\(\int x^2 (a+b \log (c (d+e \sqrt {x})^n))^3 \, dx\) [415]

Optimal result

Integrand size = 24, antiderivative size = 907 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {12 b^3 d^5 n^3 \sqrt {x}}{e^5}-\frac {12 b^3 d^5 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^6}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}-\frac {40 b^2 d^3 n^2 \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 {15 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{8 e^6}-\frac {12 b^2 d n^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{18 e^6}+\frac {6 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 )^2}{e^6}-\frac {15 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 )^2}{2 e^6}+\frac {20 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 )^2}{3 e^6}-\frac {15 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}{4 e^6}+\frac {6 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 )^2}{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 )^2}{6 e^6}-\frac {2 d^5 \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 {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6} \]

[Out]

12*b^3*d^5*n^3*x^(1/2)/e^5-15/4*b^3*d^4*n^3*(d+e*x^(1/2))^2/e^6+40/27*b^3*d^3*n^3*(d+e*x^(1/2))^3/e^6-15/32*b^
3*d^2*n^3*(d+e*x^(1/2))^4/e^6+12/125*b^3*d*n^3*(d+e*x^(1/2))^5/e^6+1/18*b^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d
+e*x^(1/2))^6/e^6-1/6*b*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^6/e^6-2*d^5*(a+b*ln(c*(d+e*x^(1/2))^n))^
3*(d+e*x^(1/2))/e^6+5*d^4*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^2/e^6-20/3*d^3*(a+b*ln(c*(d+e*x^(1/2))^n
))^3*(d+e*x^(1/2))^3/e^6+5*d^2*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^4/e^6-2*d*(a+b*ln(c*(d+e*x^(1/2))^n
))^3*(d+e*x^(1/2))^5/e^6-1/108*b^3*n^3*(d+e*x^(1/2))^6/e^6+1/3*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^6/e
^6-12*a*b^2*d^5*n^2*x^(1/2)/e^5-12*b^3*d^5*n^2*ln(c*(d+e*x^(1/2))^n)*(d+e*x^(1/2))/e^6+6*b*d^5*n*(a+b*ln(c*(d+
e*x^(1/2))^n))^2*(d+e*x^(1/2))/e^6+15/2*b^2*d^4*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^2/e^6-15/2*b*d^4
*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^2/e^6-40/9*b^2*d^3*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2)
)^3/e^6+20/3*b*d^3*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^3/e^6+15/8*b^2*d^2*n^2*(a+b*ln(c*(d+e*x^(1/2)
)^n))*(d+e*x^(1/2))^4/e^6-15/4*b*d^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^4/e^6-12/25*b^2*d*n^2*(a+b*
ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^5/e^6+6/5*b*d*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^5/e^6

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}+\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^6}{3 e^6}-\frac {b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^6}{6 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^6}{18 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {2 d \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^5}{e^6}+\frac {6 b d n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^5}{5 e^6}-\frac {12 b^2 d n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^5}{25 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {5 d^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^4}{e^6}-\frac {15 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^4}{4 e^6}+\frac {15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^4}{8 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {20 d^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^3}{3 e^6}+\frac {20 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^3}{3 e^6}-\frac {40 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^3}{9 e^6}-\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {5 d^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )^2}{e^6}-\frac {15 b d^4 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )^2}{2 e^6}+\frac {15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (d+e \sqrt {x}\right )^2}{2 e^6}-\frac {2 d^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \left (d+e \sqrt {x}\right )}{e^6}+\frac {6 b d^5 n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (d+e \sqrt {x}\right )}{e^6}-\frac {12 b^3 d^5 n^2 \log \left (c \left (d+e \sqrt {x}\right )^n\right ) \left (d+e \sqrt {x}\right )}{e^6}+\frac {12 b^3 d^5 n^3 \sqrt {x}}{e^5}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5} \]

[In]

Int[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3,x]

[Out]

