3.441 \(\int \frac{(a+b \log (c (d+\frac{e}{\sqrt{x}})^n))^3}{x^4} \, dx\)

Optimal. Leaf size=907 \[ \text{result too large to display} \]

[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)/(e^5*Sqrt[x]) - (12*b^3*d^5*n^3)/(e^5*Sqrt[x]) + (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
*Log[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
*Log[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*Lo
g[c*(d + e/Sqrt[x])^n])^3)/(3*e^6)

________________________________________________________________________________________

Rubi [A]  time = 1.01085, antiderivative size = 907, normalized size of antiderivative = 1., 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 = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \frac{b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^6}{108 e^6}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \left (d+\frac{e}{\sqrt{x}}\right )^6}{3 e^6}+\frac{b n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \left (d+\frac{e}{\sqrt{x}}\right )^6}{6 e^6}-\frac{b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \left (d+\frac{e}{\sqrt{x}}\right )^6}{18 e^6}-\frac{12 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^5}{125 e^6}+\frac{2 d \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \left (d+\frac{e}{\sqrt{x}}\right )^5}{e^6}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \left (d+\frac{e}{\sqrt{x}}\right )^5}{5 e^6}+\frac{12 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \left (d+\frac{e}{\sqrt{x}}\right )^5}{25 e^6}+\frac{15 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{32 e^6}-\frac{5 d^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{e^6}+\frac{15 b d^2 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \left (d+\frac{e}{\sqrt{x}}\right )^4}{4 e^6}-\frac{15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \left (d+\frac{e}{\sqrt{x}}\right )^4}{8 e^6}-\frac{40 b^3 d^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}+\frac{20 d^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{3 e^6}-\frac{20 b d^3 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \left (d+\frac{e}{\sqrt{x}}\right )^3}{3 e^6}+\frac{40 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \left (d+\frac{e}{\sqrt{x}}\right )^3}{9 e^6}+\frac{15 b^3 d^4 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^6}-\frac{5 d^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{e^6}+\frac{15 b d^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^6}-\frac{15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^6}+\frac{2 d^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{6 b d^5 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{12 b^3 d^5 n^2 \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{12 b^3 d^5 n^3}{e^5 \sqrt{x}}+\frac{12 a b^2 d^5 n^2}{e^5 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4,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)/(e^5*Sqrt[x]) - (12*b^3*d^5*n^3)/(e^5*Sqrt[x]) + (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
*Log[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
*Log[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*Lo
g[c*(d + e/Sqrt[x])^n])^3)/(3*e^6)

