Integrand size = 24, antiderivative size = 480 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {5 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {40 b^2 d^3 n^2 \left (d+e \sqrt {x}\right )^3}{27 e^6}+\frac {5 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4}{8 e^6}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^5}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6}{54 e^6}-\frac {4 b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right )}{3 e^6}+\frac {4 b d^5 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}-\frac {5 b d^4 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^6}+\frac {40 b d^3 n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}-\frac {5 b d^2 n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}+\frac {4 b d n \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{5 e^6}-\frac {b n \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{9 e^6}-\frac {2 b d^6 n \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^6}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \] Output:
5/2*b^2*d^4*n^2*(d+e*x^(1/2))^2/e^6-40/27*b^2*d^3*n^2*(d+e*x^(1/2))^3/e^6+ 5/8*b^2*d^2*n^2*(d+e*x^(1/2))^4/e^6-4/25*b^2*d*n^2*(d+e*x^(1/2))^5/e^6+1/5 4*b^2*n^2*(d+e*x^(1/2))^6/e^6-4*b^2*d^5*n^2*x^(1/2)/e^5+1/3*b^2*d^6*n^2*ln (d+e*x^(1/2))^2/e^6+4*b*d^5*n*(d+e*x^(1/2))*(a+b*ln(c*(d+e*x^(1/2))^n))/e^ 6-5*b*d^4*n*(d+e*x^(1/2))^2*(a+b*ln(c*(d+e*x^(1/2))^n))/e^6+40/9*b*d^3*n*( d+e*x^(1/2))^3*(a+b*ln(c*(d+e*x^(1/2))^n))/e^6-5/2*b*d^2*n*(d+e*x^(1/2))^4 *(a+b*ln(c*(d+e*x^(1/2))^n))/e^6+4/5*b*d*n*(d+e*x^(1/2))^5*(a+b*ln(c*(d+e* x^(1/2))^n))/e^6-1/9*b*n*(d+e*x^(1/2))^6*(a+b*ln(c*(d+e*x^(1/2))^n))/e^6-2 /3*b*d^6*n*ln(d+e*x^(1/2))*(a+b*ln(c*(d+e*x^(1/2))^n))/e^6+1/3*x^3*(a+b*ln (c*(d+e*x^(1/2))^n))^2
Time = 0.39 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.65 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {e \sqrt {x} \left (1800 a^2 e^5 x^{5/2}+60 a b n \left (60 d^5-30 d^4 e \sqrt {x}+20 d^3 e^2 x-15 d^2 e^3 x^{3/2}+12 d e^4 x^2-10 e^5 x^{5/2}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e \sqrt {x}-1140 d^3 e^2 x+555 d^2 e^3 x^{3/2}-264 d e^4 x^2+100 e^5 x^{5/2}\right )\right )+180 b d^6 n (-20 a+49 b n) \log \left (d+e \sqrt {x}\right )-60 b e \sqrt {x} \left (-60 a e^5 x^{5/2}+b n \left (-60 d^5+30 d^4 e \sqrt {x}-20 d^3 e^2 x+15 d^2 e^3 x^{3/2}-12 d e^4 x^2+10 e^5 x^{5/2}\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-1800 b^2 \left (d^6-e^6 x^3\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )}{5400 e^6} \] Input:
Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2,x]
Output:
(e*Sqrt[x]*(1800*a^2*e^5*x^(5/2) + 60*a*b*n*(60*d^5 - 30*d^4*e*Sqrt[x] + 2 0*d^3*e^2*x - 15*d^2*e^3*x^(3/2) + 12*d*e^4*x^2 - 10*e^5*x^(5/2)) + b^2*n^ 2*(-8820*d^5 + 2610*d^4*e*Sqrt[x] - 1140*d^3*e^2*x + 555*d^2*e^3*x^(3/2) - 264*d*e^4*x^2 + 100*e^5*x^(5/2))) + 180*b*d^6*n*(-20*a + 49*b*n)*Log[d + e*Sqrt[x]] - 60*b*e*Sqrt[x]*(-60*a*e^5*x^(5/2) + b*n*(-60*d^5 + 30*d^4*e*S qrt[x] - 20*d^3*e^2*x + 15*d^2*e^3*x^(3/2) - 12*d*e^4*x^2 + 10*e^5*x^(5/2) ))*Log[c*(d + e*Sqrt[x])^n] - 1800*b^2*(d^6 - e^6*x^3)*Log[c*(d + e*Sqrt[x ])^n]^2)/(5400*e^6)