(-15*b^3*d^4*n^3*(d + e*Sqrt[x])^2)/(4*e^6) + (40*b^3*d^3*n^3*(d + e*Sqrt[x])^3)/(27*e^6) - (15*b^3*d^2*n^3*(d
+ e*Sqrt[x])^4)/(32*e^6) + (12*b^3*d*n^3*(d + e*Sqrt[x])^5)/(125*e^6) - (b^3*n^3*(d + e*Sqrt[x])^6)/(108*e^6)
- (12*a*b^2*d^5*n^2*Sqrt[x])/e^5 + (12*b^3*d^5*n^3*Sqrt[x])/e^5 - (12*b^3*d^5*n^2*(d + e*Sqrt[x])*Log[c*(d +
e*Sqrt[x])^n])/e^6 + (15*b^2*d^4*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^6) - (40*b^2*d^3
*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(9*e^6) + (15*b^2*d^2*n^2*(d + e*Sqrt[x])^4*(a + b*Lo
g[c*(d + e*Sqrt[x])^n]))/(8*e^6) - (12*b^2*d*n^2*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(25*e^6)
+ (b^2*n^2*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(18*e^6) + (6*b*d^5*n*(d + e*Sqrt[x])*(a + b*Lo
g[c*(d + e*Sqrt[x])^n])^2)/e^6 - (15*b*d^4*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^6) + (
20*b*d^3*n*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(3*e^6) - (15*b*d^2*n*(d + e*Sqrt[x])^4*(a +
b*Log[c*(d + e*Sqrt[x])^n])^2)/(4*e^6) + (6*b*d*n*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(5*e^6
) - (b*n*(d + e*Sqrt[x])^6*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(6*e^6) - (2*d^5*(d + e*Sqrt[x])*(a + b*Log[c*(
d + e*Sqrt[x])^n])^3)/e^6 + (5*d^4*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 - (20*d^3*(d + e*
Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/(3*e^6) + (5*d^2*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])
^n])^3)/e^6 - (2*d*(d + e*Sqrt[x])^5*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^6 + ((d + e*Sqrt[x])^6*(a + b*Log[c
*(d + e*Sqrt[x])^n])^3)/(3*e^6)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[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])

