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

Optimal. Leaf size=595 \[ \frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4} \]

[Out]

-12*a*b^2*d^3*n^2*x^(1/2)/e^3+12*b^3*d^3*n^3*x^(1/2)/e^3-12*b^3*d^3*n^2*ln(c*(d+e*x^(1/2))^n)*(d+e*x^(1/2))/e^
4+6*b*d^3*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))/e^4-2*d^3*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))/
e^4-9/4*b^3*d^2*n^3*(d+e*x^(1/2))^2/e^4+9/2*b^2*d^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^2/e^4-9/2*b*
d^2*n*(a+b*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^2/e^4+3*d^2*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^2/e^
4+4/9*b^3*d*n^3*(d+e*x^(1/2))^3/e^4-4/3*b^2*d*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^3/e^4+2*b*d*n*(a+b
*ln(c*(d+e*x^(1/2))^n))^2*(d+e*x^(1/2))^3/e^4-2*d*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^3/e^4-3/64*b^3*n
^3*(d+e*x^(1/2))^4/e^4+3/16*b^2*n^2*(a+b*ln(c*(d+e*x^(1/2))^n))*(d+e*x^(1/2))^4/e^4-3/8*b*n*(a+b*ln(c*(d+e*x^(
1/2))^n))^2*(d+e*x^(1/2))^4/e^4+1/2*(a+b*ln(c*(d+e*x^(1/2))^n))^3*(d+e*x^(1/2))^4/e^4

________________________________________________________________________________________

Rubi [A]  time = 0.62, antiderivative size = 595, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac {9 b^2 d^2 n^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{2 e^4}+\frac {3 b^2 n^2 \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{16 e^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {9 b d^2 n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{2 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*b^3*d^2*n^3*(d + e*Sqrt[x])^2)/(4*e^4) + (4*b^3*d*n^3*(d + e*Sqrt[x])^3)/(9*e^4) - (3*b^3*n^3*(d + e*Sqrt[
x])^4)/(64*e^4) - (12*a*b^2*d^3*n^2*Sqrt[x])/e^3 + (12*b^3*d^3*n^3*Sqrt[x])/e^3 - (12*b^3*d^3*n^2*(d + e*Sqrt[
x])*Log[c*(d + e*Sqrt[x])^n])/e^4 + (9*b^2*d^2*n^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(2*e^4)
- (4*b^2*d*n^2*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(3*e^4) + (3*b^2*n^2*(d + e*Sqrt[x])^4*(a
+ b*Log[c*(d + e*Sqrt[x])^n]))/(16*e^4) + (6*b*d^3*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/e^4 -
(9*b*d^2*n*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(2*e^4) + (2*b*d*n*(d + e*Sqrt[x])^3*(a + b*
Log[c*(d + e*Sqrt[x])^n])^2)/e^4 - (3*b*n*(d + e*Sqrt[x])^4*(a + b*Log[c*(d + e*Sqrt[x])^n])^2)/(8*e^4) - (2*d
^3*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + (3*d^2*(d + e*Sqrt[x])^2*(a + b*Log[c*(d + e*Sqrt
[x])^n])^3)/e^4 - (2*d*(d + e*Sqrt[x])^3*(a + b*Log[c*(d + e*Sqrt[x])^n])^3)/e^4 + ((d + e*Sqrt[x])^4*(a + b*L
og[c*(d + e*Sqrt[x])^n])^3)/(2*e^4)

