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

Optimal. Leaf size=907 \[ -\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]

1/3*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(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^6-15/4*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^6-12*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+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

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} -\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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.45, size = 661, normalized size = 0.73 \begin {gather*} \frac {-36000 b^3 d^6 n^3 \log ^3\left (d+e \sqrt {x}\right )+5400 b^2 d^6 n^2 \log ^2\left (d+e \sqrt {x}\right ) \left (20 a-49 b n+20 b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )-60 b d^6 n \log \left (d+e \sqrt {x}\right ) \left (1800 a^2-8820 a b n+13489 b^2 n^2+180 b (20 a-49 b n) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+1800 b^2 \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )\right )+e \sqrt {x} \left (36000 a^3 e^5 x^{5/2}+b^3 n^3 \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 )-60 a 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 )+1800 a^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 )+60 b \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 ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+1800 b^2 \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 ^2\left (c \left (d+e \sqrt {x}\right )^n\right )+36000 b^3 e^5 x^{5/2} \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{108000 e^6} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-36000*b^3*d^6*n^3*Log[d + e*Sqrt[x]]^3 + 5400*b^2*d^6*n^2*Log[d + e*Sqrt[x]]^2*(20*a - 49*b*n + 20*b*Log[c*(
d + e*Sqrt[x])^n]) - 60*b*d^6*n*Log[d + e*Sqrt[x]]*(1800*a^2 - 8820*a*b*n + 13489*b^2*n^2 + 180*b*(20*a - 49*b
*n)*Log[c*(d + e*Sqrt[x])^n] + 1800*b^2*Log[c*(d + e*Sqrt[x])^n]^2) + e*Sqrt[x]*(36000*a^3*e^5*x^(5/2) + b^3*n
^3*(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)) - 60*a*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)) + 1800*a^2*b*n*(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)) + 60*b*(1800*a^2*e^5*x^(5/2) + 60*a*b*n*(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)) + 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)))*Log[c*(d + e*Sqrt[x])^n] + 1800*b^2*(60*a*e^5*x^(5/2) + b*n
*(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)))*Log[c*(d + e
*Sqrt[x])^n]^2 + 36000*b^3*e^5*x^(5/2)*Log[c*(d + e*Sqrt[x])^n]^3))/(108000*e^6)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]
