Integrand size = 28, antiderivative size = 750 \[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\frac {86 b^2 e^5 n^2 \sqrt {x}}{27 f^5}+\frac {a b e^4 n x}{3 f^4}-\frac {13 b^2 e^4 n^2 x}{27 f^4}+\frac {14 b^2 e^3 n^2 x^{3/2}}{81 f^3}-\frac {19 b^2 e^2 n^2 x^2}{216 f^2}+\frac {182 b^2 e n^2 x^{5/2}}{3375 f}-\frac {1}{27} b^2 n^2 x^3-\frac {2 b^2 e^6 n^2 \log \left (e+f \sqrt {x}\right )}{27 f^6}+\frac {2}{27} b^2 n^2 x^3 \log \left (d \left (e+f \sqrt {x}\right )\right )-\frac {4 b^2 e^6 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{9 f^6}+\frac {b^2 e^4 n x \log \left (c x^n\right )}{3 f^4}-\frac {14 b e^5 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{9 f^5}+\frac {b e^4 n x \left (a+b \log \left (c x^n\right )\right )}{9 f^4}-\frac {2 b e^3 n x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}+\frac {5 b e^2 n x^2 \left (a+b \log \left (c x^n\right )\right )}{36 f^2}-\frac {22 b e n x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 f}+\frac {2}{27} b n x^3 \left (a+b \log \left (c x^n\right )\right )+\frac {2 b e^6 n \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^6}-\frac {2}{9} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {e^5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{3 f^5}-\frac {e^4 x \left (a+b \log \left (c x^n\right )\right )^2}{6 f^4}+\frac {e^3 x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 f^3}-\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 f^2}+\frac {e x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^6 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^6}-\frac {4 b^2 e^6 n^2 \operatorname {PolyLog}\left (2,1+\frac {f \sqrt {x}}{e}\right )}{9 f^6}-\frac {4 b e^6 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {8 b^2 e^6 n^2 \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{3 f^6} \]
2/27*b*n*x^3*(a+b*ln(c*x^n))-2/27*b^2*e^6*n^2*ln(e+f*x^(1/2))/f^6-2/9*b*n* x^3*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2)))-1/27*b^2*n^2*x^3+1/3*a*b*e^4*n*x/f ^4+1/3*b^2*e^4*n*x*ln(c*x^n)/f^4+1/9*b*e^4*n*x*(a+b*ln(c*x^n))/f^4-2/9*b*e ^3*n*x^(3/2)*(a+b*ln(c*x^n))/f^3+5/36*b*e^2*n*x^2*(a+b*ln(c*x^n))/f^2-22/2 25*b*e*n*x^(5/2)*(a+b*ln(c*x^n))/f+2/9*b*e^6*n*(a+b*ln(c*x^n))*ln(e+f*x^(1 /2))/f^6-4/9*b^2*e^6*n^2*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/f^6-4/3*b*e^6*n* (a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/f^6-14/9*b*e^5*n*(a+b*ln(c*x^n))*x ^(1/2)/f^5-4/9*b^2*e^6*n^2*polylog(2,1+f*x^(1/2)/e)/f^6+8/3*b^2*e^6*n^2*po lylog(3,-f*x^(1/2)/e)/f^6+86/27*b^2*e^5*n^2*x^(1/2)/f^5-13/27*b^2*e^4*n^2* x/f^4+14/81*b^2*e^3*n^2*x^(3/2)/f^3-19/216*b^2*e^2*n^2*x^2/f^2+182/3375*b^ 2*e*n^2*x^(5/2)/f+1/3*x^3*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))-1/18*x^3*( a+b*ln(c*x^n))^2-1/6*e^4*x*(a+b*ln(c*x^n))^2/f^4+1/9*e^3*x^(3/2)*(a+b*ln(c *x^n))^2/f^3-1/12*e^2*x^2*(a+b*ln(c*x^n))^2/f^2+1/15*e*x^(5/2)*(a+b*ln(c*x ^n))^2/f+2/27*b^2*n^2*x^3*ln(d*(e+f*x^(1/2)))-1/3*e^6*(a+b*ln(c*x^n))^2*ln (1+f*x^(1/2)/e)/f^6+1/3*e^5*(a+b*ln(c*x^n))^2*x^(1/2)/f^5
Time = 0.58 (sec) , antiderivative size = 1319, normalized size of antiderivative = 1.76 \[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx =\text {Too large to display} \]
(a^2*e^5*Sqrt[x])/(3*f^5) - (14*a*b*e^5*n*Sqrt[x])/(9*f^5) + (86*b^2*e^5*n ^2*Sqrt[x])/(27*f^5) - (a^2*e^4*x)/(6*f^4) + (4*a*b*e^4*n*x)/(9*f^4) - (13 *b^2*e^4*n^2*x)/(27*f^4) + (a^2*e^3*x^(3/2))/(9*f^3) - (2*a*b*e^3*n*x^(3/2 ))/(9*f^3) + (14*b^2*e^3*n^2*x^(3/2))/(81*f^3) - (a^2*e^2*x^2)/(12*f^2) + (5*a*b*e^2*n*x^2)/(36*f^2) - (19*b^2*e^2*n^2*x^2)/(216*f^2) + (a^2*e*x^(5/ 2))/(15*f) - (22*a*b*e*n*x^(5/2))/(225*f) + (182*b^2*e*n^2*x^(5/2))/(3375* f) - (a^2*x^3)/18 + (2*a*b*n*x^3)/27 - (b^2*n^2*x^3)/27 - (a^2*e^6*Log[e + f*Sqrt[x]])/(3*f^6) + (2*a*b*e^6*n*Log[e + f*Sqrt[x]])/(9*f^6) - (2*b^2*e ^6*n^2*Log[e + f*Sqrt[x]])/(27*f^6) + (a^2*x^3*Log[d*(e + f*Sqrt[x])])/3 - (2*a*b*n*x^3*Log[d*(e + f*Sqrt[x])])/9 + (2*b^2*n^2*x^3*Log[d*(e + f*Sqrt [x])])/27 + (2*a*b*e^6*n*Log[e + f*Sqrt[x]]*Log[x])/(3*f^6) - (2*b^2*e^6*n ^2*Log[e + f*Sqrt[x]]*Log[x])/(9*f^6) - (2*a*b*e^6*n*Log[1 + (f*Sqrt[x])/e ]*Log[x])/(3*f^6) + (2*b^2*e^6*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x])/(9*f^6) - (b^2*e^6*n^2*Log[e + f*Sqrt[x]]*Log[x]^2)/(3*f^6) + (b^2*e^6*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2)/(3*f^6) + (2*a*b*e^5*Sqrt[x]*Log[c*x^n])/(3*f^5) - (14*b^2*e^5*n*Sqrt[x]*Log[c*x^n])/(9*f^5) - (a*b*e^4*x*Log[c*x^n])/(3*f ^4) + (4*b^2*e^4*n*x*Log[c*x^n])/(9*f^4) + (2*a*b*e^3*x^(3/2)*Log[c*x^n])/ (9*f^3) - (2*b^2*e^3*n*x^(3/2)*Log[c*x^n])/(9*f^3) - (a*b*e^2*x^2*Log[c*x^ n])/(6*f^2) + (5*b^2*e^2*n*x^2*Log[c*x^n])/(36*f^2) + (2*a*b*e*x^(5/2)*Log [c*x^n])/(15*f) - (22*b^2*e*n*x^(5/2)*Log[c*x^n])/(225*f) - (a*b*x^3*Lo...