Rule 2295

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

Rule 2296

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 2304

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 2305

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 2389

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 2390

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 2401

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 2454

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 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \, dx &=2 \operatorname {Subst}\left (\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {3 d^2 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}-\frac {3 d (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}+\frac {(d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^3}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {2 \operatorname {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^3}-\frac {(6 d) \operatorname {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^3}+\frac {\left (6 d^2\right ) \operatorname {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^3}-\frac {\left (2 d^3\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt {x}\right )}{e^3}\\ &=\frac {2 \operatorname {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {(6 d) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (6 d^2\right ) \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (2 d^3\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {(3 b n) \operatorname {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{2 e^4}+\frac {(6 b d n) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (9 b d^2 n\right ) \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (6 b d^3 n\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 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^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}+\frac {\left (3 b^2 n^2\right ) \operatorname {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{4 e^4}-\frac {\left (4 b^2 d n^2\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}+\frac {\left (9 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}-\frac {\left (12 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {9 b^2 d^2 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^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}+\frac {3 b^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 )}{16 e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 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^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}-\frac {\left (12 b^3 d^3 n^2\right ) \operatorname {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt {x}\right )}{e^4}\\ &=-\frac {9 b^3 d^2 n^3 \left (d+e \sqrt {x}\right )^2}{4 e^4}+\frac {4 b^3 d n^3 \left (d+e \sqrt {x}\right )^3}{9 e^4}-\frac {3 b^3 n^3 \left (d+e \sqrt {x}\right )^4}{64 e^4}-\frac {12 a b^2 d^3 n^2 \sqrt {x}}{e^3}+\frac {12 b^3 d^3 n^3 \sqrt {x}}{e^3}-\frac {12 b^3 d^3 n^2 \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^4}+\frac {9 b^2 d^2 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^4}-\frac {4 b^2 d n^2 \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{3 e^4}+\frac {3 b^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 )}{16 e^4}+\frac {6 b d^3 n \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {9 b d^2 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^4}+\frac {2 b d n \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^4}-\frac {3 b n \left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{8 e^4}-\frac {2 d^3 \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {3 d^2 \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}-\frac {2 d \left (d+e \sqrt {x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{e^4}+\frac {\left (d+e \sqrt {x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{2 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 433, normalized size = 0.73 \[ \frac {-288 a^3 \left (d^4-e^4 x^2\right )-12 b \left (72 a^2 \left (d^4-e^4 x^2\right )-12 a b n \left (25 d^4+12 d^3 e \sqrt {x}-6 d^2 e^2 x+4 d e^3 x^{3/2}-3 e^4 x^2\right )+b^2 n^2 \left (415 d^4+300 d^3 e \sqrt {x}-78 d^2 e^2 x+28 d e^3 x^{3/2}-9 e^4 x^2\right )\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+72 a^2 b n \left (25 d^4+12 d^3 e \sqrt {x}-6 d^2 e^2 x+4 d e^3 x^{3/2}-3 e^4 x^2\right )-72 b^2 \left (12 a \left (d^4-e^4 x^2\right )+b n \left (-25 d^4-12 d^3 e \sqrt {x}+6 d^2 e^2 x-4 d e^3 x^{3/2}+3 e^4 x^2\right )\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )+12 a b^2 n^2 \left (161 d^4-300 d^3 e \sqrt {x}+78 d^2 e^2 x-28 d e^3 x^{3/2}+9 e^4 x^2\right )-288 b^3 \left (d^4-e^4 x^2\right ) \log ^3\left (c \left (d+e \sqrt {x}\right )^n\right )+b^3 e n^3 \sqrt {x} \left (4980 d^3-690 d^2 e \sqrt {x}+148 d e^2 x-27 e^3 x^{3/2}\right )}{576 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e*n^3*Sqrt[x]*(4980*d^3 - 690*d^2*e*Sqrt[x] + 148*d*e^2*x - 27*e^3*x^(3/2)) + 72*a^2*b*n*(25*d^4 + 12*d^3
*e*Sqrt[x] - 6*d^2*e^2*x + 4*d*e^3*x^(3/2) - 3*e^4*x^2) - 288*a^3*(d^4 - e^4*x^2) + 12*a*b^2*n^2*(161*d^4 - 30
0*d^3*e*Sqrt[x] + 78*d^2*e^2*x - 28*d*e^3*x^(3/2) + 9*e^4*x^2) - 12*b*(b^2*n^2*(415*d^4 + 300*d^3*e*Sqrt[x] -
78*d^2*e^2*x + 28*d*e^3*x^(3/2) - 9*e^4*x^2) - 12*a*b*n*(25*d^4 + 12*d^3*e*Sqrt[x] - 6*d^2*e^2*x + 4*d*e^3*x^(
3/2) - 3*e^4*x^2) + 72*a^2*(d^4 - e^4*x^2))*Log[c*(d + e*Sqrt[x])^n] - 72*b^2*(12*a*(d^4 - e^4*x^2) + b*n*(-25
*d^4 - 12*d^3*e*Sqrt[x] + 6*d^2*e^2*x - 4*d*e^3*x^(3/2) + 3*e^4*x^2))*Log[c*(d + e*Sqrt[x])^n]^2 - 288*b^3*(d^
4 - e^4*x^2)*Log[c*(d + e*Sqrt[x])^n]^3)/(576*e^4)