time = 0.32, size = 655, normalized size = 0.72 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{3} + a b^{2} x^{3} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2} + a^{2} b x^{3} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{3} x^{3} - \frac {1}{60} \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + {\left (30 \, d^{4} x e - 60 \, d^{5} \sqrt {x} - 20 \, d^{3} x^{\frac {3}{2}} e^{2} + 15 \, d^{2} x^{2} e^{3} - 12 \, d x^{\frac {5}{2}} e^{4} + 10 \, x^{3} e^{5}\right )} e^{\left (-6\right )}\right )} a^{2} b n e + \frac {1}{1800} \, {\left ({\left (1800 \, d^{6} \log \left (\sqrt {x} e + d\right )^{2} + 8820 \, d^{6} \log \left (\sqrt {x} e + d\right ) - 8820 \, d^{5} \sqrt {x} e + 2610 \, d^{4} x e^{2} - 1140 \, d^{3} x^{\frac {3}{2}} e^{3} + 555 \, d^{2} x^{2} e^{4} - 264 \, d x^{\frac {5}{2}} e^{5} + 100 \, x^{3} e^{6}\right )} n^{2} e^{\left (-6\right )} - 60 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + {\left (30 \, d^{4} x e - 60 \, d^{5} \sqrt {x} - 20 \, d^{3} x^{\frac {3}{2}} e^{2} + 15 \, d^{2} x^{2} e^{3} - 12 \, d x^{\frac {5}{2}} e^{4} + 10 \, x^{3} e^{5}\right )} e^{\left (-6\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} a b^{2} - \frac {1}{108000} \, {\left (1800 \, {\left (60 \, d^{6} e^{\left (-7\right )} \log \left (\sqrt {x} e + d\right ) + {\left (30 \, d^{4} x e - 60 \, d^{5} \sqrt {x} - 20 \, d^{3} x^{\frac {3}{2}} e^{2} + 15 \, d^{2} x^{2} e^{3} - 12 \, d x^{\frac {5}{2}} e^{4} + 10 \, x^{3} e^{5}\right )} e^{\left (-6\right )}\right )} n e \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )^{2} + {\left ({\left (36000 \, d^{6} \log \left (\sqrt {x} e + d\right )^{3} + 264600 \, d^{6} \log \left (\sqrt {x} e + d\right )^{2} + 809340 \, d^{6} \log \left (\sqrt {x} e + d\right ) - 809340 \, d^{5} \sqrt {x} e + 140070 \, d^{4} x e^{2} - 41180 \, d^{3} x^{\frac {3}{2}} e^{3} + 13785 \, d^{2} x^{2} e^{4} - 4368 \, d x^{\frac {5}{2}} e^{5} + 1000 \, x^{3} e^{6}\right )} n^{2} e^{\left (-7\right )} - 60 \, {\left (1800 \, d^{6} \log \left (\sqrt {x} e + d\right )^{2} + 8820 \, d^{6} \log \left (\sqrt {x} e + d\right ) - 8820 \, d^{5} \sqrt {x} e + 2610 \, d^{4} x e^{2} - 1140 \, d^{3} x^{\frac {3}{2}} e^{3} + 555 \, d^{2} x^{2} e^{4} - 264 \, d x^{\frac {5}{2}} e^{5} + 100 \, x^{3} e^{6}\right )} n e^{\left (-7\right )} \log \left ({\left (\sqrt {x} e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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((sqrt(x)*e + d)^n*c)^3 + a*b^2*x^3*log((sqrt(x)*e + d)^n*c)^2 + a^2*b*x^3*log((sqrt(x)*e + d)^
n*c) + 1/3*a^3*x^3 - 1/60*(60*d^6*e^(-7)*log(sqrt(x)*e + d) + (30*d^4*x*e - 60*d^5*sqrt(x) - 20*d^3*x^(3/2)*e^
2 + 15*d^2*x^2*e^3 - 12*d*x^(5/2)*e^4 + 10*x^3*e^5)*e^(-6))*a^2*b*n*e + 1/1800*((1800*d^6*log(sqrt(x)*e + d)^2
+ 8820*d^6*log(sqrt(x)*e + d) - 8820*d^5*sqrt(x)*e + 2610*d^4*x*e^2 - 1140*d^3*x^(3/2)*e^3 + 555*d^2*x^2*e^4
- 264*d*x^(5/2)*e^5 + 100*x^3*e^6)*n^2*e^(-6) - 60*(60*d^6*e^(-7)*log(sqrt(x)*e + d) + (30*d^4*x*e - 60*d^5*sq
rt(x) - 20*d^3*x^(3/2)*e^2 + 15*d^2*x^2*e^3 - 12*d*x^(5/2)*e^4 + 10*x^3*e^5)*e^(-6))*n*e*log((sqrt(x)*e + d)^n
