3.533 \(\int \frac{1}{x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \, dx\)

Optimal. Leaf size=85 \[ \frac{\log (x) \left (a+b x^n\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

((a + b*x^n)*Log[x])/(a*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]) - ((a + b*x^n)*Log[a + b*x^n])/(a*n*Sqrt[a^2 + 2*
a*b*x^n + b^2*x^(2*n)])

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Rubi [A]  time = 0.0361088, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {1355, 266, 36, 29, 31} \[ \frac{\log (x) \left (a+b x^n\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]

[Out]

((a + b*x^n)*Log[x])/(a*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]) - ((a + b*x^n)*Log[a + b*x^n])/(a*n*Sqrt[a^2 + 2*
a*b*x^n + b^2*x^(2*n)])

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0148707, size = 42, normalized size = 0.49 \[ \frac{\left (a+b x^n\right ) \left (n \log (x)-\log \left (a+b x^n\right )\right )}{a n \sqrt{\left (a+b x^n\right )^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]

[Out]

((a + b*x^n)*(n*Log[x] - Log[a + b*x^n]))/(a*n*Sqrt[(a + b*x^n)^2])

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Maple [A]  time = 0.018, size = 66, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{ \left ( a+b{x}^{n} \right ) a}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}-{\frac{1}{ \left ( a+b{x}^{n} \right ) an}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}\ln \left ({x}^{n}+{\frac{a}{b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x)

[Out]

((a+b*x^n)^2)^(1/2)/(a+b*x^n)*ln(x)/a-((a+b*x^n)^2)^(1/2)/(a+b*x^n)/a/n*ln(x^n+a/b)

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Maxima [A]  time = 0.993845, size = 36, normalized size = 0.42 \begin{align*} \frac{\log \left (x\right )}{a} - \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/a - log((b*x^n + a)/b)/(a*n)

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Fricas [A]  time = 1.59713, size = 47, normalized size = 0.55 \begin{align*} \frac{n \log \left (x\right ) - \log \left (b x^{n} + a\right )}{a n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n*log(x) - log(b*x^n + a))/(a*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\left (a + b x^{n}\right )^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)

[Out]

Integral(1/(x*sqrt((a + b*x**n)**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x), x)