Time = 0.97 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.62, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2904, 2845, 2858, 27, 2772, 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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 2 \int x^{5/2} \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2d\sqrt {x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {1}{3} b e n \int \frac {x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{d+e \sqrt {x}}d\sqrt {x}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {1}{3} b n \int x^{5/2} \left (a+b \log \left (c x^{n/2}\right )\right )d\left (d+e \sqrt {x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {b n \int e^6 x^{5/2} \left (a+b \log \left (c x^{n/2}\right )\right )d\left (d+e \sqrt {x}\right )}{3 e^6}\right )\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {b n \left (-b n \int \left (\frac {\log \left (d+e \sqrt {x}\right ) d^6}{\sqrt {x}}-6 d^5+\frac {15}{2} \left (d+e \sqrt {x}\right ) d^4-\frac {20 x d^3}{3}+\frac {15}{4} x^{3/2} d^2-\frac {6 x^2 d}{5}+\frac {x^{5/2}}{6}\right )d\left (d+e \sqrt {x}\right )+d^6 \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-6 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )+\frac {15}{2} d^4 x \left (a+b \log \left (c x^{n/2}\right )\right )-\frac {20}{3} d^3 x^{3/2} \left (a+b \log \left (c x^{n/2}\right )\right )+\frac {15}{4} d^2 x^2 \left (a+b \log \left (c x^{n/2}\right )\right )-\frac {6}{5} d x^{5/2} \left (a+b \log \left (c x^{n/2}\right )\right )+\frac {1}{6} x^3 \left (a+b \log \left (c x^{n/2}\right )\right )\right )}{3 e^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\frac {1}{6} x^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2-\frac {b n \left (d^6 \log \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )-6 d^5 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c x^{n/2}\right )\right )+\frac {15}{2} d^4 x \left (a+b \log \left (c x^{n/2}\right )\right )-\frac {20}{3} d^3 x^{3/2} \left (a+b \log \left (c x^{n/2}\right )\right )+\frac {15}{4} d^2 x^2 \left (a+b \log \left (c x^{n/2}\right )\right )-\frac {6}{5} d x^{5/2} \left (a+b \log \left (c x^{n/2}\right )\right )+\frac {1}{6} x^3 \left (a+b \log \left (c x^{n/2}\right )\right )-b n \left (\frac {1}{2} d^6 \log ^2\left (d+e \sqrt {x}\right )-6 d^5 \left (d+e \sqrt {x}\right )+\frac {15 d^4 x}{4}-\frac {20}{9} d^3 x^{3/2}+\frac {15 d^2 x^2}{16}-\frac {6}{25} d x^{5/2}+\frac {x^3}{36}\right )\right )}{3 e^6}\right )\) |
Input:
Int[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2,x]
Output:
2*((x^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/6 - (b*n*(-(b*n*(-6*d^5*(d + e *Sqrt[x]) + (15*d^4*x)/4 - (20*d^3*x^(3/2))/9 + (15*d^2*x^2)/16 - (6*d*x^( 5/2))/25 + x^3/36 + (d^6*Log[d + e*Sqrt[x]]^2)/2)) - 6*d^5*(d + e*Sqrt[x]) *(a + b*Log[c*x^(n/2)]) + (15*d^4*x*(a + b*Log[c*x^(n/2)]))/2 - (20*d^3*x^ (3/2)*(a + b*Log[c*x^(n/2)]))/3 + (15*d^2*x^2*(a + b*Log[c*x^(n/2)]))/4 - (6*d*x^(5/2)*(a + b*Log[c*x^(n/2)]))/5 + (x^3*(a + b*Log[c*x^(n/2)]))/6 + d^6*Log[d + e*Sqrt[x]]*(a + b*Log[c*x^(n/2)])))/(3*e^6))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}d x\]
Input:
