3.2644 \(\int x^{-1+2 n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=44 \[ \frac{2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}-\frac{2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]

[Out]

(-2*a*(a + b*x^n)^(3/2))/(3*b^2*n) + (2*(a + b*x^n)^(5/2))/(5*b^2*n)

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Rubi [A]  time = 0.0618802, antiderivative size = 44, 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 \[ \frac{2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}-\frac{2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*(a + b*x^n)^(3/2))/(3*b^2*n) + (2*(a + b*x^n)^(5/2))/(5*b^2*n)

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Rubi in Sympy [A]  time = 8.30575, size = 37, normalized size = 0.84 \[ - \frac{2 a \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{2} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 b^{2} n} \]

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*(a + b*x**n)**(3/2)/(3*b**2*n) + 2*(a + b*x**n)**(5/2)/(5*b**2*n)

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Mathematica [A]  time = 0.0364608, size = 43, normalized size = 0.98 \[ \frac{2 \sqrt{a+b x^n} \left (-2 a^2+a b x^n+3 b^2 x^{2 n}\right )}{15 b^2 n} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[a + b*x^n]*(-2*a^2 + a*b*x^n + 3*b^2*x^(2*n)))/(15*b^2*n)

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Maple [A]  time = 0.031, size = 41, normalized size = 0.9 \[ -{\frac{-6\,{b}^{2} \left ({x}^{n} \right ) ^{2}-2\,a{x}^{n}b+4\,{a}^{2}}{15\,{b}^{2}n}\sqrt{a+b{x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(-3*b^2*(x^n)^2-a*x^n*b+2*a^2)*(a+b*x^n)^(1/2)/b^2/n

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Maxima [A]  time = 1.47734, size = 53, normalized size = 1.2 \[ \frac{2 \,{\left (3 \, b^{2} x^{2 \, n} + a b x^{n} - 2 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(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 = 0.218373, size = 53, normalized size = 1.2 \[ \frac{2 \,{\left (3 \, b^{2} x^{2 \, n} + a b x^{n} - 2 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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