3.2495 \(\int \frac{x}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=48 \[ \frac{x^2 \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2}+\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0548144, antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 6.36802, size = 44, normalized size = 0.92 \[ \frac{x^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0466798, size = 58, normalized size = 1.21 \[ \frac{x^2 \sqrt{\frac{a+b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b*x^n],x]

[Out]

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

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^n)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(b*x^n + a), x)