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

Optimal. Leaf size=285 \[ -\frac{3 b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^2}+\frac{12 a b^2 d n^2}{e \sqrt{x}}+\frac{3 b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^2}-\frac{6 b d n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^2}+\frac{12 b^3 d n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}+\frac{3 b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^2}-\frac{12 b^3 d n^3}{e \sqrt{x}} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.272381, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ -\frac{3 b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 e^2}+\frac{12 a b^2 d n^2}{e \sqrt{x}}+\frac{3 b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 e^2}-\frac{6 b d n \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{e^2}+\frac{12 b^3 d n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}+\frac{3 b^3 n^3 \left (d+\frac{e}{\sqrt{x}}\right )^2}{4 e^2}-\frac{12 b^3 d n^3}{e \sqrt{x}} \]

Antiderivative was successfully verified.

[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 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^2} \, dx &=-\left (2 \operatorname{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 \operatorname{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 \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}+\frac{(2 d) \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}\\ &=-\frac{2 \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^2}+\frac{(2 d) \operatorname{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) \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^2}-\frac{(6 b d n) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.614716, size = 558, normalized size = 1.96 \[ \frac{-6 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (e \left (2 a^2 e-2 a b n \left (e-2 d \sqrt{x}\right )+b^2 n^2 \left (e-6 d \sqrt{x}\right )\right )+2 b d^2 n x (3 b n-2 a) \log \left (d \sqrt{x}+e\right )+b d^2 n x \log (x) (2 a-3 b n)\right )+12 a^2 b d^2 n x \log \left (d \sqrt{x}+e\right )-6 a^2 b d^2 n x \log (x)-12 a^2 b d e n \sqrt{x}+6 a^2 b e^2 n-4 a^3 e^2+6 b^2 d^2 n^2 x \log ^2\left (d+\frac{e}{\sqrt{x}}\right ) \left (2 a+2 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+2 b n \log \left (d \sqrt{x}+e\right )-b n \log (x)-3 b n\right )+6 b^2 d^2 n^2 x \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (2 \log \left (d \sqrt{x}+e\right )-\log (x)\right ) \left (-2 a-2 b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+3 b n\right )+6 b^2 \log ^2\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \left (e \left (b n \left (e-2 d \sqrt{x}\right )-2 a e\right )+2 b d^2 n x \log \left (d \sqrt{x}+e\right )-b d^2 n x \log (x)\right )-36 a b^2 d^2 n^2 x \log \left (d \sqrt{x}+e\right )+18 a b^2 d^2 n^2 x \log (x)+36 a b^2 d e n^2 \sqrt{x}-6 a b^2 e^2 n^2-4 b^3 e^2 \log ^3\left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-8 b^3 d^2 n^3 x \log ^3\left (d+\frac{e}{\sqrt{x}}\right )+42 b^3 d^2 n^3 x \log \left (d \sqrt{x}+e\right )-21 b^3 d^2 n^3 x \log (x)-42 b^3 d e n^3 \sqrt{x}+3 b^3 e^2 n^3}{4 e^2 x} \]

Antiderivative was successfully verified.

[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]  time = 0.362, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.17022, size = 767, normalized size = 2.69 \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^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

________________________________________________________________________________________

Fricas [B]  time = 1.86846, size = 1162, normalized size = 4.08 \begin{align*} \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} \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^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(-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**2,x)

[Out]

Timed out

________________________________________________________________________________________

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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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