\(\int \frac {\log (c (a+b \sqrt {x})^p)}{x} \, dx\) [50]

Optimal result

Integrand size = 18, antiderivative size = 46 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \log \left (-\frac {b \sqrt {x}}{a}\right )+2 p \operatorname {PolyLog}\left (2,1+\frac {b \sqrt {x}}{a}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2504, 2441, 2352} \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=2 \log \left (-\frac {b \sqrt {x}}{a}\right ) \log \left (c \left (a+b \sqrt {x}\right )^p\right )+2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {x} b}{a}+1\right ) \]

[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 2352

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

Rule 2441

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 2504

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])

Rubi steps \begin{align*} \text {integral}& = 2 \text {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) \text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.02 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=2 \log \left (c \left (a+b \sqrt {x}\right )^p\right ) \log \left (-\frac {b \sqrt {x}}{a}\right )+2 p \operatorname {PolyLog}\left (2,\frac {a+b \sqrt {x}}{a}\right ) \]

[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]

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26

method	result	size
parts	\(\ln \left (c \left (a +b \sqrt {x}\right )^{p}\right ) \ln \left (x \right )-\frac {p b \left (\frac {4 \operatorname {dilog}\left (\frac {a +b \sqrt {x}}{a}\right )}{b}+\frac {2 \ln \left (x \right ) \ln \left (\frac {a +b \sqrt {x}}{a}\right )}{b}\right )}{2}\)	\(58\)

[In]

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

[Out]

ln(c*(a+b*x^(1/2))^p)*ln(x)-1/2*p*b*(4*dilog((a+b*x^(1/2))/a)/b+2*ln(x)*ln((a+b*x^(1/2))/a)/b)

Fricas [F]

\[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right )}{x} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=\int \frac {\log {\left (c \left (a + b \sqrt {x}\right )^{p} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (39) = 78\).

Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.72 \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=b p {\left (\frac {\log \left (b \sqrt {x} + a\right ) \log \left (x\right )}{b} - \frac {\log \left (x\right ) \log \left (\frac {b \sqrt {x}}{a} + 1\right ) + 2 \, {\rm Li}_2\left (-\frac {b \sqrt {x}}{a}\right )}{b}\right )} - p \log \left (b \sqrt {x} + a\right ) \log \left (x\right ) + \log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right ) \log \left (x\right ) \]

[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)

Giac [F]

\[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=\int { \frac {\log \left ({\left (b \sqrt {x} + a\right )}^{p} c\right )}{x} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (a+b \sqrt {x}\right )^p\right )}{x} \, dx=\int \frac {\ln \left (c\,{\left (a+b\,\sqrt {x}\right )}^p\right )}{x} \,d x \]

[In]

int(log(c*(a + b*x^(1/2))^p)/x,x)

[Out]

int(log(c*(a + b*x^(1/2))^p)/x, x)