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5}-\frac {(10 d) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^5} \\ & = \frac {2 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {(10 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (20 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (20 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (10 d^4\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (2 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = -\frac {2 d^5 \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 {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}-\frac {(b n) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {(6 b d n) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (15 b d^2 n\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (20 b d^3 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (15 b d^4 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6}+\frac {\left (6 b d^5 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = \frac {6 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 )^2}{e^6}-\frac {15 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 )^2}{2 e^6}+\frac {20 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 )^2}{3 e^6}-\frac {15 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}{4 e^6}+\frac {6 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 )^2}{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 )^2}{6 e^6}-\frac {2 d^5 \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 {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{3 e^6}-\frac {\left (12 b^2 d n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{5 e^6}+\frac {\left (15 b^2 d^2 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{2 e^6}-\frac {\left (40 b^2 d^3 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{3 e^6}+\frac {\left (15 b^2 d^4 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^6}-\frac {\left (12 b^2 d^5 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = -\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}-\frac {40 b^2 d^3 n^2 \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 {15 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{8 e^6}-\frac {12 b^2 d n^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{18 e^6}+\frac {6 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 )^2}{e^6}-\frac {15 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 )^2}{2 e^6}+\frac {20 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 )^2}{3 e^6}-\frac {15 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}{4 e^6}+\frac {6 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 )^2}{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 )^2}{6 e^6}-\frac {2 d^5 \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 {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}-\frac {\left (12 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt {x}\right )}{e^6} \\ & = -\frac {15 b^3 d^4 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^6}+\frac {40 b^3 d^3 n^3 \left (d+e \sqrt {x}\right )^3}{27 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^4}{32 e^6}+\frac {12 b^3 d n^3 \left (d+e \sqrt {x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt {x}\right )^6}{108 e^6}-\frac {12 a b^2 d^5 n^2 \sqrt {x}}{e^5}+\frac {12 b^3 d^5 n^3 \sqrt {x}}{e^5}-\frac {12 b^3 d^5 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^6}+\frac {15 b^2 d^4 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^6}-\frac {40 b^2 d^3 n^2 \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 {15 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{8 e^6}-\frac {12 b^2 d n^2 \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{25 e^6}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{18 e^6}+\frac {6 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 )^2}{e^6}-\frac {15 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 )^2}{2 e^6}+\frac {20 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 )^2}{3 e^6}-\frac {15 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}{4 e^6}+\frac {6 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 )^2}{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 )^2}{6 e^6}-\frac {2 d^5 \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 {5 d^4 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {20 d^3 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6}+\frac {5 d^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}-\frac {2 d \left (d+e \sqrt {x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^6}+\frac {\left (d+e \sqrt {x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{3 e^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.64 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt {x} \left (809340 d^5-140070 d^4 e \sqrt {x}+41180 d^3 e^2 x-13785 d^2 e^3 x^{3/2}+4368 d e^4 x^2-1000 e^5 x^{5/2}\right )+1800 a^2 b n \left (147 d^6+60 d^5 e \sqrt {x}-30 d^4 e^2 x+20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2+12 d e^5 x^{5/2}-10 e^6 x^3\right )-36000 a^3 \left (d^6-e^6 x^3\right )+60 a b^2 n^2 \left (8111 d^6-8820 d^5 e \sqrt {x}+2610 d^4 e^2 x-1140 d^3 e^3 x^{3/2}+555 d^2 e^4 x^2-264 d e^5 x^{5/2}+100 e^6 x^3\right )-60 b \left (b^2 n^2 \left (13489 d^6+8820 d^5 e \sqrt {x}-2610 d^4 e^2 x+1140 d^3 e^3 x^{3/2}-555 d^2 e^4 x^2+264 d e^5 x^{5/2}-100 e^6 x^3\right )-60 a b n \left (147 d^6+60 d^5 e \sqrt {x}-30 d^4 e^2 x+20 d^3 e^3 x^{3/2}-15 d^2 e^4 x^2+12 d e^5 x^{5/2}-10 e^6 x^3\right )+1800 a^2 \left (d^6-e^6 x^3\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^3\right )+b n \left (-147 d^6-60 d^5 e \sqrt {x}+30 d^4 e^2 x-20 d^3 e^3 x^{3/2}+15 d^2 e^4 x^2-12 d e^5 x^{5/2}+10 e^6 x^3\right )\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^3\right ) \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )}{108000 e^6} \]

[In]

Integrate[x^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^3,x]

[Out]

(b^3*e*n^3*Sqrt[x]*(809340*d^5 - 140070*d^4*e*Sqrt[x] + 41180*d^3*e^2*x - 13785*d^2*e^3*x^(3/2) + 4368*d*e^4*x
^2 - 1000*e^5*x^(5/2)) + 1800*a^2*b*n*(147*d^6 + 60*d^5*e*Sqrt[x] - 30*d^4*e^2*x + 20*d^3*e^3*x^(3/2) - 15*d^2
*e^4*x^2 + 12*d*e^5*x^(5/2) - 10*e^6*x^3) - 36000*a^3*(d^6 - e^6*x^3) + 60*a*b^2*n^2*(8111*d^6 - 8820*d^5*e*Sq
rt[x] + 2610*d^4*e^2*x - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x^2 - 264*d*e^5*x^(5/2) + 100*e^6*x^3) - 60*b*(b^2
*n^2*(13489*d^6 + 8820*d^5*e*Sqrt[x] - 2610*d^4*e^2*x + 1140*d^3*e^3*x^(3/2) - 555*d^2*e^4*x^2 + 264*d*e^5*x^(
5/2) - 100*e^6*x^3) - 60*a*b*n*(147*d^6 + 60*d^5*e*Sqrt[x] - 30*d^4*e^2*x + 20*d^3*e^3*x^(3/2) - 15*d^2*e^4*x^
2 + 12*d*e^5*x^(5/2) - 10*e^6*x^3) + 1800*a^2*(d^6 - e^6*x^3))*Log[c*(d + e*Sqrt[x])^n] - 1800*b^2*(60*a*(d^6
- e^6*x^3) + b*n*(-147*d^6 - 60*d^5*e*Sqrt[x] + 30*d^4*e^2*x - 20*d^3*e^3*x^(3/2) + 15*d^2*e^4*x^2 - 12*d*e^5*
x^(5/2) + 10*e^6*x^3))*Log[c*(d + e*Sqrt[x])^n]^2 - 36000*b^3*(d^6 - e^6*x^3)*Log[c*(d + e*Sqrt[x])^n]^3)/(108
000*e^6)