int(x^2*(a+b*ln(c*(d+e*x^(1/2))^n))^2,x)
Output:
int(x^2*(a+b*ln(c*(d+e*x^(1/2))^n))^2,x)
Time = 0.13 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.01 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {1800 \, b^{2} e^{6} x^{3} \log \left (c\right )^{2} + 100 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n + 18 \, a^{2} e^{6}\right )} x^{3} + 15 \, {\left (37 \, b^{2} d^{2} e^{4} n^{2} - 60 \, a b d^{2} e^{4} n\right )} x^{2} + 1800 \, {\left (b^{2} e^{6} n^{2} x^{3} - b^{2} d^{6} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} + 90 \, {\left (29 \, b^{2} d^{4} e^{2} n^{2} - 20 \, a b d^{4} e^{2} n\right )} x - 60 \, {\left (15 \, b^{2} d^{2} e^{4} n^{2} x^{2} + 30 \, b^{2} d^{4} e^{2} n^{2} x - 147 \, b^{2} d^{6} n^{2} + 60 \, a b d^{6} n + 10 \, {\left (b^{2} e^{6} n^{2} - 6 \, a b e^{6} n\right )} x^{3} - 60 \, {\left (b^{2} e^{6} n x^{3} - b^{2} d^{6} n\right )} \log \left (c\right ) - 4 \, {\left (3 \, b^{2} d e^{5} n^{2} x^{2} + 5 \, b^{2} d^{3} e^{3} n^{2} x + 15 \, b^{2} d^{5} e n^{2}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) - 300 \, {\left (3 \, b^{2} d^{2} e^{4} n x^{2} + 6 \, b^{2} d^{4} e^{2} n x + 2 \, {\left (b^{2} e^{6} n - 6 \, a b e^{6}\right )} x^{3}\right )} \log \left (c\right ) - 12 \, {\left (735 \, b^{2} d^{5} e n^{2} - 300 \, a b d^{5} e n + 2 \, {\left (11 \, b^{2} d e^{5} n^{2} - 30 \, a b d e^{5} n\right )} x^{2} + 5 \, {\left (19 \, b^{2} d^{3} e^{3} n^{2} - 20 \, a b d^{3} e^{3} n\right )} x - 20 \, {\left (3 \, b^{2} d e^{5} n x^{2} + 5 \, b^{2} d^{3} e^{3} n x + 15 \, b^{2} d^{5} e n\right )} \log \left (c\right )\right )} \sqrt {x}}{5400 \, e^{6}} \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^2,x, algorithm="fricas")
Output:
1/5400*(1800*b^2*e^6*x^3*log(c)^2 + 100*(b^2*e^6*n^2 - 6*a*b*e^6*n + 18*a^ 2*e^6)*x^3 + 15*(37*b^2*d^2*e^4*n^2 - 60*a*b*d^2*e^4*n)*x^2 + 1800*(b^2*e^ 6*n^2*x^3 - b^2*d^6*n^2)*log(e*sqrt(x) + d)^2 + 90*(29*b^2*d^4*e^2*n^2 - 2 0*a*b*d^4*e^2*n)*x - 60*(15*b^2*d^2*e^4*n^2*x^2 + 30*b^2*d^4*e^2*n^2*x - 1 47*b^2*d^6*n^2 + 60*a*b*d^6*n + 10*(b^2*e^6*n^2 - 6*a*b*e^6*n)*x^3 - 60*(b ^2*e^6*n*x^3 - b^2*d^6*n)*log(c) - 4*(3*b^2*d*e^5*n^2*x^2 + 5*b^2*d^3*e^3* n^2*x + 15*b^2*d^5*e*n^2)*sqrt(x))*log(e*sqrt(x) + d) - 300*(3*b^2*d^2*e^4 *n*x^2 + 6*b^2*d^4*e^2*n*x + 2*(b^2*e^6*n - 6*a*b*e^6)*x^3)*log(c) - 12*(7 35*b^2*d^5*e*n^2 - 300*a*b*d^5*e*n + 2*(11*b^2*d*e^5*n^2 - 30*a*b*d*e^5*n) *x^2 + 5*(19*b^2*d^3*e^3*n^2 - 20*a*b*d^3*e^3*n)*x - 20*(3*b^2*d*e^5*n*x^2 + 5*b^2*d^3*e^3*n*x + 15*b^2*d^5*e*n)*log(c))*sqrt(x))/e^6
\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}\, dx \] Input:
integrate(x**2*(a+b*ln(c*(d+e*x**(1/2))**n))**2,x)
Output:
Integral(x**2*(a + b*log(c*(d + e*sqrt(x))**n))**2, x)
