3.2481 \(\int \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=39 \[ \frac{x \left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{3}{2}+\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a} \]

[Out]

(x*(a + b*x^n)^(3/2)*Hypergeometric2F1[1, 3/2 + n^(-1), 1 + n^(-1), -((b*x^n)/a)
])/a

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Rubi [A]  time = 0.0406641, antiderivative size = 48, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{x \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{\sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^n],x]

[Out]

(x*Sqrt[a + b*x^n]*Hypergeometric2F1[-1/2, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/Sq
rt[1 + (b*x^n)/a]

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Rubi in Sympy [A]  time = 3.44159, size = 41, normalized size = 1.05 \[ \frac{x \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{n} \\ 1 + \frac{1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{\sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0595879, size = 66, normalized size = 1.69 \[ \frac{x \left (a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )+2 \left (a+b x^n\right )\right )}{(n+2) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^n],x]

[Out]

(x*(2*(a + b*x^n) + a*n*Sqrt[1 + (b*x^n)/a]*Hypergeometric2F1[1/2, n^(-1), 1 + n
^(-1), -((b*x^n)/a)]))/((2 + n)*Sqrt[a + b*x^n])

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int \sqrt{a+b{x}^{n}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a), x)

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Fricas [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(sqrt(b*x^n + a),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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((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. \[ \int \sqrt{b x^{n} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(b*x^n + a), x)