3.2645 \(\int x^{-1+n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=21 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b n} \]

[Out]

(2*(a + b*x^n)^(3/2))/(3*b*n)

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Rubi [A]  time = 0.0240563, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x^n)^(3/2))/(3*b*n)

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Rubi in Sympy [A]  time = 2.45398, size = 15, normalized size = 0.71 \[ \frac{2 \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(a + b*x**n)**(3/2)/(3*b*n)

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Mathematica [A]  time = 0.0172333, size = 21, normalized size = 1. \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b n} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x^n)^(3/2))/(3*b*n)

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Maple [A]  time = 0.029, size = 18, normalized size = 0.9 \[{\frac{2}{3\,bn} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(a+b*x^n)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43325, size = 23, normalized size = 1.1 \[ \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)

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Fricas [A]  time = 0.218471, size = 23, normalized size = 1.1 \[ \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.215752, size = 23, normalized size = 1.1 \[ \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b*x^n + a)^(3/2)/(b*n)