Integrand size = 28, antiderivative size = 350 \[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {7 b n \sqrt {x}}{9 d^5 f^5}+\frac {2 b n x}{9 d^4 f^4}-\frac {b n x^{3/2}}{9 d^3 f^3}+\frac {5 b n x^2}{72 d^2 f^2}-\frac {11 b n x^{5/2}}{225 d f}+\frac {1}{27} b n x^3+\frac {b n \log \left (1+d f \sqrt {x}\right )}{9 d^6 f^6}-\frac {1}{9} b n x^3 \log \left (1+d f \sqrt {x}\right )+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac {1}{3} x^3 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {2 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{3 d^6 f^6} \] Output:
-7/9*b*n*x^(1/2)/d^5/f^5+2/9*b*n*x/d^4/f^4-1/9*b*n*x^(3/2)/d^3/f^3+5/72*b* n*x^2/d^2/f^2-11/225*b*n*x^(5/2)/d/f+1/27*b*n*x^3+1/9*b*n*ln(1+d*f*x^(1/2) )/d^6/f^6-1/9*b*n*x^3*ln(1+d*f*x^(1/2))+1/3*x^(1/2)*(a+b*ln(c*x^n))/d^5/f^ 5-1/6*x*(a+b*ln(c*x^n))/d^4/f^4+1/9*x^(3/2)*(a+b*ln(c*x^n))/d^3/f^3-1/12*x ^2*(a+b*ln(c*x^n))/d^2/f^2+1/15*x^(5/2)*(a+b*ln(c*x^n))/d/f-1/18*x^3*(a+b* ln(c*x^n))-1/3*ln(1+d*f*x^(1/2))*(a+b*ln(c*x^n))/d^6/f^6+1/3*x^3*ln(1+d*f* x^(1/2))*(a+b*ln(c*x^n))-2/3*b*n*polylog(2,-d*f*x^(1/2))/d^6/f^6
Time = 0.42 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.75 \[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {600 \left (-1+d^6 f^6 x^3\right ) \log \left (1+d f \sqrt {x}\right ) \left (3 a-b n+3 b \log \left (c x^n\right )\right )+d f \sqrt {x} \left (-30 a \left (-60+30 d f \sqrt {x}-20 d^2 f^2 x+15 d^3 f^3 x^{3/2}-12 d^4 f^4 x^2+10 d^5 f^5 x^{5/2}\right )+b n \left (-4200+1200 d f \sqrt {x}-600 d^2 f^2 x+375 d^3 f^3 x^{3/2}-264 d^4 f^4 x^2+200 d^5 f^5 x^{5/2}\right )-30 b \left (-60+30 d f \sqrt {x}-20 d^2 f^2 x+15 d^3 f^3 x^{3/2}-12 d^4 f^4 x^2+10 d^5 f^5 x^{5/2}\right ) \log \left (c x^n\right )\right )-3600 b n \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{5400 d^6 f^6} \] Input:
Integrate[x^2*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]),x]
Output:
(600*(-1 + d^6*f^6*x^3)*Log[1 + d*f*Sqrt[x]]*(3*a - b*n + 3*b*Log[c*x^n]) + d*f*Sqrt[x]*(-30*a*(-60 + 30*d*f*Sqrt[x] - 20*d^2*f^2*x + 15*d^3*f^3*x^( 3/2) - 12*d^4*f^4*x^2 + 10*d^5*f^5*x^(5/2)) + b*n*(-4200 + 1200*d*f*Sqrt[x ] - 600*d^2*f^2*x + 375*d^3*f^3*x^(3/2) - 264*d^4*f^4*x^2 + 200*d^5*f^5*x^ (5/2)) - 30*b*(-60 + 30*d*f*Sqrt[x] - 20*d^2*f^2*x + 15*d^3*f^3*x^(3/2) - 12*d^4*f^4*x^2 + 10*d^5*f^5*x^(5/2))*Log[c*x^n]) - 3600*b*n*PolyLog[2, -(d *f*Sqrt[x])])/(5400*d^6*f^6)
Time = 0.61 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2823, 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.