Time = 0.05 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.68 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \, b^{2} x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + \frac {2}{3} \, a b x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{2} x^{3} - \frac {1}{90} \, a b e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} - \frac {1}{5400} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e \sqrt {x} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{3} - 12 \, d e^{4} x^{\frac {5}{2}} + 15 \, d^{2} e^{3} x^{2} - 20 \, d^{3} e^{2} x^{\frac {3}{2}} + 30 \, d^{4} e x - 60 \, d^{5} \sqrt {x}}{e^{6}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{3} - 264 \, d e^{5} x^{\frac {5}{2}} + 555 \, d^{2} e^{4} x^{2} + 1800 \, d^{6} \log \left (e \sqrt {x} + d\right )^{2} - 1140 \, d^{3} e^{3} x^{\frac {3}{2}} + 2610 \, d^{4} e^{2} x + 8820 \, d^{6} \log \left (e \sqrt {x} + d\right ) - 8820 \, d^{5} e \sqrt {x}\right )} n^{2}}{e^{6}}\right )} b^{2} \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^2,x, algorithm="maxima")
Output:
1/3*b^2*x^3*log((e*sqrt(x) + d)^n*c)^2 + 2/3*a*b*x^3*log((e*sqrt(x) + d)^n *c) + 1/3*a^2*x^3 - 1/90*a*b*e*n*(60*d^6*log(e*sqrt(x) + d)/e^7 + (10*e^5* x^3 - 12*d*e^4*x^(5/2) + 15*d^2*e^3*x^2 - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x - 60*d^5*sqrt(x))/e^6) - 1/5400*(60*e*n*(60*d^6*log(e*sqrt(x) + d)/e^7 + ( 10*e^5*x^3 - 12*d*e^4*x^(5/2) + 15*d^2*e^3*x^2 - 20*d^3*e^2*x^(3/2) + 30*d ^4*e*x - 60*d^5*sqrt(x))/e^6)*log((e*sqrt(x) + d)^n*c) - (100*e^6*x^3 - 26 4*d*e^5*x^(5/2) + 555*d^2*e^4*x^2 + 1800*d^6*log(e*sqrt(x) + d)^2 - 1140*d ^3*e^3*x^(3/2) + 2610*d^4*e^2*x + 8820*d^6*log(e*sqrt(x) + d) - 8820*d^5*e *sqrt(x))*n^2/e^6)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (412) = 824\).
Time = 0.13 (sec) , antiderivative size = 930, normalized size of antiderivative = 1.94 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^2,x, algorithm="giac")
Output:
1/5400*(1800*b^2*e*x^3*log(c)^2 + 3600*a*b*e*x^3*log(c) + 1800*a^2*e*x^3 + (1800*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)^2/e^5 - 10800*(e*sqrt(x) + d)^ 5*d*log(e*sqrt(x) + d)^2/e^5 + 27000*(e*sqrt(x) + d)^4*d^2*log(e*sqrt(x) + d)^2/e^5 - 36000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)^2/e^5 + 27000*( e*sqrt(x) + d)^2*d^4*log(e*sqrt(x) + d)^2/e^5 - 10800*(e*sqrt(x) + d)*d^5* log(e*sqrt(x) + d)^2/e^5 - 600*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 + 4320*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)/e^5 - 13500*(e*sqrt(x) + d)^4* d^2*log(e*sqrt(x) + d)/e^5 + 24000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d )/e^5 - 27000*(e*sqrt(x) + d)^2*d^4*log(e*sqrt(x) + d)/e^5 + 21600*(e*sqrt (x) + d)*d^5*log(e*sqrt(x) + d)/e^5 + 100*(e*sqrt(x) + d)^6/e^5 - 864*(e*s qrt(x) + d)^5*d/e^5 + 3375*(e*sqrt(x) + d)^4*d^2/e^5 - 8000*(e*sqrt(x) + d )^3*d^3/e^5 + 13500*(e*sqrt(x) + d)^2*d^4/e^5 - 21600*(e*sqrt(x) + d)*d^5/ e^5)*b^2*n^2 + 60*(60*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 - 360*(e*sq rt(x) + d)^5*d*log(e*sqrt(x) + d)/e^5 + 900*(e*sqrt(x) + d)^4*d^2*log(e*sq rt(x) + d)/e^5 - 1200*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)/e^5 + 900*( e*sqrt(x) + d)^2*d^4*log(e*sqrt(x) + d)/e^5 - 360*(e*sqrt(x) + d)*d^5*log( e*sqrt(x) + d)/e^5 - 10*(e*sqrt(x) + d)^6/e^5 + 72*(e*sqrt(x) + d)^5*d/e^5 - 225*(e*sqrt(x) + d)^4*d^2/e^5 + 400*(e*sqrt(x) + d)^3*d^3/e^5 - 450*(e* sqrt(x) + d)^2*d^4/e^5 + 360*(e*sqrt(x) + d)*d^5/e^5)*b^2*n*log(c) + 60*(6 0*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 - 360*(e*sqrt(x) + d)^5*d*lo...
