3.2742 \(\int \frac{x^m}{\sqrt{a+b x^{2+2 m}}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{\sqrt{b} (m+1)} \]

[Out]

ArcTanh[(Sqrt[b]*x^(1 + m))/Sqrt[a + b*x^(2*(1 + m))]]/(Sqrt[b]*(1 + m))

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Rubi [A]  time = 0.0524139, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{\sqrt{b} (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/Sqrt[a + b*x^(2 + 2*m)],x]

[Out]

ArcTanh[(Sqrt[b]*x^(1 + m))/Sqrt[a + b*x^(2*(1 + m))]]/(Sqrt[b]*(1 + m))

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Rubi in Sympy [A]  time = 7.41139, size = 56, normalized size = 1.47 \[ \frac{x^{m + 1} \sqrt{a + b x^{2 m + 2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{a \sqrt{1 + \frac{b x^{2 m + 2}}{a}} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(m + 1)*sqrt(a + b*x**(2*m + 2))*hyper((1/2, 1/2), (3/2,), -b*x**(2*m + 2)/a)
/(a*sqrt(1 + b*x**(2*m + 2)/a)*(m + 1))

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Mathematica [A]  time = 0.0837293, size = 66, normalized size = 1.74 \[ \frac{\sqrt{a} \sqrt{\frac{b x^{2 m+2}}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{b} (m+1) \sqrt{a+b x^{2 m+2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/Sqrt[a + b*x^(2 + 2*m)],x]

[Out]

(Sqrt[a]*Sqrt[1 + (b*x^(2 + 2*m))/a]*ArcSinh[(Sqrt[b]*x^(1 + m))/Sqrt[a]])/(Sqrt
[b]*(1 + m)*Sqrt[a + b*x^(2 + 2*m)])

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{{x}^{m}{\frac{1}{\sqrt{a+b{x}^{2+2\,m}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{2 \, m + 2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^(2*m + 2) + a),x, algorithm="maxima")

[Out]

integrate(x^m/sqrt(b*x^(2*m + 2) + 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/sqrt(b*x^(2*m + 2) + 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/(a+b*x**(2+2*m))**(1/2),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^(2*m + 2) + a),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*x^(2*m + 2) + a), x)