| \(\displaystyle \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2823 |
| \(\displaystyle -b n \int \left (\frac {1}{3} \log \left (d \sqrt {x} f+1\right ) x^2-\frac {x^2}{18}+\frac {x^{3/2}}{15 d f}-\frac {x}{12 d^2 f^2}+\frac {\sqrt {x}}{9 d^3 f^3}-\frac {1}{6 d^4 f^4}+\frac {1}{3 d^5 f^5 \sqrt {x}}-\frac {\log \left (d \sqrt {x} f+1\right )}{3 d^6 f^6 x}\right )dx-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}+\frac {1}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^6 f^6}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{3 d^5 f^5}-\frac {x \left (a+b \log \left (c x^n\right )\right )}{6 d^4 f^4}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d^3 f^3}-\frac {x^2 \left (a+b \log \left (c x^n\right )\right )}{12 d^2 f^2}+\frac {x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 d f}+\frac {1}{3} x^3 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )-b n \left (\frac {2 \operatorname {PolyLog}\left (2,-d f \sqrt {x}\right )}{3 d^6 f^6}-\frac {\log \left (d f \sqrt {x}+1\right )}{9 d^6 f^6}+\frac {7 \sqrt {x}}{9 d^5 f^5}-\frac {2 x}{9 d^4 f^4}+\frac {x^{3/2}}{9 d^3 f^3}-\frac {5 x^2}{72 d^2 f^2}+\frac {11 x^{5/2}}{225 d f}+\frac {1}{9} x^3 \log \left (d f \sqrt {x}+1\right )-\frac {x^3}{27}\right )\) |
Input:
Int[x^2*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]),x]
Output:
(Sqrt[x]*(a + b*Log[c*x^n]))/(3*d^5*f^5) - (x*(a + b*Log[c*x^n]))/(6*d^4*f ^4) + (x^(3/2)*(a + b*Log[c*x^n]))/(9*d^3*f^3) - (x^2*(a + b*Log[c*x^n]))/ (12*d^2*f^2) + (x^(5/2)*(a + b*Log[c*x^n]))/(15*d*f) - (x^3*(a + b*Log[c*x ^n]))/18 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*d^6*f^6) + (x^3*Lo g[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/3 - b*n*((7*Sqrt[x])/(9*d^5*f^5) - (2*x)/(9*d^4*f^4) + x^(3/2)/(9*d^3*f^3) - (5*x^2)/(72*d^2*f^2) + (11*x^(5/ 2))/(225*d*f) - x^3/27 - Log[1 + d*f*Sqrt[x]]/(9*d^6*f^6) + (x^3*Log[1 + d *f*Sqrt[x]])/9 + (2*PolyLog[2, -(d*f*Sqrt[x])])/(3*d^6*f^6))
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_. )]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :> With[{u = IntHide[(g*x)^q*Log[d* (e + f*x^m)^r], x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[1/x u, x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] && RationalQ[q])) && NeQ[q, -1]
\[\int x^{2} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )d x\]
Input:
int(x^2*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n)),x)
Output:
int(x^2*ln(d*(1/d+f*x^(1/2)))*(a+b*ln(c*x^n)),x)
\[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:
integrate(x^2*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n)),x, algorithm="fricas ")
Output:
integral((b*x^2*log(c*x^n) + a*x^2)*log(d*f*sqrt(x) + 1), x)
Timed out. \[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(x**2*ln(d*(1/d+f*x**(1/2)))*(a+b*ln(c*x**n)),x)
Output:
Timed out
\[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:
integrate(x^2*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n)),x, algorithm="maxima ")
Output:
integrate((b*log(c*x^n) + a)*x^2*log((f*sqrt(x) + 1/d)*d), x)
\[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right ) \,d x } \] Input:
