3.50 \(\int \frac{\log (c (a+b \sqrt{x})^p)}{x} \, dx\)

Optimal. Leaf size=46 \[ 2 p \text{PolyLog}\left (2,\frac{b \sqrt{x}}{a}+1\right )+2 \log \left (-\frac{b \sqrt{x}}{a}\right ) \log \left (c \left (a+b \sqrt{x}\right )^p\right ) \]

[Out]

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

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Rubi [A]  time = 0.0404115, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2454, 2394, 2315} \[ 2 p \text{PolyLog}\left (2,\frac{b \sqrt{x}}{a}+1\right )+2 \log \left (-\frac{b \sqrt{x}}{a}\right ) \log \left (c \left (a+b \sqrt{x}\right )^p\right ) \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*Sqrt[x])^p]/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0028508, size = 47, normalized size = 1.02 \[ 2 p \text{PolyLog}\left (2,\frac{a+b \sqrt{x}}{a}\right )+2 \log \left (-\frac{b \sqrt{x}}{a}\right ) \log \left (c \left (a+b \sqrt{x}\right )^p\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.266, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\ln \left ( c \left ( a+b\sqrt{x} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a+b*x^(1/2))^p)/x,x)

[Out]

int(ln(c*(a+b*x^(1/2))^p)/x,x)

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Maxima [B]  time = 1.06812, size = 107, normalized size = 2.33 \begin{align*} b p{\left (\frac{\log \left (b \sqrt{x} + a\right ) \log \left (x\right )}{b} - \frac{\log \left (x\right ) \log \left (\frac{b \sqrt{x}}{a} + 1\right ) + 2 \,{\rm Li}_2\left (-\frac{b \sqrt{x}}{a}\right )}{b}\right )} - p \log \left (b \sqrt{x} + a\right ) \log \left (x\right ) + \log \left ({\left (b \sqrt{x} + a\right )}^{p} c\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*p*(log(b*sqrt(x) + a)*log(x)/b - (log(x)*log(b*sqrt(x)/a + 1) + 2*dilog(-b*sqrt(x)/a))/b) - p*log(b*sqrt(x)
+ a)*log(x) + log((b*sqrt(x) + a)^p*c)*log(x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b*x^(1/2))^p)/x,x, algorithm="fricas")

[Out]

integral(log((b*sqrt(x) + a)^p*c)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a+b*x**(1/2))**p)/x,x)

[Out]

Integral(log(c*(a + b*sqrt(x))**p)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*sqrt(x) + a)^p*c)/x, x)