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 \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{x^4} \, dx &=-\left (2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\left (2 \operatorname{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,\frac{1}{\sqrt{x}}\right )\right )\\ &=-\frac{2 \operatorname{Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^5}+\frac{(10 d) \operatorname{Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^5}-\frac{\left (20 d^2\right ) \operatorname{Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^5}+\frac{\left (20 d^3\right ) \operatorname{Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^5}-\frac{\left (10 d^4\right ) \operatorname{Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^5}+\frac{\left (2 d^5\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac{1}{\sqrt{x}}\right )}{e^5}\\ &=-\frac{2 \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{(10 d) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{\left (20 d^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{\left (20 d^3\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{\left (10 d^4\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{\left (2 d^5\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}\\ &=\frac{2 d^5 \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{5 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{20 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}-\frac{5 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}+\frac{(b n) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{(6 b d n) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{\left (15 b d^2 n\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{\left (20 b d^3 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{\left (15 b d^4 n\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}-\frac{\left (6 b d^5 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}\\ &=-\frac{6 b d^5 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^6}+\frac{15 b d^4 n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^6}-\frac{20 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 e^6}+\frac{15 b d^2 n \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 e^6}-\frac{6 b d n \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{5 e^6}+\frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{6 e^6}+\frac{2 d^5 \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{5 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{20 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}-\frac{5 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}-\frac{\left (b^2 n^2\right ) \operatorname{Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{3 e^6}+\frac{\left (12 b^2 d n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{5 e^6}-\frac{\left (15 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 e^6}+\frac{\left (40 b^2 d^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{3 e^6}-\frac{\left (15 b^2 d^4 n^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}+\frac{\left (12 b^2 d^5 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}\\ &=\frac{15 b^3 d^4 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^6}-\frac{40 b^3 d^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}+\frac{15 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{32 e^6}-\frac{12 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^5}{125 e^6}+\frac{b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^6}{108 e^6}+\frac{12 a b^2 d^5 n^2}{e^5 \sqrt{x}}-\frac{15 b^2 d^4 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^6}+\frac{40 b^2 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{9 e^6}-\frac{15 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{8 e^6}+\frac{12 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{18 e^6}-\frac{6 b d^5 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^6}+\frac{15 b d^4 n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^6}-\frac{20 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 e^6}+\frac{15 b d^2 n \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 e^6}-\frac{6 b d n \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{5 e^6}+\frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{6 e^6}+\frac{2 d^5 \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{5 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{20 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}-\frac{5 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}+\frac{\left (12 b^3 d^5 n^2\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^6}\\ &=\frac{15 b^3 d^4 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^6}-\frac{40 b^3 d^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^3}{27 e^6}+\frac{15 b^3 d^2 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^4}{32 e^6}-\frac{12 b^3 d n^3 \left (d+\frac{e}{\sqrt{x}}\right )^5}{125 e^6}+\frac{b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^6}{108 e^6}+\frac{12 a b^2 d^5 n^2}{e^5 \sqrt{x}}-\frac{12 b^3 d^5 n^3}{e^5 \sqrt{x}}+\frac{12 b^3 d^5 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^6}-\frac{15 b^2 d^4 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^6}+\frac{40 b^2 d^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{9 e^6}-\frac{15 b^2 d^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{8 e^6}+\frac{12 b^2 d n^2 \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{25 e^6}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{18 e^6}-\frac{6 b d^5 n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^6}+\frac{15 b d^4 n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^6}-\frac{20 b d^3 n \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{3 e^6}+\frac{15 b d^2 n \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 e^6}-\frac{6 b d n \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{5 e^6}+\frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{6 e^6}+\frac{2 d^5 \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{5 d^4 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{20 d^3 \left (d+\frac{e}{\sqrt{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}-\frac{5 d^2 \left (d+\frac{e}{\sqrt{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^6}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{3 e^6}\\ \end{align*}

Mathematica [A]  time = 1.54843, size = 950, normalized size = 1.05 \[ \frac{-72000 b^3 n^3 x^3 \log ^3\left (d+\frac{e}{\sqrt{x}}\right ) d^6+809340 b^3 n^3 x^3 \log \left (\sqrt{x} d+e\right ) d^6-529200 a b^2 n^2 x^3 \log \left (\sqrt{x} d+e\right ) d^6+108000 a^2 b n x^3 \log \left (\sqrt{x} d+e\right ) d^6+5400 b^2 n^2 x^3 \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (-20 a+49 b n-20 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \left (2 \log \left (\sqrt{x} d+e\right )-\log (x)\right ) d^6-404670 b^3 n^3 x^3 \log (x) d^6+264600 a b^2 n^2 x^3 \log (x) d^6-54000 a^2 b n x^3 \log (x) d^6+5400 b^2 n^2 x^3 \log ^2\left (d+\frac{e}{\sqrt{x}}\right ) \left (20 a-49 b n+20 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+20 b n \log \left (\sqrt{x} d+e\right )-10 b n \log (x)\right ) d^6-809340 b^3 e n^3 x^{5/2} d^5+529200 a b^2 e n^2 x^{5/2} d^5-108000 a^2 b e n x^{5/2} d^5+140070 b^3 e^2 n^3 x^2 d^4-156600 a b^2 e^2 n^2 x^2 d^4+54000 a^2 b e^2 n x^2 d^4-41180 b^3 e^3 n^3 x^{3/2} d^3+68400 a b^2 e^3 n^2 x^{3/2} d^3-36000 a^2 b e^3 n x^{3/2} d^3+13785 b^3 e^4 n^3 x d^2-33300 a b^2 e^4 n^2 x d^2+27000 a^2 b e^4 n x d^2-4368 b^3 e^5 n^3 \sqrt{x} d+15840 a b^2 e^5 n^2 \sqrt{x} d-21600 a^2 b e^5 n \sqrt{x} d-36000 a^3 e^6+1000 b^3 e^6 n^3-36000 b^3 e^6 \log ^3\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-6000 a b^2 e^6 n^2+18000 a^2 b e^6 n+1800 b^2 \log ^2\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (60 b n x^3 \log \left (\sqrt{x} d+e\right ) d^6-30 b n x^3 \log (x) d^6+e \left (-60 b n x^{5/2} d^5+30 b e n x^2 d^4-20 b e^2 n x^{3/2} d^3+15 b e^3 n x d^2-12 b e^4 n \sqrt{x} d-60 a e^5+10 b e^5 n\right )\right )-60 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (180 b n (49 b n-20 a) x^3 \log \left (\sqrt{x} d+e\right ) d^6+90 b n (20 a-49 b n) x^3 \log (x) d^6+1800 a^2 e^6+b^2 e n^2 \left (-8820 x^{5/2} d^5+2610 e x^2 d^4-1140 e^2 x^{3/2} d^3+555 e^3 x d^2-264 e^4 \sqrt{x} d+100 e^5\right )-60 a b e n \left (-60 x^{5/2} d^5+30 e x^2 d^4-20 e^2 x^{3/2} d^3+15 e^3 x d^2-12 e^4 \sqrt{x} d+10 e^5\right )\right )}{108000 e^6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/Sqrt[x])^n])^3/x^4,x]

