\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt {x}})^n))^3}{x^2} \, dx\) [439]

Optimal result

Integrand size = 24, antiderivative size = 285 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {12 b^3 d n^3}{e \sqrt {x}}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {3 b^2 n^2 \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )}{2 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}+\frac {3 b n \left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{2 e^2}-\frac {6 b d n \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2}{e^2}-\frac {\left (d+\frac {e}{\sqrt {x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {2 d \left (d+\frac {e}{\sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{e^2}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}+\frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}-\frac {12 b^3 d n^3}{e \sqrt {x}} \]

[In]

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

[Out]

(3*b^3*n^3*(d + e/Sqrt[x])^2)/(4*e^2) + (12*a*b^2*d*n^2)/(e*Sqrt[x]) - (12*b^3*d*n^3)/(e*Sqrt[x]) + (12*b^3*d*
n^2*(d + e/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^2 - (3*b^2*n^2*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^
n]))/(2*e^2) - (6*b*d*n*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^2 + (3*b*n*(d + e/Sqrt[x])^2*(a
+ b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*e^2) + (2*d*(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^2 - ((d
+ e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/e^2

Rule 2332

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

Rule 2333

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

Rule 2341

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

Rule 2342

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

Rule 2436

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

Rule 2437

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

Rule 2448

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

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps \begin{align*} \text {integral}& = -\left (2 \text {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\left (2 \text {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\frac {2 \text {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}+\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt {x}}\right )}{e} \\ & = -\frac {2 \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2}+\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = \frac {2 d \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^2}-\frac {\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^2}+\frac {(3 b n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2}-\frac {(6 b d n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = -\frac {6 b d 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^2}+\frac {3 b 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^2}+\frac {2 d \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^2}-\frac {\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^2}-\frac {\left (3 b^2 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2}+\frac {\left (12 b^2 d n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = \frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {3 b^2 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^2}-\frac {6 b d 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^2}+\frac {3 b 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^2}+\frac {2 d \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^2}-\frac {\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^2}+\frac {\left (12 b^3 d n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt {x}}\right )}{e^2} \\ & = \frac {3 b^3 n^3 \left (d+\frac {e}{\sqrt {x}}\right )^2}{4 e^2}+\frac {12 a b^2 d n^2}{e \sqrt {x}}-\frac {12 b^3 d n^3}{e \sqrt {x}}+\frac {12 b^3 d n^2 \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )}{e^2}-\frac {3 b^2 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^2}-\frac {6 b d 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^2}+\frac {3 b 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^2}+\frac {2 d \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^2}-\frac {\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^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-4 a^3 e^2+6 a^2 b e^2 n-6 a b^2 e^2 n^2+3 b^3 e^2 n^3-12 a^2 b d e n \sqrt {x}+36 a b^2 d e n^2 \sqrt {x}-42 b^3 d e n^3 \sqrt {x}-8 b^3 d^2 n^3 x \log ^3\left (d+\frac {e}{\sqrt {x}}\right )-4 b^3 e^2 \log ^3\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+12 a^2 b d^2 n x \log \left (e+d \sqrt {x}\right )-36 a b^2 d^2 n^2 x \log \left (e+d \sqrt {x}\right )+42 b^3 d^2 n^3 x \log \left (e+d \sqrt {x}\right )+6 b^2 d^2 n^2 x \log \left (d+\frac {e}{\sqrt {x}}\right ) \left (-2 a+3 b n-2 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \left (2 \log \left (e+d \sqrt {x}\right )-\log (x)\right )-6 a^2 b d^2 n x \log (x)+18 a b^2 d^2 n^2 x \log (x)-21 b^3 d^2 n^3 x \log (x)+6 b^2 d^2 n^2 x \log ^2\left (d+\frac {e}{\sqrt {x}}\right ) \left (2 a-3 b n+2 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )+2 b n \log \left (e+d \sqrt {x}\right )-b n \log (x)\right )+6 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (-2 a e+b n \left (e-2 d \sqrt {x}\right )\right )+2 b d^2 n x \log \left (e+d \sqrt {x}\right )-b d^2 n x \log (x)\right )-6 b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right ) \left (e \left (2 a^2 e+b^2 n^2 \left (e-6 d \sqrt {x}\right )-2 a b n \left (e-2 d \sqrt {x}\right )\right )+2 b d^2 n (-2 a+3 b n) x \log \left (e+d \sqrt {x}\right )+b d^2 n (2 a-3 b n) x \log (x)\right )}{4 e^2 x} \]

