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

Optimal. Leaf size=51 \[ 2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+2 b n \text {Li}_2\left (\frac {\sqrt {x} e}{d}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, 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} \[ 2 b n \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right )+2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \]

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*Sqrt[x])/d)] + 2*b*n*PolyLog[2, 1 + (e*Sqrt[x])/d]

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]

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 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])

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 53, normalized size = 1.04 \[ a \log (x)+2 b \log \left (-\frac {e \sqrt {x}}{d}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )+2 b n \text {Li}_2\left (\frac {d+e \sqrt {x}}{d}\right ) \]

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

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fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) + a}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (e \sqrt {x}+d \right )^{n}\right )+a}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.14, size = 107, normalized size = 2.10 \[ -2 \, {\left (\log \left (\frac {e \sqrt {x}}{d} + 1\right ) \log \left (\sqrt {x}\right ) + {\rm Li}_2\left (-\frac {e \sqrt {x}}{d}\right )\right )} b n + \frac {b d n \log \left (e \sqrt {x} + d\right ) \log \relax (x) + {\left (b d \log \relax (c) + a d\right )} \log \relax (x) - \frac {b e n x \log \relax (x) - 2 \, b e n x}{\sqrt {x}}}{d} + \frac {2 \, {\left (b e n \sqrt {x} \log \left (\sqrt {x}\right ) - b e n \sqrt {x}\right )}}{d} \]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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