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

Optimal. Leaf size=90 \[ \frac{x^n \left (a+b x^n\right )}{b n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{a \left (a+b x^n\right ) \log \left (a+b x^n\right )}{b^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

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

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Rubi [A]  time = 0.114335, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{x^n \left (a+b x^n\right )}{b n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{a \left (a+b x^n\right ) \log \left (a+b x^n\right )}{b^2 n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 16.7741, size = 73, normalized size = 0.81 \[ - \frac{a \left (a + b x^{n}\right ) \log{\left (a + b x^{n} \right )}}{b^{2} n \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}} + \frac{\sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0479725, size = 44, normalized size = 0.49 \[ \frac{\left (a+b x^n\right ) \left (b x^n-a \log \left (a+b x^n\right )\right )}{b^2 n \sqrt{\left (a+b x^n\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.048, size = 71, normalized size = 0.8 \[{\frac{{x}^{n}}{ \left ( a+b{x}^{n} \right ) bn}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}-{\frac{a}{ \left ( a+b{x}^{n} \right ){b}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}\ln \left ({x}^{n}+{\frac{a}{b}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(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)/b/n*x^n-((a+b*x^n)^2)^(1/2)/(a+b*x^n)*a/b^2/n*ln(x
^n+a/b)

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Maxima [A]  time = 0.760084, size = 43, normalized size = 0.48 \[ \frac{x^{n}}{b n} - \frac{a \log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.272115, size = 32, normalized size = 0.36 \[ \frac{b x^{n} - a \log \left (b x^{n} + a\right )}{b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^n - a*log(b*x^n + a))/(b^2*n)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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