*c))*a*b^2 - 1/108000*(1800*(60*d^6*e^(-7)*log(sqrt(x)*e + d) + (30*d^4*x*e - 60*d^5*sqrt(x) - 20*d^3*x^(3/2)*
e^2 + 15*d^2*x^2*e^3 - 12*d*x^(5/2)*e^4 + 10*x^3*e^5)*e^(-6))*n*e*log((sqrt(x)*e + d)^n*c)^2 + ((36000*d^6*log
(sqrt(x)*e + d)^3 + 264600*d^6*log(sqrt(x)*e + d)^2 + 809340*d^6*log(sqrt(x)*e + d) - 809340*d^5*sqrt(x)*e + 1
40070*d^4*x*e^2 - 41180*d^3*x^(3/2)*e^3 + 13785*d^2*x^2*e^4 - 4368*d*x^(5/2)*e^5 + 1000*x^3*e^6)*n^2*e^(-7) -
60*(1800*d^6*log(sqrt(x)*e + d)^2 + 8820*d^6*log(sqrt(x)*e + d) - 8820*d^5*sqrt(x)*e + 2610*d^4*x*e^2 - 1140*d
^3*x^(3/2)*e^3 + 555*d^2*x^2*e^4 - 264*d*x^(5/2)*e^5 + 100*x^3*e^6)*n*e^(-7)*log((sqrt(x)*e + d)^n*c))*n*e)*b^
3

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Fricas [A]
time = 0.44, size = 1098, normalized size = 1.21 \begin {gather*} \frac {1}{108000} \, {\left (36000 \, b^{3} x^{3} e^{6} \log \left (c\right )^{3} - 1000 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n - 36 \, a^{3}\right )} x^{3} e^{6} - 15 \, {\left (919 \, b^{3} d^{2} n^{3} - 2220 \, a b^{2} d^{2} n^{2} + 1800 \, a^{2} b d^{2} n\right )} x^{2} e^{4} - 36000 \, {\left (b^{3} d^{6} n^{3} - b^{3} n^{3} x^{3} e^{6}\right )} \log \left (\sqrt {x} e + d\right )^{3} - 30 \, {\left (4669 \, b^{3} d^{4} n^{3} - 5220 \, a b^{2} d^{4} n^{2} + 1800 \, a^{2} b d^{4} n\right )} x e^{2} + 1800 \, {\left (147 \, b^{3} d^{6} n^{3} - 30 \, b^{3} d^{4} n^{3} x e^{2} - 60 \, a b^{2} d^{6} n^{2} - 15 \, b^{3} d^{2} n^{3} x^{2} e^{4} - 10 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2}\right )} x^{3} e^{6} - 60 \, {\left (b^{3} d^{6} n^{2} - b^{3} n^{2} x^{3} e^{6}\right )} \log \left (c\right ) + 4 \, {\left (15 \, b^{3} d^{5} n^{3} e + 5 \, b^{3} d^{3} n^{3} x e^{3} + 3 \, b^{3} d n^{3} x^{2} e^{5}\right )} \sqrt {x}\right )} \log \left (\sqrt {x} e + d\right )^{2} - 9000 \, {\left (6 \, b^{3} d^{4} n x e^{2} + 3 \, b^{3} d^{2} n x^{2} e^{4} + 2 \, {\left (b^{3} n - 6 \, a b^{2}\right )} x^{3} e^{6}\right )} \log \left (c\right )^{2} - 60 \, {\left (13489 \, b^{3} d^{6} n^{3} - 8820 \, a b^{2} d^{6} n^{2} + 1800 \, a^{2} b d^{6} n - 100 \, {\left (b^{3} n^{3} - 6 \, a b^{2} n^{2} + 18 \, a^{2} b n\right )} x^{3} e^{6} - 15 \, {\left (37 \, b^{3} d^{2} n^{3} - 60 \, a b^{2} d^{2} n^{2}\right )} x^{2} e^{4} - 90 \, {\left (29 \, b^{3} d^{4} n^{3} - 20 \, a b^{2} d^{4} n^{2}\right )} x e^{2} + 1800 \, {\left (b^{3} d^{6} n - b^{3} n x^{3} e^{6}\right )} \log \left (c\right )^{2} - 60 \, {\left (147 \, b^{3} d^{6} n^{2} - 30 \, b^{3} d^{4} n^{2} x e^{2} - 60 \, a b^{2} d^{6} n - 15 \, b^{3} d^{2} n^{2} x^{2} e^{4} - 10 \, {\left (b^{3} n^{2} - 6 \, a b^{2} n\right )} x^{3} e^{6}\right )} \log \left (c\right ) + 12 \, {\left (2 \, {\left (11 \, b^{3} d n^{3} - 30 \, a b^{2} d n^{2}\right )} x^{2} e^{5} + 5 \, {\left (19 \, b^{3} d^{3} n^{3} - 20 \, a b^{2} d^{3} n^{2}\right )} x e^{3} + 15 \, {\left (49 \, b^{3} d^{5} n^{3} - 20 \, a b^{2} d^{5} n^{2}\right )} e - 20 \, {\left (15 \, b^{3} d^{5} n^{2} e + 5 \, b^{3} d^{3} n^{2} x e^{3} + 3 \, b^{3} d n^{2} x^{2} e^{5}\right )} \log \left (c\right )\right )} \sqrt {x}\right )} \log \left (\sqrt {x} e + d\right ) + 300 \, {\left (20 \, {\left (b^{3} n^{2} - 6 \, a b^{2} n + 18 \, a^{2} b\right )} x^{3} e^{6} + 3 \, {\left (37 \, b^{3} d^{2} n^{2} - 60 \, a b^{2} d^{2} n\right )} x^{2} e^{4} + 18 \, {\left (29 \, b^{3} d^{4} n^{2} - 20 \, a b^{2} d^{4} n\right )} x e^{2}\right )} \log \left (c\right ) + 4 \, {\left (12 \, {\left (91 \, b^{3} d n^{3} - 330 \, a b^{2} d n^{2} + 450 \, a^{2} b d n\right )} x^{2} e^{5} + 5 \, {\left (2059 \, b^{3} d^{3} n^{3} - 3420 \, a b^{2} d^{3} n^{2} + 1800 \, a^{2} b d^{3} n\right )} x e^{3} + 1800 \, {\left (15 \, b^{3} d^{5} n e + 5 \, b^{3} d^{3} n x e^{3} + 3 \, b^{3} d n x^{2} e^{5}\right )} \log \left (c\right )^{2} + 15 \, {\left (13489 \, b^{3} d^{5} n^{3} - 8820 \, a b^{2} d^{5} n^{2} + 1800 \, a^{2} b d^{5} n\right )} e - 180 \, {\left (2 \, {\left (11 \, b^{3} d n^{2} - 30 \, a b^{2} d n\right )} x^{2} e^{5} + 5 \, {\left (19 \, b^{3} d^{3} n^{2} - 20 \, a b^{2} d^{3} n\right )} x e^{3} + 15 \, {\left (49 \, b^{3} d^{5} n^{2} - 20 \, a b^{2} d^{5} n\right )} e\right )} \log \left (c\right )\right )} \sqrt {x}\right )} e^{\left (-6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108000*(36000*b^3*x^3*e^6*log(c)^3 - 1000*(b^3*n^3 - 6*a*b^2*n^2 + 18*a^2*b*n - 36*a^3)*x^3*e^6 - 15*(919*b^
3*d^2*n^3 - 2220*a*b^2*d^2*n^2 + 1800*a^2*b*d^2*n)*x^2*e^4 - 36000*(b^3*d^6*n^3 - b^3*n^3*x^3*e^6)*log(sqrt(x)
*e + d)^3 - 30*(4669*b^3*d^4*n^3 - 5220*a*b^2*d^4*n^2 + 1800*a^2*b*d^4*n)*x*e^2 + 1800*(147*b^3*d^6*n^3 - 30*b
^3*d^4*n^3*x*e^2 - 60*a*b^2*d^6*n^2 - 15*b^3*d^2*n^3*x^2*e^4 - 10*(b^3*n^3 - 6*a*b^2*n^2)*x^3*e^6 - 60*(b^3*d^
6*n^2 - b^3*n^2*x^3*e^6)*log(c) + 4*(15*b^3*d^5*n^3*e + 5*b^3*d^3*n^3*x*e^3 + 3*b^3*d*n^3*x^2*e^5)*sqrt(x))*lo
g(sqrt(x)*e + d)^2 - 9000*(6*b^3*d^4*n*x*e^2 + 3*b^3*d^2*n*x^2*e^4 + 2*(b^3*n - 6*a*b^2)*x^3*e^6)*log(c)^2 - 6
0*(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*n^3 - 6*a*b^2*n^2 + 18*a^2*b*n)*x^3*e^
6 - 15*(37*b^3*d^2*n^3 - 60*a*b^2*d^2*n^2)*x^2*e^4 - 90*(29*b^3*d^4*n^3 - 20*a*b^2*d^4*n^2)*x*e^2 + 1800*(b^3*
d^6*n - b^3*n*x^3*e^6)*log(c)^2 - 60*(147*b^3*d^6*n^2 - 30*b^3*d^4*n^2*x*e^2 - 60*a*b^2*d^6*n - 15*b^3*d^2*n^2
*x^2*e^4 - 10*(b^3*n^2 - 6*a*b^2*n)*x^3*e^6)*log(c) + 12*(2*(11*b^3*d*n^3 - 30*a*b^2*d*n^2)*x^2*e^5 + 5*(19*b^
3*d^3*n^3 - 20*a*b^2*d^3*n^2)*x*e^3 + 15*(49*b^3*d^5*n^3 - 20*a*b^2*d^5*n^2)*e - 20*(15*b^3*d^5*n^2*e + 5*b^3*
d^3*n^2*x*e^3 + 3*b^3*d*n^2*x^2*e^5)*log(c))*sqrt(x))*log(sqrt(x)*e + d) + 300*(20*(b^3*n^2 - 6*a*b^2*n + 18*a
^2*b)*x^3*e^6 + 3*(37*b^3*d^2*n^2 - 60*a*b^2*d^2*n)*x^2*e^4 + 18*(29*b^3*d^4*n^2 - 20*a*b^2*d^4*n)*x*e^2)*log(
