3.2677 \(\int \frac{x^{m-n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=63 \[ \frac{x^{m-n+1} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{2 (m+1)}{n}-1\right );\frac{m+1}{n};-\frac{b x^n}{a}\right )}{a (m-n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0791248, antiderivative size = 72, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{m-n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m-n+1}{n};\frac{m+1}{n};-\frac{b x^n}{a}\right )}{(m-n+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.40322, size = 54, normalized size = 0.86 \[ \frac{x^{m - n + 1} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m - n + 1}{n} \\ \frac{m + 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{n}}{a}} \left (m - n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0584686, size = 72, normalized size = 1.14 \[ \frac{x^{m-n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m-n+1}{n};\frac{m+1}{n};-\frac{b x^n}{a}\right )}{(m-n+1) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{{x}^{m-n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(m-n)/(a+b*x^n)^(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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