3.2653 \(\int \frac{x^{-1+2 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=42 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac{2 a \sqrt{a+b x^n}}{b^2 n} \]

[Out]

(-2*a*Sqrt[a + b*x^n])/(b^2*n) + (2*(a + b*x^n)^(3/2))/(3*b^2*n)

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Rubi [A]  time = 0.0630181, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac{2 a \sqrt{a+b x^n}}{b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*Sqrt[a + b*x^n])/(b^2*n) + (2*(a + b*x^n)^(3/2))/(3*b^2*n)

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Rubi in Sympy [A]  time = 8.21604, size = 36, normalized size = 0.86 \[ - \frac{2 a \sqrt{a + b x^{n}}}{b^{2} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*sqrt(a + b*x**n)/(b**2*n) + 2*(a + b*x**n)**(3/2)/(3*b**2*n)

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Mathematica [A]  time = 0.0366854, size = 30, normalized size = 0.71 \[ \frac{2 \left (b x^n-2 a\right ) \sqrt{a+b x^n}}{3 b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-2*a + b*x^n)*Sqrt[a + b*x^n])/(3*b^2*n)

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Maple [A]  time = 0.031, size = 28, normalized size = 0.7 \[ -{\frac{-2\,b{x}^{n}+4\,a}{3\,{b}^{2}n}\sqrt{a+b{x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-b*x^n+2*a)*(a+b*x^n)^(1/2)/b^2/n

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Maxima [A]  time = 1.463, size = 53, normalized size = 1.26 \[ \frac{2 \,{\left (b^{2} x^{2 \, n} - a b x^{n} - 2 \, a^{2}\right )}}{3 \, \sqrt{b x^{n} + a} b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^2*x^(2*n) - a*b*x^n - 2*a^2)/(sqrt(b*x^n + a)*b^2*n)

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Fricas [A]  time = 0.232123, size = 35, normalized size = 0.83 \[ \frac{2 \, \sqrt{b x^{n} + a}{\left (b x^{n} - 2 \, a\right )}}{3 \, b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*sqrt(b*x^n + a)*(b*x^n - 2*a)/(b^2*n)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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