Time = 1.13 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2824 |
| \(\displaystyle -2 b n \int \left (-\frac {\log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) e^6}{3 f^6 x}+\frac {\left (a+b \log \left (c x^n\right )\right ) e^5}{3 f^5 \sqrt {x}}-\frac {\left (a+b \log \left (c x^n\right )\right ) e^4}{6 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right ) e^3}{9 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right ) e^2}{12 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right ) e}{15 f}-\frac {1}{18} x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {1}{3} x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )\right )dx+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^6 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^6}+\frac {e^5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{3 f^5}-\frac {e^4 x \left (a+b \log \left (c x^n\right )\right )^2}{6 f^4}+\frac {e^3 x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 f^3}-\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 f^2}+\frac {e x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 b n \left (\frac {1}{9} x^3 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 e^6 \operatorname {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}-\frac {e^6 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 b f^6 n}+\frac {e^6 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{6 b f^6 n}-\frac {e^6 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{9 f^6}+\frac {7 e^5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{9 f^5}-\frac {e^4 x \left (a+b \log \left (c x^n\right )\right )}{18 f^4}+\frac {e^3 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac {5 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )}{72 f^2}+\frac {11 e x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{225 f}-\frac {1}{27} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac {a e^4 x}{6 f^4}-\frac {b e^4 x \log \left (c x^n\right )}{6 f^4}-\frac {1}{27} b n x^3 \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {2 b e^6 n \operatorname {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right )}{9 f^6}-\frac {4 b e^6 n \operatorname {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{3 f^6}+\frac {b e^6 n \log \left (e+f \sqrt {x}\right )}{27 f^6}+\frac {2 b e^6 n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{9 f^6}-\frac {43 b e^5 n \sqrt {x}}{27 f^5}+\frac {13 b e^4 n x}{54 f^4}-\frac {7 b e^3 n x^{3/2}}{81 f^3}+\frac {19 b e^2 n x^2}{432 f^2}-\frac {91 b e n x^{5/2}}{3375 f}+\frac {1}{54} b n x^3\right )+\frac {1}{3} x^3 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {e^6 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{3 f^6}+\frac {e^5 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{3 f^5}-\frac {e^4 x \left (a+b \log \left (c x^n\right )\right )^2}{6 f^4}+\frac {e^3 x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{9 f^3}-\frac {e^2 x^2 \left (a+b \log \left (c x^n\right )\right )^2}{12 f^2}+\frac {e x^{5/2} \left (a+b \log \left (c x^n\right )\right )^2}{15 f}-\frac {1}{18} x^3 \left (a+b \log \left (c x^n\right )\right )^2\) |
(e^5*Sqrt[x]*(a + b*Log[c*x^n])^2)/(3*f^5) - (e^4*x*(a + b*Log[c*x^n])^2)/ (6*f^4) + (e^3*x^(3/2)*(a + b*Log[c*x^n])^2)/(9*f^3) - (e^2*x^2*(a + b*Log [c*x^n])^2)/(12*f^2) + (e*x^(5/2)*(a + b*Log[c*x^n])^2)/(15*f) - (x^3*(a + b*Log[c*x^n])^2)/18 - (e^6*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(3*f^ 6) + (x^3*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/3 - 2*b*n*((-43*b*e ^5*n*Sqrt[x])/(27*f^5) - (a*e^4*x)/(6*f^4) + (13*b*e^4*n*x)/(54*f^4) - (7* b*e^3*n*x^(3/2))/(81*f^3) + (19*b*e^2*n*x^2)/(432*f^2) - (91*b*e*n*x^(5/2) )/(3375*f) + (b*n*x^3)/54 + (b*e^6*n*Log[e + f*Sqrt[x]])/(27*f^6) - (b*n*x ^3*Log[d*(e + f*Sqrt[x])])/27 + (2*b*e^6*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqr t[x])/e)])/(9*f^6) - (b*e^4*x*Log[c*x^n])/(6*f^4) + (7*e^5*Sqrt[x]*(a + b* Log[c*x^n]))/(9*f^5) - (e^4*x*(a + b*Log[c*x^n]))/(18*f^4) + (e^3*x^(3/2)* (a + b*Log[c*x^n]))/(9*f^3) - (5*e^2*x^2*(a + b*Log[c*x^n]))/(72*f^2) + (1 1*e*x^(5/2)*(a + b*Log[c*x^n]))/(225*f) - (x^3*(a + b*Log[c*x^n]))/27 - (e ^6*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(9*f^6) + (x^3*Log[d*(e + f*Sqrt [x])]*(a + b*Log[c*x^n]))/9 - (e^6*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n])^2 )/(6*b*f^6*n) + (e^6*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(6*b*f^6 *n) + (2*b*e^6*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(9*f^6) + (2*e^6*(a + b*Lo g[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/(3*f^6) - (4*b*e^6*n*PolyLog[3, -( (f*Sqrt[x])/e)])/(3*f^6))
3.2.22.3.1 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]))
\[\int x^{2} {\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2} \ln \left (d \left (e +f \sqrt {x}\right )\right )d x\]
\[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
\[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right ) \,d x } \]
Timed out. \[ \int x^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx=\int x^2\,\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2 \,d x \]