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 = 0.101774, 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 \[ \frac{\log (x) \left (a+b x^n\right )}{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{\left (a+b x^n\right ) \log \left (a+b x^n\right )}{a n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

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)])

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Rubi in Sympy [A]  time = 20.457, size = 90, normalized size = 1.06 \[ \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}} \log{\left (x^{n} \right )}}{a n \left (a b + b^{2} x^{n}\right )} - \frac{b \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}} \log{\left (a + b x^{n} \right )}}{a n \left (a b + b^{2} x^{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)

[Out]

b*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))*log(x**n)/(a*n*(a*b + b**2*x**n)) - b*
sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))*log(a + b*x**n)/(a*n*(a*b + b**2*x**n))

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Mathematica [A]  time = 0.0352295, 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 = 0.029, size = 66, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{ \left ( a+b{x}^{n} \right ) a}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}-{\frac{1}{ \left ( a+b{x}^{n} \right ) an}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}\ln \left ({x}^{n}+{\frac{a}{b}} \right ) } \]

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 = 0.763262, size = 36, normalized size = 0.42 \[ \frac{\log \left (x\right )}{a} - \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/a - log((b*x^n + a)/b)/(a*n)

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Fricas [A]  time = 0.281927, size = 30, normalized size = 0.35 \[ \frac{n \log \left (x\right ) - \log \left (b x^{n} + a\right )}{a n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x),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. \[ \int \frac{1}{x \sqrt{\left (a + b x^{n}\right )^{2}}}\, dx \]

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/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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