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fricas [A]  time = 0.54, size = 861, normalized size = 1.45 \[ \frac {288 \, b^{3} e^{4} x^{2} \log \relax (c)^{3} + 288 \, {\left (b^{3} e^{4} n^{3} x^{2} - b^{3} d^{4} n^{3}\right )} \log \left (e \sqrt {x} + d\right )^{3} - 9 \, {\left (3 \, b^{3} e^{4} n^{3} - 12 \, a b^{2} e^{4} n^{2} + 24 \, a^{2} b e^{4} n - 32 \, a^{3} e^{4}\right )} x^{2} - 72 \, {\left (6 \, b^{3} d^{2} e^{2} n^{3} x - 25 \, b^{3} d^{4} n^{3} + 12 \, a b^{2} d^{4} n^{2} + 3 \, {\left (b^{3} e^{4} n^{3} - 4 \, a b^{2} e^{4} n^{2}\right )} x^{2} - 12 \, {\left (b^{3} e^{4} n^{2} x^{2} - b^{3} d^{4} n^{2}\right )} \log \relax (c) - 4 \, {\left (b^{3} d e^{3} n^{3} x + 3 \, b^{3} d^{3} e n^{3}\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right )^{2} - 216 \, {\left (2 \, b^{3} d^{2} e^{2} n x + {\left (b^{3} e^{4} n - 4 \, a b^{2} e^{4}\right )} x^{2}\right )} \log \relax (c)^{2} - 6 \, {\left (115 \, b^{3} d^{2} e^{2} n^{3} - 156 \, a b^{2} d^{2} e^{2} n^{2} + 72 \, a^{2} b d^{2} e^{2} n\right )} x - 12 \, {\left (415 \, b^{3} d^{4} n^{3} - 300 \, a b^{2} d^{4} n^{2} + 72 \, a^{2} b d^{4} n - 9 \, {\left (b^{3} e^{4} n^{3} - 4 \, a b^{2} e^{4} n^{2} + 8 \, a^{2} b e^{4} n\right )} x^{2} - 72 \, {\left (b^{3} e^{4} n x^{2} - b^{3} d^{4} n\right )} \log \relax (c)^{2} - 6 \, {\left (13 \, b^{3} d^{2} e^{2} n^{3} - 12 \, a b^{2} d^{2} e^{2} n^{2}\right )} x + 12 \, {\left (6 \, b^{3} d^{2} e^{2} n^{2} x - 25 \, b^{3} d^{4} n^{2} + 12 \, a b^{2} d^{4} n + 3 \, {\left (b^{3} e^{4} n^{2} - 4 \, a b^{2} e^{4} n\right )} x^{2}\right )} \log \relax (c) + 4 \, {\left (75 \, b^{3} d^{3} e n^{3} - 36 \, a b^{2} d^{3} e n^{2} + {\left (7 \, b^{3} d e^{3} n^{3} - 12 \, a b^{2} d e^{3} n^{2}\right )} x - 12 \, {\left (b^{3} d e^{3} n^{2} x + 3 \, b^{3} d^{3} e n^{2}\right )} \log \relax (c)\right )} \sqrt {x}\right )} \log \left (e \sqrt {x} + d\right ) + 36 \, {\left (3 \, {\left (b^{3} e^{4} n^{2} - 4 \, a b^{2} e^{4} n + 8 \, a^{2} b e^{4}\right )} x^{2} + 2 \, {\left (13 \, b^{3} d^{2} e^{2} n^{2} - 12 \, a b^{2} d^{2} e^{2} n\right )} x\right )} \log \relax (c) + 4 \, {\left (1245 \, b^{3} d^{3} e n^{3} - 900 \, a b^{2} d^{3} e n^{2} + 216 \, a^{2} b d^{3} e n + 72 \, {\left (b^{3} d e^{3} n x + 3 \, b^{3} d^{3} e n\right )} \log \relax (c)^{2} + {\left (37 \, b^{3} d e^{3} n^{3} - 84 \, a b^{2} d e^{3} n^{2} + 72 \, a^{2} b d e^{3} n\right )} x - 12 \, {\left (75 \, b^{3} d^{3} e n^{2} - 36 \, a b^{2} d^{3} e n + {\left (7 \, b^{3} d e^{3} n^{2} - 12 \, a b^{2} d e^{3} n\right )} x\right )} \log \relax (c)\right )} \sqrt {x}}{576 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(288*b^3*e^4*x^2*log(c)^3 + 288*(b^3*e^4*n^3*x^2 - b^3*d^4*n^3)*log(e*sqrt(x) + d)^3 - 9*(3*b^3*e^4*n^3
- 12*a*b^2*e^4*n^2 + 24*a^2*b*e^4*n - 32*a^3*e^4)*x^2 - 72*(6*b^3*d^2*e^2*n^3*x - 25*b^3*d^4*n^3 + 12*a*b^2*d^
4*n^2 + 3*(b^3*e^4*n^3 - 4*a*b^2*e^4*n^2)*x^2 - 12*(b^3*e^4*n^2*x^2 - b^3*d^4*n^2)*log(c) - 4*(b^3*d*e^3*n^3*x
+ 3*b^3*d^3*e*n^3)*sqrt(x))*log(e*sqrt(x) + d)^2 - 216*(2*b^3*d^2*e^2*n*x + (b^3*e^4*n - 4*a*b^2*e^4)*x^2)*lo
g(c)^2 - 6*(115*b^3*d^2*e^2*n^3 - 156*a*b^2*d^2*e^2*n^2 + 72*a^2*b*d^2*e^2*n)*x - 12*(415*b^3*d^4*n^3 - 300*a*
b^2*d^4*n^2 + 72*a^2*b*d^4*n - 9*(b^3*e^4*n^3 - 4*a*b^2*e^4*n^2 + 8*a^2*b*e^4*n)*x^2 - 72*(b^3*e^4*n*x^2 - b^3
*d^4*n)*log(c)^2 - 6*(13*b^3*d^2*e^2*n^3 - 12*a*b^2*d^2*e^2*n^2)*x + 12*(6*b^3*d^2*e^2*n^2*x - 25*b^3*d^4*n^2
+ 12*a*b^2*d^4*n + 3*(b^3*e^4*n^2 - 4*a*b^2*e^4*n)*x^2)*log(c) + 4*(75*b^3*d^3*e*n^3 - 36*a*b^2*d^3*e*n^2 + (7
*b^3*d*e^3*n^3 - 12*a*b^2*d*e^3*n^2)*x - 12*(b^3*d*e^3*n^2*x + 3*b^3*d^3*e*n^2)*log(c))*sqrt(x))*log(e*sqrt(x)
+ d) + 36*(3*(b^3*e^4*n^2 - 4*a*b^2*e^4*n + 8*a^2*b*e^4)*x^2 + 2*(13*b^3*d^2*e^2*n^2 - 12*a*b^2*d^2*e^2*n)*x)
*log(c) + 4*(1245*b^3*d^3*e*n^3 - 900*a*b^2*d^3*e*n^2 + 216*a^2*b*d^3*e*n + 72*(b^3*d*e^3*n*x + 3*b^3*d^3*e*n)
*log(c)^2 + (37*b^3*d*e^3*n^3 - 84*a*b^2*d*e^3*n^2 + 72*a^2*b*d*e^3*n)*x - 12*(75*b^3*d^3*e*n^2 - 36*a*b^2*d^3
*e*n + (7*b^3*d*e^3*n^2 - 12*a*b^2*d*e^3*n)*x)*log(c))*sqrt(x))/e^4