[In]

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

[Out]

(-4*a^3*e^2 + 6*a^2*b*e^2*n - 6*a*b^2*e^2*n^2 + 3*b^3*e^2*n^3 - 12*a^2*b*d*e*n*Sqrt[x] + 36*a*b^2*d*e*n^2*Sqrt
[x] - 42*b^3*d*e*n^3*Sqrt[x] - 8*b^3*d^2*n^3*x*Log[d + e/Sqrt[x]]^3 - 4*b^3*e^2*Log[c*(d + e/Sqrt[x])^n]^3 + 1
2*a^2*b*d^2*n*x*Log[e + d*Sqrt[x]] - 36*a*b^2*d^2*n^2*x*Log[e + d*Sqrt[x]] + 42*b^3*d^2*n^3*x*Log[e + d*Sqrt[x
]] + 6*b^2*d^2*n^2*x*Log[d + e/Sqrt[x]]*(-2*a + 3*b*n - 2*b*Log[c*(d + e/Sqrt[x])^n])*(2*Log[e + d*Sqrt[x]] -
Log[x]) - 6*a^2*b*d^2*n*x*Log[x] + 18*a*b^2*d^2*n^2*x*Log[x] - 21*b^3*d^2*n^3*x*Log[x] + 6*b^2*d^2*n^2*x*Log[d
+ e/Sqrt[x]]^2*(2*a - 3*b*n + 2*b*Log[c*(d + e/Sqrt[x])^n] + 2*b*n*Log[e + d*Sqrt[x]] - b*n*Log[x]) + 6*b^2*L
og[c*(d + e/Sqrt[x])^n]^2*(e*(-2*a*e + b*n*(e - 2*d*Sqrt[x])) + 2*b*d^2*n*x*Log[e + d*Sqrt[x]] - b*d^2*n*x*Log
[x]) - 6*b*Log[c*(d + e/Sqrt[x])^n]*(e*(2*a^2*e + b^2*n^2*(e - 6*d*Sqrt[x]) - 2*a*b*n*(e - 2*d*Sqrt[x])) + 2*b
*d^2*n*(-2*a + 3*b*n)*x*Log[e + d*Sqrt[x]] + b*d^2*n*(2*a - 3*b*n)*x*Log[x]))/(4*e^2*x)

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (249) = 498\).