[Out]

(-36000*a^3*e^6 + 18000*a^2*b*e^6*n - 6000*a*b^2*e^6*n^2 + 1000*b^3*e^6*n^3 - 21600*a^2*b*d*e^5*n*Sqrt[x] + 15
840*a*b^2*d*e^5*n^2*Sqrt[x] - 4368*b^3*d*e^5*n^3*Sqrt[x] + 27000*a^2*b*d^2*e^4*n*x - 33300*a*b^2*d^2*e^4*n^2*x
+ 13785*b^3*d^2*e^4*n^3*x - 36000*a^2*b*d^3*e^3*n*x^(3/2) + 68400*a*b^2*d^3*e^3*n^2*x^(3/2) - 41180*b^3*d^3*e
^3*n^3*x^(3/2) + 54000*a^2*b*d^4*e^2*n*x^2 - 156600*a*b^2*d^4*e^2*n^2*x^2 + 140070*b^3*d^4*e^2*n^3*x^2 - 10800
0*a^2*b*d^5*e*n*x^(5/2) + 529200*a*b^2*d^5*e*n^2*x^(5/2) - 809340*b^3*d^5*e*n^3*x^(5/2) - 72000*b^3*d^6*n^3*x^
3*Log[d + e/Sqrt[x]]^3 - 36000*b^3*e^6*Log[c*(d + e/Sqrt[x])^n]^3 + 108000*a^2*b*d^6*n*x^3*Log[e + d*Sqrt[x]]
- 529200*a*b^2*d^6*n^2*x^3*Log[e + d*Sqrt[x]] + 809340*b^3*d^6*n^3*x^3*Log[e + d*Sqrt[x]] + 5400*b^2*d^6*n^2*x
^3*Log[d + e/Sqrt[x]]*(-20*a + 49*b*n - 20*b*Log[c*(d + e/Sqrt[x])^n])*(2*Log[e + d*Sqrt[x]] - Log[x]) - 54000
*a^2*b*d^6*n*x^3*Log[x] + 264600*a*b^2*d^6*n^2*x^3*Log[x] - 404670*b^3*d^6*n^3*x^3*Log[x] + 5400*b^2*d^6*n^2*x
^3*Log[d + e/Sqrt[x]]^2*(20*a - 49*b*n + 20*b*Log[c*(d + e/Sqrt[x])^n] + 20*b*n*Log[e + d*Sqrt[x]] - 10*b*n*Lo
g[x]) + 1800*b^2*Log[c*(d + e/Sqrt[x])^n]^2*(e*(-60*a*e^5 + 10*b*e^5*n - 12*b*d*e^4*n*Sqrt[x] + 15*b*d^2*e^3*n
*x - 20*b*d^3*e^2*n*x^(3/2) + 30*b*d^4*e*n*x^2 - 60*b*d^5*n*x^(5/2)) + 60*b*d^6*n*x^3*Log[e + d*Sqrt[x]] - 30*
b*d^6*n*x^3*Log[x]) - 60*b*Log[c*(d + e/Sqrt[x])^n]*(1800*a^2*e^6 + b^2*e*n^2*(100*e^5 - 264*d*e^4*Sqrt[x] + 5
55*d^2*e^3*x - 1140*d^3*e^2*x^(3/2) + 2610*d^4*e*x^2 - 8820*d^5*x^(5/2)) - 60*a*b*e*n*(10*e^5 - 12*d*e^4*Sqrt[
x] + 15*d^2*e^3*x - 20*d^3*e^2*x^(3/2) + 30*d^4*e*x^2 - 60*d^5*x^(5/2)) + 180*b*d^6*n*(-20*a + 49*b*n)*x^3*Log
[e + d*Sqrt[x]] + 90*b*d^6*n*(20*a - 49*b*n)*x^3*Log[x]))/(108000*e^6*x^3)

