n - d^2*f^2*log(c))*b^3)*x*lo g(x^n)^2 + 3*(a^2*b*d^2*f^2 - 2*(d^2*f^2*n - d^2*f^2*log(c))*a*b^2 + (2*d^ 2*f^2*n^2 - 2*d^2*f^2*n*log(c) + d^2*f^2*log(c)^2)*b^3)*x*log(x^n) + (a^3* d^2*f^2 - 3*(d^2*f^2*n - d^2*f^2*log(c))*a^2*b + 3*(2*d^2*f^2*n^2 - 2*d^2* f^2*n*log(c) + d^2*f^2*log(c)^2)*a*b^2 - (6*d^2*f^2*n^3 - 6*d^2*f^2*n^2*lo g(c) + 3*d^2*f^2*n*log(c)^2 - d^2*f^2*log(c)^3)*b^3)*x)/(d*f*sqrt(x) + 1), x)
\[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \]
Timed out. \[ \int \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx=\int \ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3 \,d x \]