Maple [F]

\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{3}d x\]

[In]

int(x^2*(a+b*ln(c*(d+e*x^(1/2))^n))^3,x)

[Out]

int(x^2*(a+b*ln(c*(d+e*x^(1/2))^n))^3,x)

Fricas [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 1197, normalized size of antiderivative = 1.32 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="fricas")

[Out]

1/108000*(36000*b^3*e^6*x^3*log(c)^3 - 1000*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n - 36*a^3*e^6)*x^3
+ 36000*(b^3*e^6*n^3*x^3 - b^3*d^6*n^3)*log(e*sqrt(x) + d)^3 - 15*(919*b^3*d^2*e^4*n^3 - 2220*a*b^2*d^2*e^4*n^
2 + 1800*a^2*b*d^2*e^4*n)*x^2 - 1800*(15*b^3*d^2*e^4*n^3*x^2 + 30*b^3*d^4*e^2*n^3*x - 147*b^3*d^6*n^3 + 60*a*b
^2*d^6*n^2 + 10*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2)*x^3 - 60*(b^3*e^6*n^2*x^3 - b^3*d^6*n^2)*log(c) - 4*(3*b^3*d*e
^5*n^3*x^2 + 5*b^3*d^3*e^3*n^3*x + 15*b^3*d^5*e*n^3)*sqrt(x))*log(e*sqrt(x) + d)^2 - 9000*(3*b^3*d^2*e^4*n*x^2
+ 6*b^3*d^4*e^2*n*x + 2*(b^3*e^6*n - 6*a*b^2*e^6)*x^3)*log(c)^2 - 30*(4669*b^3*d^4*e^2*n^3 - 5220*a*b^2*d^4*e
^2*n^2 + 1800*a^2*b*d^4*e^2*n)*x - 60*(13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^6*n - 100*(b^3*e^
6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n)*x^3 - 15*(37*b^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x^2 - 1800*(b^3
*e^6*n*x^3 - b^3*d^6*n)*log(c)^2 - 90*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)*x + 60*(15*b^3*d^2*e^4*n^2*x
^2 + 30*b^3*d^4*e^2*n^2*x - 147*b^3*d^6*n^2 + 60*a*b^2*d^6*n + 10*(b^3*e^6*n^2 - 6*a*b^2*e^6*n)*x^3)*log(c) +
12*(735*b^3*d^5*e*n^3 - 300*a*b^2*d^5*e*n^2 + 2*(11*b^3*d*e^5*n^3 - 30*a*b^2*d*e^5*n^2)*x^2 + 5*(19*b^3*d^3*e^
3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x - 20*(3*b^3*d*e^5*n^2*x^2 + 5*b^3*d^3*e^3*n^2*x + 15*b^3*d^5*e*n^2)*log(c))*sq
rt(x))*log(e*sqrt(x) + d) + 300*(20*(b^3*e^6*n^2 - 6*a*b^2*e^6*n + 18*a^2*b*e^6)*x^3 + 3*(37*b^3*d^2*e^4*n^2 -
60*a*b^2*d^2*e^4*n)*x^2 + 18*(29*b^3*d^4*e^2*n^2 - 20*a*b^2*d^4*e^2*n)*x)*log(c) + 4*(202335*b^3*d^5*e*n^3 -
132300*a*b^2*d^5*e*n^2 + 27000*a^2*b*d^5*e*n + 12*(91*b^3*d*e^5*n^3 - 330*a*b^2*d*e^5*n^2 + 450*a^2*b*d*e^5*n)
*x^2 + 1800*(3*b^3*d*e^5*n*x^2 + 5*b^3*d^3*e^3*n*x + 15*b^3*d^5*e*n)*log(c)^2 + 5*(2059*b^3*d^3*e^3*n^3 - 3420
*a*b^2*d^3*e^3*n^2 + 1800*a^2*b*d^3*e^3*n)*x - 180*(735*b^3*d^5*e*n^2 - 300*a*b^2*d^5*e*n + 2*(11*b^3*d*e^5*n^
2 - 30*a*b^2*d*e^5*n)*x^2 + 5*(19*b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x)*log(c))*sqrt(x))/e^6