________________________________________________________________________________________

Maple [F]  time = 0.391, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3/x^4,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/2))^n))^3/x^4,x)

________________________________________________________________________________________

Maxima [A]  time = 1.16268, size = 1166, normalized size = 1.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="maxima")

[Out]

1/60*a^2*b*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^
2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3)) + 1/1800*(60*e*n*(60*d^6*log(d*sqrt(x) + e)/e
^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x)
- 10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt(x))^n) - (1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4
410*d^6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x - 264*d*e^5*
sqrt(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e))*n^2/(e^6*x^3))*a*b^2 + 1/108000*(
1800*e*n*(60*d^6*log(d*sqrt(x) + e)/e^7 - 30*d^6*log(x)/e^7 - (60*d^5*x^(5/2) - 30*d^4*e*x^2 + 20*d^3*e^2*x^(3
/2) - 15*d^2*e^3*x + 12*d*e^4*sqrt(x) - 10*e^5)/(e^6*x^3))*log(c*(d + e/sqrt(x))^n)^2 + e*n*((36000*d^6*x^3*lo
g(d*sqrt(x) + e)^3 - 4500*d^6*x^3*log(x)^3 + 66150*d^6*x^3*log(x)^2 - 404670*d^6*x^3*log(x) - 809340*d^5*e*x^(
5/2) + 140070*d^4*e^2*x^2 - 41180*d^3*e^3*x^(3/2) + 13785*d^2*e^4*x - 4368*d*e^5*sqrt(x) + 1000*e^6 - 5400*(10
*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e)^2 + 60*(450*d^6*x^3*log(x)^2 - 4410*d^6*x^3*log(x) + 13489*d^
6*x^3)*log(d*sqrt(x) + e))*n^2/(e^7*x^3) - 60*(1800*d^6*x^3*log(d*sqrt(x) + e)^2 + 450*d^6*x^3*log(x)^2 - 4410
*d^6*x^3*log(x) - 8820*d^5*e*x^(5/2) + 2610*d^4*e^2*x^2 - 1140*d^3*e^3*x^(3/2) + 555*d^2*e^4*x - 264*d*e^5*sqr
t(x) + 100*e^6 - 180*(10*d^6*x^3*log(x) - 49*d^6*x^3)*log(d*sqrt(x) + e))*n*log(c*(d + e/sqrt(x))^n)/(e^7*x^3)
))*b^3 - 1/3*b^3*log(c*(d + e/sqrt(x))^n)^3/x^3 - a*b^2*log(c*(d + e/sqrt(x))^n)^2/x^3 - a^2*b*log(c*(d + e/sq
rt(x))^n)/x^3 - 1/3*a^3/x^3

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Fricas [A]  time = 1.98959, size = 2684, normalized size = 2.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="fricas")

[Out]