Time = 0.34 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.90 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {3 \, b^{3} e^{2} n^{3} - 4 \, b^{3} e^{2} \log \left (c\right )^{3} - 6 \, a b^{2} e^{2} n^{2} + 6 \, a^{2} b e^{2} n - 4 \, a^{3} e^{2} + 4 \, {\left (b^{3} d^{2} n^{3} x - b^{3} e^{2} n^{3}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{3} + 6 \, {\left (b^{3} e^{2} n - 2 \, a b^{2} e^{2}\right )} \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{3} d e n^{3} \sqrt {x} - b^{3} e^{2} n^{3} + 2 \, a b^{2} e^{2} n^{2} + {\left (3 \, b^{3} d^{2} n^{3} - 2 \, a b^{2} d^{2} n^{2}\right )} x - 2 \, {\left (b^{3} d^{2} n^{2} x - b^{3} e^{2} n^{2}\right )} \log \left (c\right )\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right )^{2} - 6 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n + 2 \, a^{2} b e^{2}\right )} \log \left (c\right ) - 6 \, {\left (b^{3} e^{2} n^{3} - 2 \, a b^{2} e^{2} n^{2} + 2 \, a^{2} b e^{2} n - 2 \, {\left (b^{3} d^{2} n x - b^{3} e^{2} n\right )} \log \left (c\right )^{2} - {\left (7 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2} + 2 \, a^{2} b d^{2} n\right )} x - 2 \, {\left (b^{3} e^{2} n^{2} - 2 \, a b^{2} e^{2} n - {\left (3 \, b^{3} d^{2} n^{2} - 2 \, a b^{2} d^{2} n\right )} x\right )} \log \left (c\right ) - 2 \, {\left (3 \, b^{3} d e n^{3} - 2 \, b^{3} d e n^{2} \log \left (c\right ) - 2 \, a b^{2} d e n^{2}\right )} \sqrt {x}\right )} \log \left (\frac {d x + e \sqrt {x}}{x}\right ) - 6 \, {\left (7 \, b^{3} d e n^{3} + 2 \, b^{3} d e n \log \left (c\right )^{2} - 6 \, a b^{2} d e n^{2} + 2 \, a^{2} b d e n - 2 \, {\left (3 \, b^{3} d e n^{2} - 2 \, a b^{2} d e n\right )} \log \left (c\right )\right )} \sqrt {x}}{4 \, e^{2} x} \]

[In]

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

[Out]

1/4*(3*b^3*e^2*n^3 - 4*b^3*e^2*log(c)^3 - 6*a*b^2*e^2*n^2 + 6*a^2*b*e^2*n - 4*a^3*e^2 + 4*(b^3*d^2*n^3*x - b^3
*e^2*n^3)*log((d*x + e*sqrt(x))/x)^3 + 6*(b^3*e^2*n - 2*a*b^2*e^2)*log(c)^2 - 6*(2*b^3*d*e*n^3*sqrt(x) - b^3*e
^2*n^3 + 2*a*b^2*e^2*n^2 + (3*b^3*d^2*n^3 - 2*a*b^2*d^2*n^2)*x - 2*(b^3*d^2*n^2*x - b^3*e^2*n^2)*log(c))*log((
d*x + e*sqrt(x))/x)^2 - 6*(b^3*e^2*n^2 - 2*a*b^2*e^2*n + 2*a^2*b*e^2)*log(c) - 6*(b^3*e^2*n^3 - 2*a*b^2*e^2*n^
2 + 2*a^2*b*e^2*n - 2*(b^3*d^2*n*x - b^3*e^2*n)*log(c)^2 - (7*b^3*d^2*n^3 - 6*a*b^2*d^2*n^2 + 2*a^2*b*d^2*n)*x
- 2*(b^3*e^2*n^2 - 2*a*b^2*e^2*n - (3*b^3*d^2*n^2 - 2*a*b^2*d^2*n)*x)*log(c) - 2*(3*b^3*d*e*n^3 - 2*b^3*d*e*n
^2*log(c) - 2*a*b^2*d*e*n^2)*sqrt(x))*log((d*x + e*sqrt(x))/x) - 6*(7*b^3*d*e*n^3 + 2*b^3*d*e*n*log(c)^2 - 6*a
*b^2*d*e*n^2 + 2*a^2*b*d*e*n - 2*(3*b^3*d*e*n^2 - 2*a*b^2*d*e*n)*log(c))*sqrt(x))/(e^2*x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (249) = 498\).