Sympy [F]

\[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}\, dx \]

[In]

integrate(x**2*(a+b*ln(c*(d+e*x**(1/2))**n))**3,x)

[Out]

Integral(x**2*(a + b*log(c*(d + e*sqrt(x))**n))**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 666, normalized size of antiderivative = 0.73 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {1}{3} \, b^{3} x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + a b^{2} x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + a^{2} b x^{3} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{3} x^{3} - \frac {1}{60} \, a^{2} 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}{1800} \, {\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 )} a b^{2} - \frac {1}{108000} \, {\left (1800 \, 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 )^{2} + e n {\left (\frac {{\left (1000 \, e^{6} x^{3} + 36000 \, d^{6} \log \left (e \sqrt {x} + d\right )^{3} - 4368 \, d e^{5} x^{\frac {5}{2}} + 13785 \, d^{2} e^{4} x^{2} + 264600 \, d^{6} \log \left (e \sqrt {x} + d\right )^{2} - 41180 \, d^{3} e^{3} x^{\frac {3}{2}} + 140070 \, d^{4} e^{2} x + 809340 \, d^{6} \log \left (e \sqrt {x} + d\right ) - 809340 \, d^{5} e \sqrt {x}\right )} n^{2}}{e^{7}} - \frac {60 \, {\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 \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{e^{7}}\right )}\right )} b^{3} \]

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="maxima")

[Out]

1/3*b^3*x^3*log((e*sqrt(x) + d)^n*c)^3 + a*b^2*x^3*log((e*sqrt(x) + d)^n*c)^2 + a^2*b*x^3*log((e*sqrt(x) + d)^
n*c) + 1/3*a^3*x^3 - 1/60*a^2*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) - 1/1800*(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)*l
og((e*sqrt(x) + d)^n*c) - (100*e^6*x^3 - 264*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)*a*b^2 - 1/
108000*(1800*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)^2 + e*n*((1000*e^6*x^3 + 36000*d^6*log(e
*sqrt(x) + d)^3 - 4368*d*e^5*x^(5/2) + 13785*d^2*e^4*x^2 + 264600*d^6*log(e*sqrt(x) + d)^2 - 41180*d^3*e^3*x^(
3/2) + 140070*d^4*e^2*x + 809340*d^6*log(e*sqrt(x) + d) - 809340*d^5*e*sqrt(x))*n^2/e^7 - 60*(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 + 88
20*d^6*log(e*sqrt(x) + d) - 8820*d^5*e*sqrt(x))*n*log((e*sqrt(x) + d)^n*c)/e^7))*b^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2160 vs. \(2 (787) = 1574\).

