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

Optimal. Leaf size=93 \[ -4 b n \text {Li}_2\left (\frac {e}{d \sqrt {x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )-2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2+4 b^2 n^2 \text {Li}_3\left (\frac {e}{d \sqrt {x}}+1\right ) \]

[Out]

-2*(a+b*ln(c*(d+e/x^(1/2))^n))^2*ln(-e/d/x^(1/2))-4*b*n*(a+b*ln(c*(d+e/x^(1/2))^n))*polylog(2,1+e/d/x^(1/2))+4
*b^2*n^2*polylog(3,1+e/d/x^(1/2))

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Rubi [A]  time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {2454, 2396, 2433, 2374, 6589} \[ -4 b n \text {PolyLog}\left (2,\frac {e}{d \sqrt {x}}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )+4 b^2 n^2 \text {PolyLog}\left (3,\frac {e}{d \sqrt {x}}+1\right )-2 \log \left (-\frac {e}{d \sqrt {x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 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 )^2}{x} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{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 )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )+(4 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 )}{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 )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )+(4 b n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \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 )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )-4 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )+\left (4 b^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\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 )^2 \log \left (-\frac {e}{d \sqrt {x}}\right )-4 b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )\right ) \text {Li}_2\left (1+\frac {e}{d \sqrt {x}}\right )+4 b^2 n^2 \text {Li}_3\left (1+\frac {e}{d \sqrt {x}}\right )\\ \end {align*}

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Mathematica [B]  time = 0.36, size = 386, normalized size = 4.15 \[ 2 b n \left (2 \text {Li}_2\left (-\frac {e}{d \sqrt {x}}\right )+\log (x) \left (\log \left (d+\frac {e}{\sqrt {x}}\right )-\log \left (\frac {e}{d \sqrt {x}}+1\right )\right )\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-b n \log \left (d+\frac {e}{\sqrt {x}}\right )\right )+\log (x) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt {x}}\right )^n\right )-b n \log \left (d+\frac {e}{\sqrt {x}}\right )\right )^2+\frac {1}{12} b^2 n^2 \left (-48 \text {Li}_3\left (\frac {\sqrt {x} d}{e}+1\right )-48 \text {Li}_3\left (-\frac {d \sqrt {x}}{e}\right )+48 \text {Li}_2\left (\frac {\sqrt {x} d}{e}+1\right ) \log \left (\frac {e}{d}+\sqrt {x}\right )-48 \text {Li}_2\left (-\frac {d \sqrt {x}}{e}\right ) \left (\log \left (d+\frac {e}{\sqrt {x}}\right )-\log \left (\frac {e}{d}+\sqrt {x}\right )\right )+6 \log ^2(x) \log \left (d+\frac {e}{\sqrt {x}}\right )-6 \log ^2(x) \log \left (\frac {d \sqrt {x}}{e}+1\right )+12 \log (x) \log ^2\left (d+\frac {e}{\sqrt {x}}\right )-12 \log (x) \log ^2\left (\frac {e}{d}+\sqrt {x}\right )+24 \log ^2\left (\frac {e}{d}+\sqrt {x}\right ) \log \left (-\frac {d \sqrt {x}}{e}\right )-24 \log (x) \log \left (d+\frac {e}{\sqrt {x}}\right ) \log \left (\frac {d \sqrt {x}}{e}+1\right )+24 \log (x) \log \left (\frac {e}{d}+\sqrt {x}\right ) \log \left (\frac {d \sqrt {x}}{e}+1\right )+\log ^3(x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])^2*Log[x] + 2*b*n*(a - b*n*Log[d + e/Sqrt[x]] + b*Log
[c*(d + e/Sqrt[x])^n])*((Log[d + e/Sqrt[x]] - Log[1 + e/(d*Sqrt[x])])*Log[x] + 2*PolyLog[2, -(e/(d*Sqrt[x]))])
+ (b^2*n^2*(24*Log[e/d + Sqrt[x]]^2*Log[-((d*Sqrt[x])/e)] + 12*Log[d + e/Sqrt[x]]^2*Log[x] - 12*Log[e/d + Sqr
t[x]]^2*Log[x] - 24*Log[d + e/Sqrt[x]]*Log[1 + (d*Sqrt[x])/e]*Log[x] + 24*Log[e/d + Sqrt[x]]*Log[1 + (d*Sqrt[x
])/e]*Log[x] + 6*Log[d + e/Sqrt[x]]*Log[x]^2 - 6*Log[1 + (d*Sqrt[x])/e]*Log[x]^2 + Log[x]^3 + 48*Log[e/d + Sqr
t[x]]*PolyLog[2, 1 + (d*Sqrt[x])/e] - 48*(Log[d + e/Sqrt[x]] - Log[e/d + Sqrt[x]])*PolyLog[2, -((d*Sqrt[x])/e)
] - 48*PolyLog[3, 1 + (d*Sqrt[x])/e] - 48*PolyLog[3, -((d*Sqrt[x])/e)]))/12

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left (c \left (\frac {d x + e \sqrt {x}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac {d x + e \sqrt {x}}{x}\right )^{n}\right ) + a^{2}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c {\left (d + \frac {e}{\sqrt {x}}\right )}^{n}\right ) + a\right )}^{2}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x,x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (d +\frac {e}{\sqrt {x}}\right )^{n}\right )+a \right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b^{2} n^{2} \log \left (d \sqrt {x} + e\right )^{2} \log \relax (x) - \int \frac {{\left (b^{2} d n x \log \relax (x) - 2 \, {\left (b^{2} d \log \relax (c) + a b d\right )} x + 2 \, {\left (b^{2} d x + b^{2} e \sqrt {x}\right )} \log \left (x^{\frac {1}{2} \, n}\right ) - 2 \, {\left (b^{2} e \log \relax (c) + a b e\right )} \sqrt {x}\right )} n \log \left (d \sqrt {x} + e\right ) - {\left (b^{2} d x + b^{2} e \sqrt {x}\right )} \log \left (x^{\frac {1}{2} \, n}\right )^{2} - {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} x + 2 \, {\left ({\left (b^{2} d \log \relax (c) + a b d\right )} x + {\left (b^{2} e \log \relax (c) + a b e\right )} \sqrt {x}\right )} \log \left (x^{\frac {1}{2} \, n}\right ) - {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} \sqrt {x}}{d x^{2} + e x^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(1/2))^n))^2/x,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{\sqrt {x}}\right )}^n\right )\right )}^2}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt {x}}\right )^{n} \right )}\right )^{2}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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