c) + 4*(12*(91*b^3*d*n^3 - 330*a*b^2*d*n^2 + 450*a^2*b*d*n)*x^2*e^5 + 5*(2059*b^3*d^3*n^3 - 3420*a*b^2*d^3*n^2
+ 1800*a^2*b*d^3*n)*x*e^3 + 1800*(15*b^3*d^5*n*e + 5*b^3*d^3*n*x*e^3 + 3*b^3*d*n*x^2*e^5)*log(c)^2 + 15*(1348
9*b^3*d^5*n^3 - 8820*a*b^2*d^5*n^2 + 1800*a^2*b*d^5*n)*e - 180*(2*(11*b^3*d*n^2 - 30*a*b^2*d*n)*x^2*e^5 + 5*(1
9*b^3*d^3*n^2 - 20*a*b^2*d^3*n)*x*e^3 + 15*(49*b^3*d^5*n^2 - 20*a*b^2*d^5*n)*e)*log(c))*sqrt(x))*e^(-6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2223 vs. \(2 (803) = 1606\).
time = 4.97, size = 2223, normalized size = 2.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108000*(36000*b^3*x^3*e*log(c)^3 + 108000*a*b^2*x^3*e*log(c)^2 + 108000*a^2*b*x^3*e*log(c) + (36000*(sqrt(x)
*e + d)^6*e^(-5)*log(sqrt(x)*e + d)^3 - 216000*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d)^3 + 540000*(sqrt(
x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d)^3 - 720000*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d)^3 + 54000
0*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d)^3 - 216000*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d)^3 -
18000*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d)^2 + 129600*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d)^2
- 405000*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d)^2 + 720000*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e
+ d)^2 - 810000*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d)^2 + 648000*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqr
t(x)*e + d)^2 + 6000*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d) - 51840*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x
)*e + d) + 202500*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) - 480000*(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sq
rt(x)*e + d) + 810000*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) - 1296000*(sqrt(x)*e + d)*d^5*e^(-5)*log
(sqrt(x)*e + d) - 1000*(sqrt(x)*e + d)^6*e^(-5) + 10368*(sqrt(x)*e + d)^5*d*e^(-5) - 50625*(sqrt(x)*e + d)^4*d
^2*e^(-5) + 160000*(sqrt(x)*e + d)^3*d^3*e^(-5) - 405000*(sqrt(x)*e + d)^2*d^4*e^(-5) + 1296000*(sqrt(x)*e + d
)*d^5*e^(-5))*b^3*n^3 + 36000*a^3*x^3*e + 60*(1800*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d)^2 - 10800*(sqrt
(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d)^2 - 36000*(
sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d)^2 - 1
0800*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d)^2 - 600*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d) + 4320*
(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) - 13500*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) + 24000*
(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) - 27000*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) + 2160
0*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) + 100*(sqrt(x)*e + d)^6*e^(-5) - 864*(sqrt(x)*e + d)^5*d*e^(-5