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giac [B]  time = 0.30, size = 1483, normalized size = 2.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(288*b^3*x^2*e*log(c)^3 + 864*a*b^2*x^2*e*log(c)^2 + (288*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d)^3
- 1152*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d)^3 + 1728*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d)^
3 - 1152*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d)^3 - 216*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d)^2 +
1152*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d)^2 - 2592*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d)^2
+ 3456*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d)^2 + 108*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) - 76
8*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) + 2592*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) - 6912*
(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) - 27*(sqrt(x)*e + d)^4*e^(-3) + 256*(sqrt(x)*e + d)^3*d*e^(-3) -
1296*(sqrt(x)*e + d)^2*d^2*e^(-3) + 6912*(sqrt(x)*e + d)*d^3*e^(-3))*b^3*n^3 + 12*(72*(sqrt(x)*e + d)^4*e^(-3
)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d)^2 + 432*(sqrt(x)*e + d)^2*d^2*e^(-3
)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d)^2 - 36*(sqrt(x)*e + d)^4*e^(-3)*log
(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) - 432*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqr
t(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) + 9*(sqrt(x)*e + d)^4*e^(-3) - 64*(sqrt(x)*e +
d)^3*d*e^(-3) + 216*(sqrt(x)*e + d)^2*d^2*e^(-3) - 576*(sqrt(x)*e + d)*d^3*e^(-3))*b^3*n^2*log(c) + 864*a^2*b
*x^2*e*log(c) + 72*(12*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)
*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d
) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16*(sqrt(x)*e + d)^3*d*e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*e^(-3) + 48*(sqrt(x)
*e + d)*d^3*e^(-3))*b^3*n*log(c)^2 + 12*(72*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d
)^3*d*e^(-3)*log(sqrt(x)*e + d)^2 + 432*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d
)*d^3*e^(-3)*log(sqrt(x)*e + d)^2 - 36*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e
^(-3)*log(sqrt(x)*e + d) - 432*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-3
)*log(sqrt(x)*e + d) + 9*(sqrt(x)*e + d)^4*e^(-3) - 64*(sqrt(x)*e + d)^3*d*e^(-3) + 216*(sqrt(x)*e + d)^2*d^2*
e^(-3) - 576*(sqrt(x)*e + d)*d^3*e^(-3))*a*b^2*n^2 + 288*a^3*x^2*e + 144*(12*(sqrt(x)*e + d)^4*e^(-3)*log(sqrt
(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-3)*log(sqrt(x)*e
+ d) - 48*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16*(sqrt(x)*e + d)^3*d*
e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*e^(-3) + 48*(sqrt(x)*e + d)*d^3*e^(-3))*a*b^2*n*log(c) + 72*(12*(sqrt(x)*e +
d)^4*e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-3)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*