integrate(x^2*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n)),x, algorithm="giac")
Output:
integrate((b*log(c*x^n) + a)*x^2*log((f*sqrt(x) + 1/d)*d), x)
Timed out. \[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^2\,\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \] Input:
int(x^2*log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n)),x)
Output:
int(x^2*log(d*(f*x^(1/2) + 1/d))*(a + b*log(c*x^n)), x)
\[ \int x^2 \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {360 \sqrt {x}\, a \,d^{5} f^{5} x^{2}+600 \sqrt {x}\, a \,d^{3} f^{3} x -4200 \sqrt {x}\, b d f n +1800 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a \,d^{6} f^{6} x^{3}-300 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{6} f^{6} x^{3}-450 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{4} f^{4} x^{2}-900 \,\mathrm {log}\left (x^{n} c \right ) b \,d^{2} f^{2} x +200 b \,d^{6} f^{6} n \,x^{3}+375 b \,d^{4} f^{4} n \,x^{2}+1200 b \,d^{2} f^{2} n x +1800 \sqrt {x}\, a d f -300 a \,d^{6} f^{6} x^{3}-450 a \,d^{4} f^{4} x^{2}-900 a \,d^{2} f^{2} x +360 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,d^{5} f^{5} x^{2}+600 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b \,d^{3} f^{3} x -264 \sqrt {x}\, b \,d^{5} f^{5} n \,x^{2}-600 \sqrt {x}\, b \,d^{3} f^{3} n x +1800 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b \,d^{6} f^{6} x^{3}-600 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b \,d^{6} f^{6} n \,x^{3}-1800 \left (\int \frac {\mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b n +1800 \left (\int \frac {\sqrt {x}\, \mathrm {log}\left (\sqrt {x}\, d f +1\right )}{d^{2} f^{2} x^{2}-x}d x \right ) b d f n +1800 \mathrm {log}\left (\sqrt {x}\, d f +1\right )^{2} b n -1800 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) \mathrm {log}\left (x^{n} c \right ) b +600 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) b n +1800 \sqrt {x}\, \mathrm {log}\left (x^{n} c \right ) b d f -1800 \,\mathrm {log}\left (\sqrt {x}\, d f +1\right ) a}{5400 d^{6} f^{6}} \] Input:
int(x^2*log(d*(1/d+f*x^(1/2)))*(a+b*log(c*x^n)),x)
Output:
(360*sqrt(x)*log(x**n*c)*b*d**5*f**5*x**2 + 600*sqrt(x)*log(x**n*c)*b*d**3 *f**3*x + 1800*sqrt(x)*log(x**n*c)*b*d*f + 360*sqrt(x)*a*d**5*f**5*x**2 + 600*sqrt(x)*a*d**3*f**3*x + 1800*sqrt(x)*a*d*f - 264*sqrt(x)*b*d**5*f**5*n *x**2 - 600*sqrt(x)*b*d**3*f**3*n*x - 4200*sqrt(x)*b*d*f*n - 1800*int(log( sqrt(x)*d*f + 1)/(d**2*f**2*x**2 - x),x)*b*n + 1800*int((sqrt(x)*log(sqrt( x)*d*f + 1))/(d**2*f**2*x**2 - x),x)*b*d*f*n + 1800*log(sqrt(x)*d*f + 1)** 2*b*n + 1800*log(sqrt(x)*d*f + 1)*log(x**n*c)*b*d**6*f**6*x**3 - 1800*log( sqrt(x)*d*f + 1)*log(x**n*c)*b + 1800*log(sqrt(x)*d*f + 1)*a*d**6*f**6*x** 3 - 1800*log(sqrt(x)*d*f + 1)*a - 600*log(sqrt(x)*d*f + 1)*b*d**6*f**6*n*x **3 + 600*log(sqrt(x)*d*f + 1)*b*n - 300*log(x**n*c)*b*d**6*f**6*x**3 - 45 0*log(x**n*c)*b*d**4*f**4*x**2 - 900*log(x**n*c)*b*d**2*f**2*x - 300*a*d** 6*f**6*x**3 - 450*a*d**4*f**4*x**2 - 900*a*d**2*f**2*x + 200*b*d**6*f**6*n *x**3 + 375*b*d**4*f**4*n*x**2 + 1200*b*d**2*f**2*n*x)/(5400*d**6*f**6)