Time = 0.35 (sec) , antiderivative size = 2160, normalized size of antiderivative = 2.38 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/2))^n))^3,x, algorithm="giac")

[Out]

1/108000*(36000*b^3*e*x^3*log(c)^3 + 108000*a*b^2*e*x^3*log(c)^2 + 108000*a^2*b*e*x^3*log(c) + (36000*(e*sqrt(
x) + d)^6*log(e*sqrt(x) + d)^3/e^5 - 216000*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)^3/e^5 + 540000*(e*sqrt(x) +
d)^4*d^2*log(e*sqrt(x) + d)^3/e^5 - 720000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)^3/e^5 + 540000*(e*sqrt(x)
+ d)^2*d^4*log(e*sqrt(x) + d)^3/e^5 - 216000*(e*sqrt(x) + d)*d^5*log(e*sqrt(x) + d)^3/e^5 - 18000*(e*sqrt(x)
+ d)^6*log(e*sqrt(x) + d)^2/e^5 + 129600*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)^2/e^5 - 405000*(e*sqrt(x) + d)
^4*d^2*log(e*sqrt(x) + d)^2/e^5 + 720000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)^2/e^5 - 810000*(e*sqrt(x) +
d)^2*d^4*log(e*sqrt(x) + d)^2/e^5 + 648000*(e*sqrt(x) + d)*d^5*log(e*sqrt(x) + d)^2/e^5 + 6000*(e*sqrt(x) + d)
^6*log(e*sqrt(x) + d)/e^5 - 51840*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)/e^5 + 202500*(e*sqrt(x) + d)^4*d^2*lo
g(e*sqrt(x) + d)/e^5 - 480000*(e*sqrt(x) + d)^3*d^3*log(e*sqrt(x) + d)/e^5 + 810000*(e*sqrt(x) + d)^2*d^4*log(
e*sqrt(x) + d)/e^5 - 1296000*(e*sqrt(x) + d)*d^5*log(e*sqrt(x) + d)/e^5 - 1000*(e*sqrt(x) + d)^6/e^5 + 10368*(
e*sqrt(x) + d)^5*d/e^5 - 50625*(e*sqrt(x) + d)^4*d^2/e^5 + 160000*(e*sqrt(x) + d)^3*d^3/e^5 - 405000*(e*sqrt(x
) + d)^2*d^4/e^5 + 1296000*(e*sqrt(x) + d)*d^5/e^5)*b^3*n^3 + 36000*a^3*e*x^3 + 60*(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*sqrt(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*s
qrt(x) + d)*d^5/e^5)*b^3*n^2*log(c) + 1800*(60*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 - 360*(e*sqrt(x) + d)^
5*d*log(e*sqrt(x) + d)/e^5 + 900*(e*sqrt(x) + d)^4*d^2*log(e*sqrt(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*sqr
t(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^3*n*log(c)^2 + 60*(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*sqr
t(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)*a*b^2*n^2 + 3600*(60*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 - 360*(
e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)/e^5 + 900*(e*sqrt(x) + d)^4*d^2*log(e*sqrt(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)*a*b^2*n*log(
c) + 1800*(60*(e*sqrt(x) + d)^6*log(e*sqrt(x) + d)/e^5 - 360*(e*sqrt(x) + d)^5*d*log(e*sqrt(x) + d)/e^5 + 900*
(e*sqrt(x) + d)^4*d^2*log(e*sqrt(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)*a^2*b*n)/e

Mupad [B] (verification not implemented)