e^(-3)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-3)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-3) + 16
*(sqrt(x)*e + d)^3*d*e^(-3) - 36*(sqrt(x)*e + d)^2*d^2*e^(-3) + 48*(sqrt(x)*e + d)*d^3*e^(-3))*a^2*b*n)*e^(-1)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a \right )^{3} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.57, size = 536, normalized size = 0.90 \[ \frac {1}{2} \, b^{3} x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {1}{8} \, a^{2} b e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} + \frac {3}{2} \, a^{2} b x^{2} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{3} x^{2} - \frac {1}{48} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - \frac {{\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 28 \, d e^{3} x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 300 \, d^{3} e \sqrt {x}\right )} n^{2}}{e^{4}}\right )} a b^{2} - \frac {1}{576} \, {\left (72 \, e n {\left (\frac {12 \, d^{4} \log \left (e \sqrt {x} + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{2} - 4 \, d e^{2} x^{\frac {3}{2}} + 6 \, d^{2} e x - 12 \, d^{3} \sqrt {x}}{e^{4}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (288 \, d^{4} \log \left (e \sqrt {x} + d\right )^{3} + 27 \, e^{4} x^{2} + 1800 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 148 \, d e^{3} x^{\frac {3}{2}} + 690 \, d^{2} e^{2} x + 4980 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 4980 \, d^{3} e \sqrt {x}\right )} n^{2}}{e^{5}} - \frac {12 \, {\left (9 \, e^{4} x^{2} + 72 \, d^{4} \log \left (e \sqrt {x} + d\right )^{2} - 28 \, d e^{3} x^{\frac {3}{2}} + 78 \, d^{2} e^{2} x + 300 \, d^{4} \log \left (e \sqrt {x} + d\right ) - 300 \, d^{3} e \sqrt {x}\right )} n \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )}{e^{5}}\right )}\right )} b^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2*log((e*sqrt(x) + d)^n*c)^3 + 3/2*a*b^2*x^2*log((e*sqrt(x) + d)^n*c)^2 - 1/8*a^2*b*e*n*(12*d^4*log(
e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4*d*e^2*x^(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4) + 3/2*a^2*b*x^2*log((e*sq
rt(x) + d)^n*c) + 1/2*a^3*x^2 - 1/48*(12*e*n*(12*d^4*log(e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4*d*e^2*x^(3/2) + 6
*d^2*e*x - 12*d^3*sqrt(x))/e^4)*log((e*sqrt(x) + d)^n*c) - (9*e^4*x^2 + 72*d^4*log(e*sqrt(x) + d)^2 - 28*d*e^3
*x^(3/2) + 78*d^2*e^2*x + 300*d^4*log(e*sqrt(x) + d) - 300*d^3*e*sqrt(x))*n^2/e^4)*a*b^2 - 1/576*(72*e*n*(12*d
^4*log(e*sqrt(x) + d)/e^5 + (3*e^3*x^2 - 4*d*e^2*x^(3/2) + 6*d^2*e*x - 12*d^3*sqrt(x))/e^4)*log((e*sqrt(x) + d
)^n*c)^2 + e*n*((288*d^4*log(e*sqrt(x) + d)^3 + 27*e^4*x^2 + 1800*d^4*log(e*sqrt(x) + d)^2 - 148*d*e^3*x^(3/2)
+ 690*d^2*e^2*x + 4980*d^4*log(e*sqrt(x) + d) - 4980*d^3*e*sqrt(x))*n^2/e^5 - 12*(9*e^4*x^2 + 72*d^4*log(e*sq
rt(x) + d)^2 - 28*d*e^3*x^(3/2) + 78*d^2*e^2*x + 300*d^4*log(e*sqrt(x) + d) - 300*d^3*e*sqrt(x))*n*log((e*sqrt
(x) + d)^n*c)/e^5))*b^3