Time = 0.23 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {3}{2} \, a^{2} b e n {\left (\frac {2 \, d^{2} \log \left (d \sqrt {x} + e\right )}{e^{3}} - \frac {d^{2} \log \left (x\right )}{e^{3}} - \frac {2 \, d \sqrt {x} - e}{e^{2} x}\right )} - \frac {b^{3} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{3}}{x} + \frac {3}{4} \, {\left (4 \, e n {\left (\frac {2 \, d^{2} \log \left (d \sqrt {x} + e\right )}{e^{3}} - \frac {d^{2} \log \left (x\right )}{e^{3}} - \frac {2 \, d \sqrt {x} - e}{e^{2} x}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) - \frac {{\left (4 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d e \sqrt {x} + 2 \, e^{2} - 4 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{2} x}\right )} a b^{2} + \frac {1}{8} \, {\left (12 \, e n {\left (\frac {2 \, d^{2} \log \left (d \sqrt {x} + e\right )}{e^{3}} - \frac {d^{2} \log \left (x\right )}{e^{3}} - \frac {2 \, d \sqrt {x} - e}{e^{2} x}\right )} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2} + e n {\left (\frac {{\left (8 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{3} - d^{2} x \log \left (x\right )^{3} + 9 \, d^{2} x \log \left (x\right )^{2} - 42 \, d^{2} x \log \left (x\right ) - 12 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )^{2} - 84 \, d e \sqrt {x} + 6 \, e^{2} + 6 \, {\left (d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) + 14 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )\right )} n^{2}}{e^{3} x} - \frac {6 \, {\left (4 \, d^{2} x \log \left (d \sqrt {x} + e\right )^{2} + d^{2} x \log \left (x\right )^{2} - 6 \, d^{2} x \log \left (x\right ) - 12 \, d e \sqrt {x} + 2 \, e^{2} - 4 \, {\left (d^{2} x \log \left (x\right ) - 3 \, d^{2} x\right )} \log \left (d \sqrt {x} + e\right )\right )} n \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{e^{3} x}\right )}\right )} b^{3} - \frac {3 \, a b^{2} \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )^{2}}{x} - \frac {3 \, a^{2} b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right )}{x} - \frac {a^{3}}{x} \]

[In]

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

[Out]

3/2*a^2*b*e*n*(2*d^2*log(d*sqrt(x) + e)/e^3 - d^2*log(x)/e^3 - (2*d*sqrt(x) - e)/(e^2*x)) - b^3*log(c*(d + e/s
qrt(x))^n)^3/x + 3/4*(4*e*n*(2*d^2*log(d*sqrt(x) + e)/e^3 - d^2*log(x)/e^3 - (2*d*sqrt(x) - e)/(e^2*x))*log(c*
(d + e/sqrt(x))^n) - (4*d^2*x*log(d*sqrt(x) + e)^2 + d^2*x*log(x)^2 - 6*d^2*x*log(x) - 12*d*e*sqrt(x) + 2*e^2
- 4*(d^2*x*log(x) - 3*d^2*x)*log(d*sqrt(x) + e))*n^2/(e^2*x))*a*b^2 + 1/8*(12*e*n*(2*d^2*log(d*sqrt(x) + e)/e^
3 - d^2*log(x)/e^3 - (2*d*sqrt(x) - e)/(e^2*x))*log(c*(d + e/sqrt(x))^n)^2 + e*n*((8*d^2*x*log(d*sqrt(x) + e)^
3 - d^2*x*log(x)^3 + 9*d^2*x*log(x)^2 - 42*d^2*x*log(x) - 12*(d^2*x*log(x) - 3*d^2*x)*log(d*sqrt(x) + e)^2 - 8
4*d*e*sqrt(x) + 6*e^2 + 6*(d^2*x*log(x)^2 - 6*d^2*x*log(x) + 14*d^2*x)*log(d*sqrt(x) + e))*n^2/(e^3*x) - 6*(4*
d^2*x*log(d*sqrt(x) + e)^2 + d^2*x*log(x)^2 - 6*d^2*x*log(x) - 12*d*e*sqrt(x) + 2*e^2 - 4*(d^2*x*log(x) - 3*d^
2*x)*log(d*sqrt(x) + e))*n*log(c*(d + e/sqrt(x))^n)/(e^3*x)))*b^3 - 3*a*b^2*log(c*(d + e/sqrt(x))^n)^2/x - 3*a
^2*b*log(c*(d + e/sqrt(x))^n)/x - a^3/x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 543 vs. \(2 (249) = 498\).