Time = 9.29 (sec) , antiderivative size = 976, normalized size of antiderivative = 1.08 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx=\frac {a^3\,x^3}{3}+\frac {b^3\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3}{3}-\frac {b^3\,n^3\,x^3}{108}+a\,b^2\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2-\frac {b^3\,n\,x^3\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{6}+\frac {b^3\,n^2\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{18}+\frac {a\,b^2\,n^2\,x^3}{18}-\frac {b^3\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3}{3\,e^6}+a^2\,b\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )-\frac {a^2\,b\,n\,x^3}{6}-\frac {a\,b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3}-\frac {13489\,b^3\,d^6\,n^3\,\ln \left (d+e\,\sqrt {x}\right )}{1800\,e^6}-\frac {919\,b^3\,d^2\,n^3\,x^2}{7200\,e^2}+\frac {2059\,b^3\,d^3\,n^3\,x^{3/2}}{5400\,e^3}+\frac {13489\,b^3\,d^5\,n^3\,\sqrt {x}}{1800\,e^5}-\frac {a\,b^2\,d^6\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{e^6}+\frac {49\,b^3\,d^6\,n\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{20\,e^6}+\frac {91\,b^3\,d\,n^3\,x^{5/2}}{2250\,e}-\frac {4669\,b^3\,d^4\,n^3\,x}{3600\,e^4}-\frac {a^2\,b\,d^6\,n\,\ln \left (d+e\,\sqrt {x}\right )}{e^6}+\frac {b^3\,d\,n\,x^{5/2}\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{5\,e}-\frac {11\,b^3\,d\,n^2\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{75\,e}-\frac {b^3\,d^4\,n\,x\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{2\,e^4}+\frac {29\,b^3\,d^4\,n^2\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{20\,e^4}-\frac {a^2\,b\,d^2\,n\,x^2}{4\,e^2}-\frac {11\,a\,b^2\,d\,n^2\,x^{5/2}}{75\,e}+\frac {29\,a\,b^2\,d^4\,n^2\,x}{20\,e^4}+\frac {a^2\,b\,d^3\,n\,x^{3/2}}{3\,e^3}+\frac {a^2\,b\,d^5\,n\,\sqrt {x}}{e^5}+\frac {49\,a\,b^2\,d^6\,n^2\,\ln \left (d+e\,\sqrt {x}\right )}{10\,e^6}-\frac {b^3\,d^2\,n\,x^2\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{4\,e^2}+\frac {37\,b^3\,d^2\,n^2\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{120\,e^2}+\frac {b^3\,d^3\,n\,x^{3/2}\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{3\,e^3}-\frac {19\,b^3\,d^3\,n^2\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{30\,e^3}+\frac {b^3\,d^5\,n\,\sqrt {x}\,{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2}{e^5}-\frac {49\,b^3\,d^5\,n^2\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{10\,e^5}+\frac {37\,a\,b^2\,d^2\,n^2\,x^2}{120\,e^2}-\frac {19\,a\,b^2\,d^3\,n^2\,x^{3/2}}{30\,e^3}-\frac {49\,a\,b^2\,d^5\,n^2\,\sqrt {x}}{10\,e^5}+\frac {a^2\,b\,d\,n\,x^{5/2}}{5\,e}-\frac {a^2\,b\,d^4\,n\,x}{2\,e^4}+\frac {2\,a\,b^2\,d\,n\,x^{5/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{5\,e}-\frac {a\,b^2\,d^4\,n\,x\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{e^4}-\frac {a\,b^2\,d^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{2\,e^2}+\frac {2\,a\,b^2\,d^3\,n\,x^{3/2}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{3\,e^3}+\frac {2\,a\,b^2\,d^5\,n\,\sqrt {x}\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{e^5} \]

[In]

int(x^2*(a + b*log(c*(d + e*x^(1/2))^n))^3,x)

[Out]