Time = 16.24 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.90 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {a^2\,x^3}{3}+\frac {b^2\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3}+\frac {b^2\,n^2\,x^3}{54}+\frac {2\,a\,b\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3}-\frac {b^2\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3\,e^6}-\frac {a\,b\,n\,x^3}{9}-\frac {b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{9}+\frac {49\,b^2\,d^6\,n^2\,\ln \left (d+e\,\sqrt {x}\right )}{30\,e^6}+\frac {37\,b^2\,d^2\,n^2\,x^2}{360\,e^2}-\frac {19\,b^2\,d^3\,n^2\,x^{3/2}}{90\,e^3}-\frac {49\,b^2\,d^5\,n^2\,\sqrt {x}}{30\,e^5}-\frac {11\,b^2\,d\,n^2\,x^{5/2}}{225\,e}+\frac {29\,b^2\,d^4\,n^2\,x}{60\,e^4}-\frac {b^2\,d^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{6\,e^2}+\frac {2\,b^2\,d^3\,n\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{9\,e^3}+\frac {2\,b^2\,d^5\,n\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^5}+\frac {2\,a\,b\,d\,n\,x^{5/2}}{15\,e}-\frac {a\,b\,d^4\,n\,x}{3\,e^4}-\frac {2\,a\,b\,d^6\,n\,\ln \left (d+e\,\sqrt {x}\right )}{3\,e^6}+\frac {2\,b^2\,d\,n\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{15\,e}-\frac {b^2\,d^4\,n\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^4}-\frac {a\,b\,d^2\,n\,x^2}{6\,e^2}+\frac {2\,a\,b\,d^3\,n\,x^{3/2}}{9\,e^3}+\frac {2\,a\,b\,d^5\,n\,\sqrt {x}}{3\,e^5} \] Input:
int(x^2*(a + b*log(c*(d + e*x^(1/2))^n))^2,x)
Output:
(a^2*x^3)/3 + (b^2*x^3*log(c*(d + e*x^(1/2))^n)^2)/3 + (b^2*n^2*x^3)/54 + (2*a*b*x^3*log(c*(d + e*x^(1/2))^n))/3 - (b^2*d^6*log(c*(d + e*x^(1/2))^n) ^2)/(3*e^6) - (a*b*n*x^3)/9 - (b^2*n*x^3*log(c*(d + e*x^(1/2))^n))/9 + (49 *b^2*d^6*n^2*log(d + e*x^(1/2)))/(30*e^6) + (37*b^2*d^2*n^2*x^2)/(360*e^2) - (19*b^2*d^3*n^2*x^(3/2))/(90*e^3) - (49*b^2*d^5*n^2*x^(1/2))/(30*e^5) - (11*b^2*d*n^2*x^(5/2))/(225*e) + (29*b^2*d^4*n^2*x)/(60*e^4) - (b^2*d^2*n *x^2*log(c*(d + e*x^(1/2))^n))/(6*e^2) + (2*b^2*d^3*n*x^(3/2)*log(c*(d + e *x^(1/2))^n))/(9*e^3) + (2*b^2*d^5*n*x^(1/2)*log(c*(d + e*x^(1/2))^n))/(3* e^5) + (2*a*b*d*n*x^(5/2))/(15*e) - (a*b*d^4*n*x)/(3*e^4) - (2*a*b*d^6*n*l og(d + e*x^(1/2)))/(3*e^6) + (2*b^2*d*n*x^(5/2)*log(c*(d + e*x^(1/2))^n))/ (15*e) - (b^2*d^4*n*x*log(c*(d + e*x^(1/2))^n))/(3*e^4) - (a*b*d^2*n*x^2)/ (6*e^2) + (2*a*b*d^3*n*x^(3/2))/(9*e^3) + (2*a*b*d^5*n*x^(1/2))/(3*e^5)
