3.338 \(\int \frac{\left (x+\sqrt{a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=59 \[ \frac{4 \left (\sqrt{a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac{n+2}{2};\frac{n+4}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]

[Out]

(4*(x + Sqrt[a + x^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, -((x +
Sqrt[a + x^2])^2/a)])/(a^2*(2 + n))

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Rubi [A]  time = 0.120211, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{4 \left (\sqrt{a+x^2}+x\right )^{n+2} \, _2F_1\left (2,\frac{n+2}{2};\frac{n+4}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a^2 (n+2)} \]

Antiderivative was successfully verified.

[In]  Int[(x + Sqrt[a + x^2])^n/(a + x^2)^(3/2),x]

[Out]

(4*(x + Sqrt[a + x^2])^(2 + n)*Hypergeometric2F1[2, (2 + n)/2, (4 + n)/2, -((x +
Sqrt[a + x^2])^2/a)])/(a^2*(2 + n))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 4 \int ^{x + \sqrt{a + x^{2}}} \frac{x x^{n}}{\left (a + x^{2}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+(x**2+a)**(1/2))**n/(x**2+a)**(3/2),x)

[Out]

4*Integral(x*x**n/(a + x**2)**2, (x, x + sqrt(a + x**2)))

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Mathematica [A]  time = 0.0484099, size = 0, normalized size = 0. \[ \int \frac{\left (x+\sqrt{a+x^2}\right )^n}{\left (a+x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(x + Sqrt[a + x^2])^n/(a + x^2)^(3/2),x]

[Out]

Integrate[(x + Sqrt[a + x^2])^n/(a + x^2)^(3/2), x]

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \[ \int{1 \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n} \left ({x}^{2}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(x^2+a)^(1/2))^n/(x^2+a)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + sqrt(x^2 + a))^n/(x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

integrate((x + sqrt(x^2 + a))^n/(x^2 + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + sqrt(x^2 + a))^n/(x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

integral((x + sqrt(x^2 + a))^n/(x^2 + a)^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + \sqrt{a + x^{2}}\right )^{n}}{\left (a + x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x+(x**2+a)**(1/2))**n/(x**2+a)**(3/2),x)

[Out]

Integral((x + sqrt(a + x**2))**n/(a + x**2)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{{\left (x^{2} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + sqrt(x^2 + a))^n/(x^2 + a)^(3/2),x, algorithm="giac")

[Out]

integrate((x + sqrt(x^2 + a))^n/(x^2 + a)^(3/2), x)