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

Optimal. Leaf size=51 \[ -2 b n \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )-2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \]

[Out]

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

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Rubi [A]  time = 0.0502189, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {2454, 2394, 2315} \[ -2 b n \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )-2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))] - 2*b*n*PolyLog[2, 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 2394

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.0029514, size = 53, normalized size = 1.04 \[ -2 b n \text{PolyLog}\left (2,\frac{d+\frac{e}{\sqrt{x}}}{d}\right )+a \log (x)-2 b \log \left (-\frac{e}{d \sqrt{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.346, 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 ) }\, 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))/x,x)

[Out]

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

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Maxima [B]  time = 2.36046, size = 167, normalized size = 3.27 \begin{align*} -2 \,{\left (\log \left (\frac{d \sqrt{x}}{e} + 1\right ) \log \left (\sqrt{x}\right ) +{\rm Li}_2\left (-\frac{d \sqrt{x}}{e}\right )\right )} b n + \frac{b e n \log \left (x\right )^{2} + 4 \, b d n \sqrt{x} \log \left (x\right ) + 4 \, b e \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right ) \log \left (x\right ) - 4 \, b e \log \left (x\right ) \log \left (x^{\frac{1}{2} \, n}\right ) - 8 \, b d n \sqrt{x} + 4 \,{\left (b e \log \left (c\right ) + a e\right )} \log \left (x\right ) - \frac{4 \,{\left (b d n x \log \left (x\right ) - 2 \, b d n x\right )}}{\sqrt{x}}}{4 \, e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(log(d*sqrt(x)/e + 1)*log(sqrt(x)) + dilog(-d*sqrt(x)/e))*b*n + 1/4*(b*e*n*log(x)^2 + 4*b*d*n*sqrt(x)*log(x
) + 4*b*e*log((d*sqrt(x) + e)^n)*log(x) - 4*b*e*log(x)*log(x^(1/2*n)) - 8*b*d*n*sqrt(x) + 4*(b*e*log(c) + a*e)
*log(x) - 4*(b*d*n*x*log(x) - 2*b*d*n*x)/sqrt(x))/e

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

[Out]

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

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

[Out]

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

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

[Out]

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