Time = 0.16 (sec) , antiderivative size = 439, normalized size of antiderivative = 0.91 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {3600 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{5} e n +1200 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{3} e^{3} n x +720 \sqrt {x}\, \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d \,e^{5} n \,x^{2}+3600 \sqrt {x}\, a b \,d^{5} e n +1200 \sqrt {x}\, a b \,d^{3} e^{3} n x +720 \sqrt {x}\, a b d \,e^{5} n \,x^{2}-8820 \sqrt {x}\, b^{2} d^{5} e \,n^{2}-1140 \sqrt {x}\, b^{2} d^{3} e^{3} n^{2} x -264 \sqrt {x}\, b^{2} d \,e^{5} n^{2} x^{2}-1800 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{2} d^{6}+1800 \mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right )^{2} b^{2} e^{6} x^{3}-3600 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a b \,d^{6}+3600 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) a b \,e^{6} x^{3}+8820 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{6} n -1800 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{4} e^{2} n x -900 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} d^{2} e^{4} n \,x^{2}-600 \,\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{n} c \right ) b^{2} e^{6} n \,x^{3}+1800 a^{2} e^{6} x^{3}-1800 a b \,d^{4} e^{2} n x -900 a b \,d^{2} e^{4} n \,x^{2}-600 a b \,e^{6} n \,x^{3}+2610 b^{2} d^{4} e^{2} n^{2} x +555 b^{2} d^{2} e^{4} n^{2} x^{2}+100 b^{2} e^{6} n^{2} x^{3}}{5400 e^{6}} \] Input:
int(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^2,x)
Output:
(3600*sqrt(x)*log((sqrt(x)*e + d)**n*c)*b**2*d**5*e*n + 1200*sqrt(x)*log(( sqrt(x)*e + d)**n*c)*b**2*d**3*e**3*n*x + 720*sqrt(x)*log((sqrt(x)*e + d)* *n*c)*b**2*d*e**5*n*x**2 + 3600*sqrt(x)*a*b*d**5*e*n + 1200*sqrt(x)*a*b*d* *3*e**3*n*x + 720*sqrt(x)*a*b*d*e**5*n*x**2 - 8820*sqrt(x)*b**2*d**5*e*n** 2 - 1140*sqrt(x)*b**2*d**3*e**3*n**2*x - 264*sqrt(x)*b**2*d*e**5*n**2*x**2 - 1800*log((sqrt(x)*e + d)**n*c)**2*b**2*d**6 + 1800*log((sqrt(x)*e + d)* *n*c)**2*b**2*e**6*x**3 - 3600*log((sqrt(x)*e + d)**n*c)*a*b*d**6 + 3600*l og((sqrt(x)*e + d)**n*c)*a*b*e**6*x**3 + 8820*log((sqrt(x)*e + d)**n*c)*b* *2*d**6*n - 1800*log((sqrt(x)*e + d)**n*c)*b**2*d**4*e**2*n*x - 900*log((s qrt(x)*e + d)**n*c)*b**2*d**2*e**4*n*x**2 - 600*log((sqrt(x)*e + d)**n*c)* b**2*e**6*n*x**3 + 1800*a**2*e**6*x**3 - 1800*a*b*d**4*e**2*n*x - 900*a*b* d**2*e**4*n*x**2 - 600*a*b*e**6*n*x**3 + 2610*b**2*d**4*e**2*n**2*x + 555* b**2*d**2*e**4*n**2*x**2 + 100*b**2*e**6*n**2*x**3)/(5400*e**6)