) + 3375*(sqrt(x)*e + d)^4*d^2*e^(-5) - 8000*(sqrt(x)*e + d)^3*d^3*e^(-5) + 13500*(sqrt(x)*e + d)^2*d^4*e^(-5)
- 21600*(sqrt(x)*e + d)*d^5*e^(-5))*b^3*n^2*log(c) + 1800*(60*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d) - 3
60*(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) - 1200*
(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) - 360*(s
qrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-5) + 72*(sqrt(x)*e + d)^5*d*e^(-5) - 22
5*(sqrt(x)*e + d)^4*d^2*e^(-5) + 400*(sqrt(x)*e + d)^3*d^3*e^(-5) - 450*(sqrt(x)*e + d)^2*d^4*e^(-5) + 360*(sq
rt(x)*e + d)*d^5*e^(-5))*b^3*n*log(c)^2 + 60*(1800*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d)^2 - 10800*(sqrt
(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d)^2 - 36000*(
sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d)^2 + 27000*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d)^2 - 1
0800*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d)^2 - 600*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d) + 4320*
(sqrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) - 13500*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) + 24000*
(sqrt(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) - 27000*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) + 2160
0*(sqrt(x)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) + 100*(sqrt(x)*e + d)^6*e^(-5) - 864*(sqrt(x)*e + d)^5*d*e^(-5
) + 3375*(sqrt(x)*e + d)^4*d^2*e^(-5) - 8000*(sqrt(x)*e + d)^3*d^3*e^(-5) + 13500*(sqrt(x)*e + d)^2*d^4*e^(-5)
- 21600*(sqrt(x)*e + d)*d^5*e^(-5))*a*b^2*n^2 + 3600*(60*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d) - 360*(s
qrt(x)*e + d)^5*d*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) - 1200*(sqrt
(x)*e + d)^3*d^3*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) - 360*(sqrt(x
)*e + d)*d^5*e^(-5)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-5) + 72*(sqrt(x)*e + d)^5*d*e^(-5) - 225*(sq
rt(x)*e + d)^4*d^2*e^(-5) + 400*(sqrt(x)*e + d)^3*d^3*e^(-5) - 450*(sqrt(x)*e + d)^2*d^4*e^(-5) + 360*(sqrt(x)
*e + d)*d^5*e^(-5))*a*b^2*n*log(c) + 1800*(60*(sqrt(x)*e + d)^6*e^(-5)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d
)^5*d*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^4*d^2*e^(-5)*log(sqrt(x)*e + d) - 1200*(sqrt(x)*e + d)^3
*d^3*e^(-5)*log(sqrt(x)*e + d) + 900*(sqrt(x)*e + d)^2*d^4*e^(-5)*log(sqrt(x)*e + d) - 360*(sqrt(x)*e + d)*d^5
*e^(-5)*log(sqrt(x)*e + d) - 10*(sqrt(x)*e + d)^6*e^(-5) + 72*(sqrt(x)*e + d)^5*d*e^(-5) - 225*(sqrt(x)*e + d)
^4*d^2*e^(-5) + 400*(sqrt(x)*e + d)^3*d^3*e^(-5) - 450*(sqrt(x)*e + d)^2*d^4*e^(-5) + 360*(sqrt(x)*e + d)*d^5*
e^(-5))*a^2*b*n)*e^(-1)

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Mupad [B]
time = 8.18, size = 976, normalized size = 1.08 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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