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

Optimal. Leaf size=152 \[ -\frac{2 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt{x}}}\right )}{d^2}+\frac{2 b e^2 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}+\frac{2 b e n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2+\frac{b^2 e^2 n^2 \log (x)}{d^2} \]

[Out]

(2*b*e*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/d^2 + (2*b*e^2*n*Log[1 - d/(d + e/Sqrt[x])]
*(a + b*Log[c*(d + e/Sqrt[x])^n]))/d^2 + x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2 + (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 = 0.354466, antiderivative size = 174, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {2451, 2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ \frac{2 b^2 e^2 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )}{d^2}-\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{d^2}+\frac{2 b e^2 n \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}+\frac{2 b e n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2+\frac{b^2 e^2 n^2 \log (x)}{d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbol] :> With[{k = Denominator[n]}, Di
st[k, Subst[Int[x^(k - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p,
q}, x] && FractionQ[n]

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

Mathematica [A]  time = 0.122766, size = 170, normalized size = 1.12 \[ \frac{b e n \left (b e n \left (\log \left (d \sqrt{x}+e\right ) \left (\log \left (d \sqrt{x}+e\right )-2 \log \left (-\frac{d \sqrt{x}}{e}\right )\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{x}}{e}+1\right )\right )-2 e \log \left (d \sqrt{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )+2 a d \sqrt{x}+2 b d \sqrt{x} \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )+b e n \left (2 \log \left (d+\frac{e}{\sqrt{x}}\right )+\log (x)\right )\right )}{d^2}+x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.336, size = 0, normalized size = 0. \begin{align*} \int \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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -2 \,{\left (e n{\left (\frac{e \log \left (d \sqrt{x} + e\right )}{d^{2}} - \frac{\sqrt{x}}{d}\right )} - x \log \left (c{\left (d + \frac{e}{\sqrt{x}}\right )}^{n}\right )\right )} a b +{\left (x \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{2} - \int -\frac{d x \log \left (c\right )^{2} + e \sqrt{x} \log \left (c\right )^{2} +{\left (d x + e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} -{\left (d n x - 2 \, d x \log \left (c\right ) - 2 \, e \sqrt{x} \log \left (c\right ) + 2 \,{\left (d x + e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right ) - 2 \,{\left (d x \log \left (c\right ) + e \sqrt{x} \log \left (c\right )\right )} \log \left (x^{\frac{1}{2} \, n}\right )}{d x + e \sqrt{x}}\,{d x}\right )} b^{2} + 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, algorithm="maxima")

[Out]

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

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

[Out]

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

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

[Out]

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

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

[Out]

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