(a^3*x^3)/3 + (b^3*x^3*log(c*(d + e*x^(1/2))^n)^3)/3 - (b^3*n^3*x^3)/108 + a*b^2*x^3*log(c*(d + e*x^(1/2))^n)^
2 - (b^3*n*x^3*log(c*(d + e*x^(1/2))^n)^2)/6 + (b^3*n^2*x^3*log(c*(d + e*x^(1/2))^n))/18 + (a*b^2*n^2*x^3)/18
- (b^3*d^6*log(c*(d + e*x^(1/2))^n)^3)/(3*e^6) + a^2*b*x^3*log(c*(d + e*x^(1/2))^n) - (a^2*b*n*x^3)/6 - (a*b^2
*n*x^3*log(c*(d + e*x^(1/2))^n))/3 - (13489*b^3*d^6*n^3*log(d + e*x^(1/2)))/(1800*e^6) - (919*b^3*d^2*n^3*x^2)
/(7200*e^2) + (2059*b^3*d^3*n^3*x^(3/2))/(5400*e^3) + (13489*b^3*d^5*n^3*x^(1/2))/(1800*e^5) - (a*b^2*d^6*log(
c*(d + e*x^(1/2))^n)^2)/e^6 + (49*b^3*d^6*n*log(c*(d + e*x^(1/2))^n)^2)/(20*e^6) + (91*b^3*d*n^3*x^(5/2))/(225
0*e) - (4669*b^3*d^4*n^3*x)/(3600*e^4) - (a^2*b*d^6*n*log(d + e*x^(1/2)))/e^6 + (b^3*d*n*x^(5/2)*log(c*(d + e*
x^(1/2))^n)^2)/(5*e) - (11*b^3*d*n^2*x^(5/2)*log(c*(d + e*x^(1/2))^n))/(75*e) - (b^3*d^4*n*x*log(c*(d + e*x^(1
/2))^n)^2)/(2*e^4) + (29*b^3*d^4*n^2*x*log(c*(d + e*x^(1/2))^n))/(20*e^4) - (a^2*b*d^2*n*x^2)/(4*e^2) - (11*a*
b^2*d*n^2*x^(5/2))/(75*e) + (29*a*b^2*d^4*n^2*x)/(20*e^4) + (a^2*b*d^3*n*x^(3/2))/(3*e^3) + (a^2*b*d^5*n*x^(1/
2))/e^5 + (49*a*b^2*d^6*n^2*log(d + e*x^(1/2)))/(10*e^6) - (b^3*d^2*n*x^2*log(c*(d + e*x^(1/2))^n)^2)/(4*e^2)
+ (37*b^3*d^2*n^2*x^2*log(c*(d + e*x^(1/2))^n))/(120*e^2) + (b^3*d^3*n*x^(3/2)*log(c*(d + e*x^(1/2))^n)^2)/(3*
e^3) - (19*b^3*d^3*n^2*x^(3/2)*log(c*(d + e*x^(1/2))^n))/(30*e^3) + (b^3*d^5*n*x^(1/2)*log(c*(d + e*x^(1/2))^n
)^2)/e^5 - (49*b^3*d^5*n^2*x^(1/2)*log(c*(d + e*x^(1/2))^n))/(10*e^5) + (37*a*b^2*d^2*n^2*x^2)/(120*e^2) - (19
*a*b^2*d^3*n^2*x^(3/2))/(30*e^3) - (49*a*b^2*d^5*n^2*x^(1/2))/(10*e^5) + (a^2*b*d*n*x^(5/2))/(5*e) - (a^2*b*d^
4*n*x)/(2*e^4) + (2*a*b^2*d*n*x^(5/2)*log(c*(d + e*x^(1/2))^n))/(5*e) - (a*b^2*d^4*n*x*log(c*(d + e*x^(1/2))^n
))/e^4 - (a*b^2*d^2*n*x^2*log(c*(d + e*x^(1/2))^n))/(2*e^2) + (2*a*b^2*d^3*n*x^(3/2)*log(c*(d + e*x^(1/2))^n))
/(3*e^3) + (2*a*b^2*d^5*n*x^(1/2)*log(c*(d + e*x^(1/2))^n))/e^5