1/108000*(1000*b^3*e^6*n^3 - 36000*b^3*e^6*log(c)^3 - 6000*a*b^2*e^6*n^2 + 18000*a^2*b*e^6*n - 36000*a^3*e^6 +
36000*(b^3*d^6*n^3*x^3 - b^3*e^6*n^3)*log((d*x + e*sqrt(x))/x)^3 + 30*(4669*b^3*d^4*e^2*n^3 - 5220*a*b^2*d^4*
e^2*n^2 + 1800*a^2*b*d^4*e^2*n)*x^2 + 9000*(6*b^3*d^4*e^2*n*x^2 + 3*b^3*d^2*e^4*n*x + 2*b^3*e^6*n - 12*a*b^2*e
^6)*log(c)^2 + 1800*(30*b^3*d^4*e^2*n^3*x^2 + 15*b^3*d^2*e^4*n^3*x + 10*b^3*e^6*n^3 - 60*a*b^2*e^6*n^2 - 3*(49
*b^3*d^6*n^3 - 20*a*b^2*d^6*n^2)*x^3 + 60*(b^3*d^6*n^2*x^3 - b^3*e^6*n^2)*log(c) - 4*(15*b^3*d^5*e*n^3*x^2 + 5
*b^3*d^3*e^3*n^3*x + 3*b^3*d*e^5*n^3)*sqrt(x))*log((d*x + e*sqrt(x))/x)^2 + 15*(919*b^3*d^2*e^4*n^3 - 2220*a*b
^2*d^2*e^4*n^2 + 1800*a^2*b*d^2*e^4*n)*x - 300*(20*b^3*e^6*n^2 - 120*a*b^2*e^6*n + 360*a^2*b*e^6 + 18*(29*b^3*
d^4*e^2*n^2 - 20*a*b^2*d^4*e^2*n)*x^2 + 3*(37*b^3*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n)*x)*log(c) - 60*(100*b^3*e^
6*n^3 - 600*a*b^2*e^6*n^2 + 1800*a^2*b*e^6*n - (13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^6*n)*x^3
+ 90*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)*x^2 - 1800*(b^3*d^6*n*x^3 - b^3*e^6*n)*log(c)^2 + 15*(37*b^3
*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x - 60*(30*b^3*d^4*e^2*n^2*x^2 + 15*b^3*d^2*e^4*n^2*x + 10*b^3*e^6*n^2 -
60*a*b^2*e^6*n - 3*(49*b^3*d^6*n^2 - 20*a*b^2*d^6*n)*x^3)*log(c) - 12*(22*b^3*d*e^5*n^3 - 60*a*b^2*d*e^5*n^2 +
15*(49*b^3*d^5*e*n^3 - 20*a*b^2*d^5*e*n^2)*x^2 + 5*(19*b^3*d^3*e^3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x - 20*(15*b^3
*d^5*e*n^2*x^2 + 5*b^3*d^3*e^3*n^2*x + 3*b^3*d*e^5*n^2)*log(c))*sqrt(x))*log((d*x + e*sqrt(x))/x) - 4*(1092*b^
3*d*e^5*n^3 - 3960*a*b^2*d*e^5*n^2 + 5400*a^2*b*d*e^5*n + 15*(13489*b^3*d^5*e*n^3 - 8820*a*b^2*d^5*e*n^2 + 180
0*a^2*b*d^5*e*n)*x^2 + 1800*(15*b^3*d^5*e*n*x^2 + 5*b^3*d^3*e^3*n*x + 3*b^3*d*e^5*n)*log(c)^2 + 5*(2059*b^3*d^
3*e^3*n^3 - 3420*a*b^2*d^3*e^3*n^2 + 1800*a^2*b*d^3*e^3*n)*x - 180*(22*b^3*d*e^5*n^2 - 60*a*b^2*d*e^5*n + 15*(
49*b^3*d^5*e*n^2 - 20*a*b^2*d^5*e*n)*x^2 + 5*(19*b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x)*log(c))*sqrt(x))/(e^
6*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e/x**(1/2))**n))**3/x**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right ) + a\right )}^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^3/x^4,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/sqrt(x))^n) + a)^3/x^4, x)