Time = 0.40 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.91 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {4 \, {\left (\frac {2 \, {\left (d \sqrt {x} + e\right )} b^{3} d n^{3}}{e \sqrt {x}} - \frac {{\left (d \sqrt {x} + e\right )}^{2} b^{3} n^{3}}{e x}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{3} + 6 \, {\left (\frac {{\left (b^{3} n^{3} - 2 \, b^{3} n^{2} \log \left (c\right ) - 2 \, a b^{2} n^{2}\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e x} - \frac {4 \, {\left (b^{3} d n^{3} - b^{3} d n^{2} \log \left (c\right ) - a b^{2} d n^{2}\right )} {\left (d \sqrt {x} + e\right )}}{e \sqrt {x}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right )^{2} - 6 \, {\left (\frac {{\left (b^{3} n^{3} - 2 \, b^{3} n^{2} \log \left (c\right ) + 2 \, b^{3} n \log \left (c\right )^{2} - 2 \, a b^{2} n^{2} + 4 \, a b^{2} n \log \left (c\right ) + 2 \, a^{2} b n\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e x} - \frac {4 \, {\left (2 \, b^{3} d n^{3} - 2 \, b^{3} d n^{2} \log \left (c\right ) + b^{3} d n \log \left (c\right )^{2} - 2 \, a b^{2} d n^{2} + 2 \, a b^{2} d n \log \left (c\right ) + a^{2} b d n\right )} {\left (d \sqrt {x} + e\right )}}{e \sqrt {x}}\right )} \log \left (\frac {d \sqrt {x} + e}{\sqrt {x}}\right ) + \frac {{\left (3 \, b^{3} n^{3} - 6 \, b^{3} n^{2} \log \left (c\right ) + 6 \, b^{3} n \log \left (c\right )^{2} - 4 \, b^{3} \log \left (c\right )^{3} - 6 \, a b^{2} n^{2} + 12 \, a b^{2} n \log \left (c\right ) - 12 \, a b^{2} \log \left (c\right )^{2} + 6 \, a^{2} b n - 12 \, a^{2} b \log \left (c\right ) - 4 \, a^{3}\right )} {\left (d \sqrt {x} + e\right )}^{2}}{e x} - \frac {8 \, {\left (6 \, b^{3} d n^{3} - 6 \, b^{3} d n^{2} \log \left (c\right ) + 3 \, b^{3} d n \log \left (c\right )^{2} - b^{3} d \log \left (c\right )^{3} - 6 \, a b^{2} d n^{2} + 6 \, a b^{2} d n \log \left (c\right ) - 3 \, a b^{2} d \log \left (c\right )^{2} + 3 \, a^{2} b d n - 3 \, a^{2} b d \log \left (c\right ) - a^{3} d\right )} {\left (d \sqrt {x} + e\right )}}{e \sqrt {x}}}{4 \, e} \]

[In]

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

[Out]

1/4*(4*(2*(d*sqrt(x) + e)*b^3*d*n^3/(e*sqrt(x)) - (d*sqrt(x) + e)^2*b^3*n^3/(e*x))*log((d*sqrt(x) + e)/sqrt(x)
)^3 + 6*((b^3*n^3 - 2*b^3*n^2*log(c) - 2*a*b^2*n^2)*(d*sqrt(x) + e)^2/(e*x) - 4*(b^3*d*n^3 - b^3*d*n^2*log(c)
- a*b^2*d*n^2)*(d*sqrt(x) + e)/(e*sqrt(x)))*log((d*sqrt(x) + e)/sqrt(x))^2 - 6*((b^3*n^3 - 2*b^3*n^2*log(c) +
2*b^3*n*log(c)^2 - 2*a*b^2*n^2 + 4*a*b^2*n*log(c) + 2*a^2*b*n)*(d*sqrt(x) + e)^2/(e*x) - 4*(2*b^3*d*n^3 - 2*b^
3*d*n^2*log(c) + b^3*d*n*log(c)^2 - 2*a*b^2*d*n^2 + 2*a*b^2*d*n*log(c) + a^2*b*d*n)*(d*sqrt(x) + e)/(e*sqrt(x)
))*log((d*sqrt(x) + e)/sqrt(x)) + (3*b^3*n^3 - 6*b^3*n^2*log(c) + 6*b^3*n*log(c)^2 - 4*b^3*log(c)^3 - 6*a*b^2*
n^2 + 12*a*b^2*n*log(c) - 12*a*b^2*log(c)^2 + 6*a^2*b*n - 12*a^2*b*log(c) - 4*a^3)*(d*sqrt(x) + e)^2/(e*x) - 8
*(6*b^3*d*n^3 - 6*b^3*d*n^2*log(c) + 3*b^3*d*n*log(c)^2 - b^3*d*log(c)^3 - 6*a*b^2*d*n^2 + 6*a*b^2*d*n*log(c)
- 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*n - 3*a^2*b*d*log(c) - a^3*d)*(d*sqrt(x) + e)/(e*sqrt(x)))/e

