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

Optimal. Leaf size=135 \[ 12 b^2 n^2 \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-6 b n \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-12 b^3 n^3 \text{PolyLog}\left (4,\frac{e}{d \sqrt{x}}+1\right )-2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \]

[Out]

-2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3*Log[-(e/(d*Sqrt[x]))] - 6*b*n*(a + b*Log[c*(d + e/Sqrt[x])^n])^2*PolyLog
[2, 1 + e/(d*Sqrt[x])] + 12*b^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[3, 1 + e/(d*Sqrt[x])] - 12*b^3*n^
3*PolyLog[4, 1 + e/(d*Sqrt[x])]

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Rubi [A]  time = 0.197535, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2396, 2433, 2374, 2383, 6589} \[ 12 b^2 n^2 \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )-6 b n \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2-12 b^3 n^3 \text{PolyLog}\left (4,\frac{e}{d \sqrt{x}}+1\right )-2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3*Log[-(e/(d*Sqrt[x]))] - 6*b*n*(a + b*Log[c*(d + e/Sqrt[x])^n])^2*PolyLog
[2, 1 + e/(d*Sqrt[x])] + 12*b^2*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[3, 1 + e/(d*Sqrt[x])] - 12*b^3*n^
3*PolyLog[4, 1 + e/(d*Sqrt[x])]

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 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rule 6589

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{x} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \log \left (-\frac{e}{d \sqrt{x}}\right )+(6 b e n) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \log \left (-\frac{e}{d \sqrt{x}}\right )+(6 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \log \left (-\frac{e}{d \sqrt{x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )+\left (12 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \log \left (-\frac{e}{d \sqrt{x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )+12 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \text{Li}_3\left (1+\frac{e}{d \sqrt{x}}\right )-\left (12 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=-2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \log \left (-\frac{e}{d \sqrt{x}}\right )-6 b n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )+12 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \text{Li}_3\left (1+\frac{e}{d \sqrt{x}}\right )-12 b^3 n^3 \text{Li}_4\left (1+\frac{e}{d \sqrt{x}}\right )\\ \end{align*}

Mathematica [B]  time = 0.262637, size = 532, normalized size = 3.94 \[ 6 b^2 n^2 \left (-2 \text{PolyLog}\left (3,\frac{d \sqrt{x}}{e}+1\right )-2 \text{PolyLog}\left (3,-\frac{d \sqrt{x}}{e}\right )+2 \log \left (\frac{e}{d}+\sqrt{x}\right ) \text{PolyLog}\left (2,\frac{d \sqrt{x}}{e}+1\right )-2 \left (\log \left (d+\frac{e}{\sqrt{x}}\right )-\log \left (\frac{e}{d}+\sqrt{x}\right )\right ) \text{PolyLog}\left (2,-\frac{d \sqrt{x}}{e}\right )+\frac{1}{4} \log ^2(x) \log \left (d+\frac{e}{\sqrt{x}}\right )-\frac{1}{4} \log ^2(x) \log \left (\frac{d \sqrt{x}}{e}+1\right )+\frac{1}{2} \log (x) \log ^2\left (d+\frac{e}{\sqrt{x}}\right )-\frac{1}{2} \log (x) \log ^2\left (\frac{e}{d}+\sqrt{x}\right )+\log ^2\left (\frac{e}{d}+\sqrt{x}\right ) \log \left (-\frac{d \sqrt{x}}{e}\right )-\log (x) \log \left (d+\frac{e}{\sqrt{x}}\right ) \log \left (\frac{d \sqrt{x}}{e}+1\right )+\log (x) \log \left (\frac{e}{d}+\sqrt{x}\right ) \log \left (\frac{d \sqrt{x}}{e}+1\right )+\frac{\log ^3(x)}{24}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )+3 b n \left (2 \text{PolyLog}\left (2,-\frac{e}{d \sqrt{x}}\right )+\log (x) \left (\log \left (d+\frac{e}{\sqrt{x}}\right )-\log \left (\frac{e}{d \sqrt{x}}+1\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2-2 b^3 n^3 \left (6 \text{PolyLog}\left (4,\frac{e}{d \sqrt{x}}+1\right )+3 \log ^2\left (d+\frac{e}{\sqrt{x}}\right ) \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )-6 \log \left (d+\frac{e}{\sqrt{x}}\right ) \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right )+\log \left (-\frac{e}{d \sqrt{x}}\right ) \log ^3\left (d+\frac{e}{\sqrt{x}}\right )\right )+\log (x) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])^3*Log[x] + 3*b*n*(a - b*n*Log[d + e/Sqrt[x]] + b*Log
[c*(d + e/Sqrt[x])^n])^2*((Log[d + e/Sqrt[x]] - Log[1 + e/(d*Sqrt[x])])*Log[x] + 2*PolyLog[2, -(e/(d*Sqrt[x]))
]) + 6*b^2*n^2*(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])*(Log[e/d + Sqrt[x]]^2*Log[-((d*Sqrt[x
])/e)] + (Log[d + e/Sqrt[x]]^2*Log[x])/2 - (Log[e/d + Sqrt[x]]^2*Log[x])/2 - Log[d + e/Sqrt[x]]*Log[1 + (d*Sqr
t[x])/e]*Log[x] + Log[e/d + Sqrt[x]]*Log[1 + (d*Sqrt[x])/e]*Log[x] + (Log[d + e/Sqrt[x]]*Log[x]^2)/4 - (Log[1
+ (d*Sqrt[x])/e]*Log[x]^2)/4 + Log[x]^3/24 + 2*Log[e/d + Sqrt[x]]*PolyLog[2, 1 + (d*Sqrt[x])/e] - 2*(Log[d + e
/Sqrt[x]] - Log[e/d + Sqrt[x]])*PolyLog[2, -((d*Sqrt[x])/e)] - 2*PolyLog[3, 1 + (d*Sqrt[x])/e] - 2*PolyLog[3,
-((d*Sqrt[x])/e)]) - 2*b^3*n^3*(Log[d + e/Sqrt[x]]^3*Log[-(e/(d*Sqrt[x]))] + 3*Log[d + e/Sqrt[x]]^2*PolyLog[2,
1 + e/(d*Sqrt[x])] - 6*Log[d + e/Sqrt[x]]*PolyLog[3, 1 + e/(d*Sqrt[x])] + 6*PolyLog[4, 1 + e/(d*Sqrt[x])])

