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

Optimal. Leaf size=195 \[ \frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{4 a b d n}{e \sqrt{x}}-\frac{4 b^2 d n \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}+\frac{4 b^2 d n^2}{e \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.1977, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{b n \left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{e^2}-\frac{\left (d+\frac{e}{\sqrt{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}+\frac{2 d \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{e^2}-\frac{4 a b d n}{e \sqrt{x}}-\frac{4 b^2 d n \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}+\frac{4 b^2 d n^2}{e \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*n^2*(d + e/Sqrt[x])^2)/(2*e^2) - (4*a*b*d*n)/(e*Sqrt[x]) + (4*b^2*d*n^2)/(e*Sqrt[x]) - (4*b^2*d*n*(d + e
/Sqrt[x])*Log[c*(d + e/Sqrt[x])^n])/e^2 + (b*n*(d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n]))/e^2 + (2*d*
(d + e/Sqrt[x])*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/e^2 - ((d + e/Sqrt[x])^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^
2)/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 )^2}{x^2} \, dx &=-\left (2 \operatorname{Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, 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 )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^2 \, 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 )^2 \, 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 )^2 \, 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 )^2 \, 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 )^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 )^2}{e^2}+\frac{(2 b n) \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{(4 b d n) \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{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}-\frac{4 a b d n}{e \sqrt{x}}+\frac{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 )}{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 )^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 )^2}{e^2}-\frac{\left (4 b^2 d n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{e^2}\\ &=-\frac{b^2 n^2 \left (d+\frac{e}{\sqrt{x}}\right )^2}{2 e^2}-\frac{4 a b d n}{e \sqrt{x}}+\frac{4 b^2 d n^2}{e \sqrt{x}}-\frac{4 b^2 d n \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )}{e^2}+\frac{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 )}{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 )^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 )^2}{e^2}\\ \end{align*}

Mathematica [C]  time = 0.29817, size = 298, normalized size = 1.53 \[ -\frac{\frac{b n \left (-4 b d^2 n x \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+2 b d^2 n x \left (\log \left (d \sqrt{x}+e\right ) \left (\log \left (d \sqrt{x}+e\right )-2 \log \left (-\frac{d \sqrt{x}}{e}\right )\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{x}}{e}+1\right )\right )-4 d^2 x \log \left (d \sqrt{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-4 d^2 x \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-2 e^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+4 a d e \sqrt{x}+4 b d \sqrt{x} \left (d \sqrt{x}+e\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+b n \left (2 d^2 x \log \left (d+\frac{e}{\sqrt{x}}\right )+e \left (e-2 d \sqrt{x}\right )\right )-4 b d e n \sqrt{x}\right )}{e^2}+2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (b*n*(4*a*d*e*Sqrt[x] - 4*b*d*e*n*Sqrt[x] + b*n*(e*(e - 2*d*Sqrt[x])
+ 2*d^2*x*Log[d + e/Sqrt[x]]) + 4*b*d*(e + d*Sqrt[x])*Sqrt[x]*Log[c*(d + e/Sqrt[x])^n] - 2*e^2*(a + b*Log[c*(d
+ e/Sqrt[x])^n]) - 4*d^2*x*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[e + d*Sqrt[x]] - 4*d^2*x*(a + b*Log[c*(d + e/
Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))] - 4*b*d^2*n*x*PolyLog[2, 1 + e/(d*Sqrt[x])] + 2*b*d^2*n*x*(Log[e + d*Sqrt[x
]]*(Log[e + d*Sqrt[x]] - 2*Log[-((d*Sqrt[x])/e)]) - 2*PolyLog[2, 1 + (d*Sqrt[x])/e])))/e^2)/(2*x)

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Maple [F]  time = 0.331, 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 ) ^{2}}\, 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))^2/x^2,x)

[Out]

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

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Maxima [A]  time = 1.08132, size = 335, normalized size = 1.72 \begin{align*} a 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{1}{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 )} b^{2} - \frac{b^{2} \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )^{2}}{x} - \frac{2 \, a b \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )}{x} - \frac{a^{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))^2/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73133, size = 521, normalized size = 2.67 \begin{align*} -\frac{b^{2} e^{2} n^{2} + 2 \, b^{2} e^{2} \log \left (c\right )^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2} - 2 \,{\left (b^{2} d^{2} n^{2} x - b^{2} e^{2} n^{2}\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right )^{2} - 2 \,{\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} \log \left (c\right ) + 2 \,{\left (2 \, b^{2} d e n^{2} \sqrt{x} - b^{2} e^{2} n^{2} + 2 \, a b e^{2} n +{\left (3 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n\right )} x - 2 \,{\left (b^{2} d^{2} n x - b^{2} e^{2} n\right )} \log \left (c\right )\right )} \log \left (\frac{d x + e \sqrt{x}}{x}\right ) - 2 \,{\left (3 \, b^{2} d e n^{2} - 2 \, b^{2} d e n \log \left (c\right ) - 2 \, a b d e n\right )} \sqrt{x}}{2 \, 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))^2/x^2,x, algorithm="fricas")

[Out]

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

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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))**2/x**2,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 )}^{2}}{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))^2/x^2,x, algorithm="giac")

[Out]

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