Mupad [B] (verification not implemented)

Time = 1.82 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {\frac {d\,\left (2\,a^3-3\,a^2\,b\,n+3\,a\,b^2\,n^2-\frac {3\,b^3\,n^3}{2}\right )}{e}-\frac {d\,\left (2\,a^3-6\,a\,b^2\,n^2+9\,b^3\,n^3\right )}{e}}{\sqrt {x}}-{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^3\,\left (\frac {b^3}{x}-\frac {b^3\,d^2}{e^2}\right )+\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\,\left (\frac {\frac {3\,b\,d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {6\,b\,d\,\left (a^2-b^2\,n^2\right )}{e}}{\sqrt {x}}-\frac {3\,b\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{2\,x}\right )+{\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )}^2\,\left (\frac {\frac {3\,b^2\,d\,\left (2\,a-b\,n\right )}{e}-\frac {6\,a\,b^2\,d}{e}}{\sqrt {x}}-\frac {3\,b^2\,\left (2\,a-b\,n\right )}{2\,x}+\frac {3\,d\,\left (2\,a\,b^2\,d-3\,b^3\,d\,n\right )}{2\,e^2}\right )-\frac {a^3-\frac {3\,a^2\,b\,n}{2}+\frac {3\,a\,b^2\,n^2}{2}-\frac {3\,b^3\,n^3}{4}}{x}+\frac {\ln \left (d+\frac {e}{\sqrt {x}}\right )\,\left (6\,a^2\,b\,d^2\,n-18\,a\,b^2\,d^2\,n^2+21\,b^3\,d^2\,n^3\right )}{2\,e^2} \]

[In]

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

[Out]

((d*(2*a^3 - (3*b^3*n^3)/2 + 3*a*b^2*n^2 - 3*a^2*b*n))/e - (d*(2*a^3 + 9*b^3*n^3 - 6*a*b^2*n^2))/e)/x^(1/2) -
log(c*(d + e/x^(1/2))^n)^3*(b^3/x - (b^3*d^2)/e^2) + log(c*(d + e/x^(1/2))^n)*(((3*b*d*(2*a^2 + b^2*n^2 - 2*a*
b*n))/e - (6*b*d*(a^2 - b^2*n^2))/e)/x^(1/2) - (3*b*(2*a^2 + b^2*n^2 - 2*a*b*n))/(2*x)) + log(c*(d + e/x^(1/2)
)^n)^2*(((3*b^2*d*(2*a - b*n))/e - (6*a*b^2*d)/e)/x^(1/2) - (3*b^2*(2*a - b*n))/(2*x) + (3*d*(2*a*b^2*d - 3*b^
3*d*n))/(2*e^2)) - (a^3 - (3*b^3*n^3)/4 + (3*a*b^2*n^2)/2 - (3*a^2*b*n)/2)/x + (log(d + e/x^(1/2))*(21*b^3*d^2
*n^3 - 18*a*b^2*d^2*n^2 + 6*a^2*b*d^2*n))/(2*e^2)