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Maple [F]  time = 0.347, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x} \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,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} b^{3} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{3} \log \left (x\right ) - \int \frac{2 \,{\left (b^{3} d x + b^{3} e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{3} + 3 \,{\left (b^{3} d n x \log \left (x\right ) - 2 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x + 2 \,{\left (b^{3} d x + b^{3} e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right ) - 2 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} \sqrt{x}\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{2} - 6 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} - 2 \,{\left (b^{3} d \log \left (c\right )^{3} + 3 \, a b^{2} d \log \left (c\right )^{2} + 3 \, a^{2} b d \log \left (c\right ) + a^{3} d\right )} x - 6 \,{\left ({\left (b^{3} d x + b^{3} e \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} +{\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x - 2 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right ) +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} \sqrt{x}\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right ) + 6 \,{\left ({\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} \sqrt{x}\right )} \log \left (x^{\frac{1}{2} \, n}\right ) - 2 \,{\left (b^{3} e \log \left (c\right )^{3} + 3 \, a b^{2} e \log \left (c\right )^{2} + 3 \, a^{2} b e \log \left (c\right ) + a^{3} e\right )} \sqrt{x}}{2 \,{\left (d x^{2} + e x^{\frac{3}{2}}\right )}}\,{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,x, algorithm="maxima")

[Out]

b^3*log((d*sqrt(x) + e)^n)^3*log(x) - integrate(1/2*(2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n))^3 + 3*(b^3*d*n
*x*log(x) - 2*(b^3*d*log(c) + a*b^2*d)*x + 2*(b^3*d*x + b^3*e*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c) + a*b^
2*e)*sqrt(x))*log((d*sqrt(x) + e)^n)^2 - 6*((b^3*d*log(c) + a*b^2*d)*x + (b^3*e*log(c) + a*b^2*e)*sqrt(x))*log
(x^(1/2*n))^2 - 2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x - 6*((b^3*d*x + b^3*e*sqr
t(x))*log(x^(1/2*n))^2 + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x - 2*((b^3*d*log(c) + a*b^2*d)*x + (b^
3*e*log(c) + a*b^2*e)*sqrt(x))*log(x^(1/2*n)) + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*sqrt(x))*log((d*
sqrt(x) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^
2*b*e)*sqrt(x))*log(x^(1/2*n)) - 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*sqrt(x))/(
d*x^2 + e*x^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{3} \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right ) + a^{3}}{x}, 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))^3/x,x, algorithm="fricas")

[Out]

integral((b^3*log(c*((d*x + e*sqrt(x))/x)^n)^3 + 3*a*b^2*log(c*((d*x + e*sqrt(x))/x)^n)^2 + 3*a^2*b*log(c*((d*
x + e*sqrt(x))/x)^n) + a^3)/x, 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))**3/x,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 )}^{3}}{x}\,{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,x, algorithm="giac")

[Out]

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