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

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

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.619151, antiderivative size = 595, normalized size of antiderivative = 1., 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 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])

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 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 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 2295

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

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*}

Mathematica [A]  time = 0.297533, size = 433, normalized size = 0.73 \[ \frac{-12 b \left (72 a^2 \left (d^4-e^4 x^2\right )-12 a b n \left (-6 d^2 e^2 x+12 d^3 e \sqrt{x}+25 d^4+4 d e^3 x^{3/2}-3 e^4 x^2\right )+b^2 n^2 \left (-78 d^2 e^2 x+300 d^3 e \sqrt{x}+415 d^4+28 d e^3 x^{3/2}-9 e^4 x^2\right )\right ) \log \left (c \left (d+e \sqrt{x}\right )^n\right )+72 a^2 b n \left (-6 d^2 e^2 x+12 d^3 e \sqrt{x}+25 d^4+4 d e^3 x^{3/2}-3 e^4 x^2\right )-288 a^3 \left (d^4-e^4 x^2\right )-72 b^2 \left (12 a \left (d^4-e^4 x^2\right )+b n \left (6 d^2 e^2 x-12 d^3 e \sqrt{x}-25 d^4-4 d e^3 x^{3/2}+3 e^4 x^2\right )\right ) \log ^2\left (c \left (d+e \sqrt{x}\right )^n\right )+12 a b^2 n^2 \left (78 d^2 e^2 x-300 d^3 e \sqrt{x}+161 d^4-28 d e^3 x^{3/2}+9 e^4 x^2\right )-288 b^3 \left (d^4-e^4 x^2\right ) \log ^3\left (c \left (d+e \sqrt{x}\right )^n\right )+b^3 e n^3 \sqrt{x} \left (-690 d^2 e \sqrt{x}+4980 d^3+148 d e^2 x-27 e^3 x^{3/2}\right )}{576 e^4} \]

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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Maple [F]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+e\sqrt{x} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.12213, size = 724, normalized size = 1.22 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 2.26034, size = 1875, normalized size = 3.15 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [B]  time = 1.26749, size = 2283, normalized size = 3.84 \begin{align*} \text{result too large to display} \end{align*}

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*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d)^3 - 1152*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d)
^3 + 1728*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d)^3 - 1152*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e +
d)^3 - 216*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d)^2 + 1152*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d)^
2 - 2592*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d)^2 + 3456*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d
)^2 + 108*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) - 768*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d) + 25
92*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) - 6912*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d) - 27*(
sqrt(x)*e + d)^4*e^(-2) + 256*(sqrt(x)*e + d)^3*d*e^(-2) - 1296*(sqrt(x)*e + d)^2*d^2*e^(-2) + 6912*(sqrt(x)*e
+ d)*d^3*e^(-2))*b^3*n^3*e^(-1) + 12*(72*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)^
3*d*e^(-2)*log(sqrt(x)*e + d)^2 + 432*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)*
d^3*e^(-2)*log(sqrt(x)*e + d)^2 - 36*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e^(
-2)*log(sqrt(x)*e + d) - 432*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-2)*
log(sqrt(x)*e + d) + 9*(sqrt(x)*e + d)^4*e^(-2) - 64*(sqrt(x)*e + d)^3*d*e^(-2) + 216*(sqrt(x)*e + d)^2*d^2*e^
(-2) - 576*(sqrt(x)*e + d)*d^3*e^(-2))*b^3*n^2*e^(-1)*log(c) + 72*(12*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e +
d) - 48*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) -
48*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-2) + 16*(sqrt(x)*e + d)^3*d*e^(-2)
- 36*(sqrt(x)*e + d)^2*d^2*e^(-2) + 48*(sqrt(x)*e + d)*d^3*e^(-2))*b^3*n*e^(-1)*log(c)^2 + 288*((sqrt(x)*e + d
)^4 - 4*(sqrt(x)*e + d)^3*d + 6*(sqrt(x)*e + d)^2*d^2 - 4*(sqrt(x)*e + d)*d^3)*b^3*e^(-3)*log(c)^3 + 12*(72*(s
qrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d)^2 + 432*(sqrt(
x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d)^2 - 288*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d)^2 - 36*(sqrt(x
)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) + 192*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d) - 432*(sqrt(x)*e + d)
^2*d^2*e^(-2)*log(sqrt(x)*e + d) + 576*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d) + 9*(sqrt(x)*e + d)^4*e^(
-2) - 64*(sqrt(x)*e + d)^3*d*e^(-2) + 216*(sqrt(x)*e + d)^2*d^2*e^(-2) - 576*(sqrt(x)*e + d)*d^3*e^(-2))*a*b^2
*n^2*e^(-1) + 144*(12*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*
e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) - 48*(sqrt(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d)
- 3*(sqrt(x)*e + d)^4*e^(-2) + 16*(sqrt(x)*e + d)^3*d*e^(-2) - 36*(sqrt(x)*e + d)^2*d^2*e^(-2) + 48*(sqrt(x)*
e + d)*d^3*e^(-2))*a*b^2*n*e^(-1)*log(c) + 864*((sqrt(x)*e + d)^4 - 4*(sqrt(x)*e + d)^3*d + 6*(sqrt(x)*e + d)^
2*d^2 - 4*(sqrt(x)*e + d)*d^3)*a*b^2*e^(-3)*log(c)^2 + 72*(12*(sqrt(x)*e + d)^4*e^(-2)*log(sqrt(x)*e + d) - 48
*(sqrt(x)*e + d)^3*d*e^(-2)*log(sqrt(x)*e + d) + 72*(sqrt(x)*e + d)^2*d^2*e^(-2)*log(sqrt(x)*e + d) - 48*(sqrt
(x)*e + d)*d^3*e^(-2)*log(sqrt(x)*e + d) - 3*(sqrt(x)*e + d)^4*e^(-2) + 16*(sqrt(x)*e + d)^3*d*e^(-2) - 36*(sq
rt(x)*e + d)^2*d^2*e^(-2) + 48*(sqrt(x)*e + d)*d^3*e^(-2))*a^2*b*n*e^(-1) + 864*((sqrt(x)*e + d)^4 - 4*(sqrt(x
)*e + d)^3*d + 6*(sqrt(x)*e + d)^2*d^2 - 4*(sqrt(x)*e + d)*d^3)*a^2*b*e^(-3)*log(c) + 288*((sqrt(x)*e + d)^4 -
4*(sqrt(x)*e + d)^3*d + 6*(sqrt(x)*e + d)^2*d^2 - 4*(sqrt(x)*e + d)*d^3)*a^3*e^(-3))*e^(-1)