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mupad [B]  time = 0.77, size = 840, normalized size = 1.41 \[ {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^3\,\left (\frac {b^3\,x^2}{2}-\frac {b^3\,d^4}{2\,e^4}\right )-x^{3/2}\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{3\,e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{36\,e}\right )-{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (\frac {x^{3/2}\,\left (\frac {b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {4\,a\,b^2\,d}{e}\right )}{2}-\frac {3\,b^2\,x^2\,\left (4\,a-b\,n\right )}{8}+\frac {d\,\left (12\,a\,b^2\,d^3-25\,b^3\,d^3\,n\right )}{8\,e^4}+\frac {d^2\,\sqrt {x}\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{4\,e^2}-\frac {d\,x\,\left (\frac {6\,b^2\,d\,\left (4\,a-b\,n\right )}{e}-\frac {24\,a\,b^2\,d}{e}\right )}{8\,e}\right )+x\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{2\,e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{16\,e^2}\right )-\sqrt {x}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (2\,a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{4}-\frac {3\,b^3\,n^3}{16}\right )}{e}-\frac {d\,\left (24\,a^3-12\,a\,b^2\,n^2+7\,b^3\,n^3\right )}{12\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (12\,a-13\,b\,n\right )}{8\,e^2}\right )}{e}+\frac {b^2\,d^3\,n^2\,\left (12\,a-25\,b\,n\right )}{4\,e^3}\right )+x^2\,\left (\frac {a^3}{2}-\frac {3\,a^2\,b\,n}{8}+\frac {3\,a\,b^2\,n^2}{16}-\frac {3\,b^3\,n^3}{64}\right )+\frac {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\frac {x^{3/2}\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{12\,e^2}-\frac {x\,\left (\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2\right )}{8\,e^2}+\frac {\sqrt {x}\,\left (\frac {d\,\left (\frac {d\,\left (16\,b\,d\,e^3\,\left (6\,a^2-b^2\,n^2\right )-12\,b\,d\,e^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )\right )}{e}-24\,b^3\,d^2\,e^2\,n^2\right )}{e}-48\,b^3\,d^3\,e\,n^2\right )}{4\,e^2}+\frac {3\,b\,e^2\,x^2\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{4}\right )}{4\,e^2}-\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (72\,a^2\,b\,d^4\,n-300\,a\,b^2\,d^4\,n^2+415\,b^3\,d^4\,n^3\right )}{48\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(c*(d + e*x^(1/2))^n)^3*((b^3*x^2)/2 - (b^3*d^4)/(2*e^4)) - x^(3/2)*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*
n^2)/4 - (3*a^2*b*n)/2))/(3*e) - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(36*e)) - log(c*(d + e*x^(1/2))^n)^2*
((x^(3/2)*((b^2*d*(4*a - b*n))/e - (4*a*b^2*d)/e))/2 - (3*b^2*x^2*(4*a - b*n))/8 + (d*(12*a*b^2*d^3 - 25*b^3*d
^3*n))/(8*e^4) + (d^2*x^(1/2)*((6*b^2*d*(4*a - b*n))/e - (24*a*b^2*d)/e))/(4*e^2) - (d*x*((6*b^2*d*(4*a - b*n)
)/e - (24*a*b^2*d)/e))/(8*e)) + x*((d*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d*(
24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(12*e)))/(2*e) + (b^2*d^2*n^2*(12*a - 13*b*n))/(16*e^2)) - x^(1/2)*((d*((d
*((d*(2*a^3 - (3*b^3*n^3)/16 + (3*a*b^2*n^2)/4 - (3*a^2*b*n)/2))/e - (d*(24*a^3 + 7*b^3*n^3 - 12*a*b^2*n^2))/(
12*e)))/e + (b^2*d^2*n^2*(12*a - 13*b*n))/(8*e^2)))/e + (b^2*d^3*n^2*(12*a - 25*b*n))/(4*e^3)) + x^2*(a^3/2 -
(3*b^3*n^3)/64 + (3*a*b^2*n^2)/16 - (3*a^2*b*n)/8) + (log(c*(d + e*x^(1/2))^n)*((x^(3/2)*(16*b*d*e^3*(6*a^2 -
b^2*n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/(12*e^2) - (x*((d*(16*b*d*e^3*(6*a^2 - b^2*n^2) - 12*b*d*e
^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2))/(8*e^2) + (x^(1/2)*((d*((d*(16*b*d*e^3*(6*a^2 - b^2*
n^2) - 12*b*d*e^3*(8*a^2 + b^2*n^2 - 4*a*b*n)))/e - 24*b^3*d^2*e^2*n^2))/e - 48*b^3*d^3*e*n^2))/(4*e^2) + (3*b
*e^2*x^2*(8*a^2 + b^2*n^2 - 4*a*b*n))/4))/(4*e^2) - (log(d + e*x^(1/2))*(415*b^3*d^4*n^3 - 300*a*b^2*d^4*n^2 +
72*a^2*b*d^4*n))/(48*e^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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