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

Optimal. Leaf size=155 \[ \frac{2 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt{x}}\right )}{d^2}-\frac{2 b e^2 n \log \left (1-\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}-\frac{2 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}+\frac{b^2 e^2 n^2 \log (x)}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.35124, antiderivative size = 176, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 10, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac{2 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{d^2}+\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2}-\frac{2 b e^2 n \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}-\frac{2 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}+\frac{b^2 e^2 n^2 \log (x)}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*e*n*(d + e*Sqrt[x])*(a + b*Log[c*(d + e*Sqrt[x])^n]))/(d^2*Sqrt[x]) + (e^2*(a + b*Log[c*(d + e*Sqrt[x])^
n])^2)/d^2 - (a + b*Log[c*(d + e*Sqrt[x])^n])^2/x - (2*b*e^2*n*(a + b*Log[c*(d + e*Sqrt[x])^n])*Log[-((e*Sqrt[
x])/d)])/d^2 + (b^2*e^2*n^2*Log[x])/d^2 - (2*b^2*e^2*n^2*PolyLog[2, 1 + (e*Sqrt[x])/d])/d^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 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2317

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2314

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}+(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}+(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}+\frac{(2 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+e \sqrt{x}\right )}{d}-\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+e \sqrt{x}\right )}{d}\\ &=-\frac{2 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2 \sqrt{x}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}-\frac{(2 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt{x}\right )}{d^2}+\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{d^2}+\frac{\left (2 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+e \sqrt{x}\right )}{d^2}\\ &=-\frac{2 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2 \sqrt{x}}+\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}-\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{b^2 e^2 n^2 \log (x)}{d^2}+\frac{\left (2 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+e \sqrt{x}\right )}{d^2}\\ &=-\frac{2 b e n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2 \sqrt{x}}+\frac{e^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{d^2}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{x}-\frac{2 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \log \left (-\frac{e \sqrt{x}}{d}\right )}{d^2}+\frac{b^2 e^2 n^2 \log (x)}{d^2}-\frac{2 b^2 e^2 n^2 \text{Li}_2\left (1+\frac{e \sqrt{x}}{d}\right )}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.163453, size = 188, normalized size = 1.21 \[ 2 \left (b e n \left (-\frac{b e n \text{PolyLog}\left (2,\frac{d+e \sqrt{x}}{d}\right )}{d^2}+\frac{e \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 b d^2 n}-\frac{e \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^2}-\frac{a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )}{d \sqrt{x}}+\frac{b e n \left (\frac{\log (x)}{2 d}-\frac{\log \left (d+e \sqrt{x}\right )}{d}\right )}{d}\right )-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 x}\right ) \]

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

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Maple [F]  time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+e\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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left (\log \left (\frac{e \sqrt{x}}{d} + 1\right ) \log \left (\sqrt{x}\right ) +{\rm Li}_2\left (-\frac{e \sqrt{x}}{d}\right )\right )} b^{2} e^{2} n^{2}}{d^{2}} + \frac{2 \,{\left (a b e^{2} n -{\left (e^{2} n^{2} - e^{2} n \log \left (c\right )\right )} b^{2}\right )} \log \left (e \sqrt{x} + d\right )}{d^{2}} - \frac{2 \,{\left (b^{2} e^{2} n \log \left (c\right ) + a b e^{2} n\right )} \log \left (\sqrt{x}\right )}{d^{2}} + \frac{b^{2} e^{4} n^{2} x + b^{2} d^{2} e^{2} n^{2} \log \left (x\right )}{d^{4}} + \frac{2 \, b^{2} e^{5} n^{2} x^{\frac{3}{2}} - 6 \, b^{2} d^{2} e^{3} n^{2} \sqrt{x} \log \left (\sqrt{x}\right ) - 3 \, b^{2} d e^{4} n^{2} x + 12 \, b^{2} d^{2} e^{3} n^{2} \sqrt{x}}{3 \, d^{5}} - \frac{3 \, b^{2} d^{3} e^{2} n^{2} x^{\frac{3}{2}} \log \left (e \sqrt{x} + d\right )^{2} + 2 \, b^{2} e^{5} n^{2} x^{3} - 3 \, b^{2} d^{2} e^{3} n^{2} x^{2} \log \left (x\right ) + 12 \, b^{2} d^{2} e^{3} n^{2} x^{2} + 3 \, b^{2} d^{5} \sqrt{x} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{2} + 6 \,{\left (b^{2} d^{4} e n \log \left (c\right ) + a b d^{4} e n\right )} x - 3 \,{\left (2 \, b^{2} d^{3} e^{2} n x^{\frac{3}{2}} \log \left (e \sqrt{x} + d\right ) - 2 \, b^{2} d^{4} e n x -{\left (b^{2} d^{3} e^{2} n x \log \left (x\right ) + 2 \, b^{2} d^{5} \log \left (c\right ) + 2 \, a b d^{5}\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )}{3 \, d^{5} x^{\frac{3}{2}}} \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]

2*(log(e*sqrt(x)/d + 1)*log(sqrt(x)) + dilog(-e*sqrt(x)/d))*b^2*e^2*n^2/d^2 + 2*(a*b*e^2*n - (e^2*n^2 - e^2*n*
log(c))*b^2)*log(e*sqrt(x) + d)/d^2 - 2*(b^2*e^2*n*log(c) + a*b*e^2*n)*log(sqrt(x))/d^2 + integrate((b^2*e^4*n
^2*x + b^2*d^2*e^2*n^2)/x, x)/d^4 + 1/3*(2*b^2*e^5*n^2*x^(3/2) - 6*b^2*d^2*e^3*n^2*sqrt(x)*log(sqrt(x)) - 3*b^
2*d*e^4*n^2*x + 12*b^2*d^2*e^3*n^2*sqrt(x))/d^5 - 1/3*(3*b^2*d^3*e^2*n^2*x^(3/2)*log(e*sqrt(x) + d)^2 + 2*b^2*
e^5*n^2*x^3 - 3*b^2*d^2*e^3*n^2*x^2*log(x) + 12*b^2*d^2*e^3*n^2*x^2 + 3*b^2*d^5*sqrt(x)*log((e*sqrt(x) + d)^n)
^2 + 6*(b^2*d^4*e*n*log(c) + a*b*d^4*e*n)*x - 3*(2*b^2*d^3*e^2*n*x^(3/2)*log(e*sqrt(x) + d) - 2*b^2*d^4*e*n*x
- (b^2*d^3*e^2*n*x*log(x) + 2*b^2*d^5*log(c) + 2*a*b*d^5)*sqrt(x))*log((e*sqrt(x) + d)^n))/(d^5*x^(3/2))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\right ) + a^{2}}{x^{2}}, x\right ) \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]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \log{\left (c \left (d + e \sqrt{x}\right )^{n} \right )}\right )^{2}}{x^{2}}\, dx \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]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e \sqrt{x} + d\right )}^{n} c\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((e*sqrt(x) + d)^n*c) + a)^2/x^2, x)