Optimal. Leaf size=59 \[ \frac{2 \left (\sqrt{a+x^2}+x\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a (n+1)} \]
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Rubi [A] time = 0.122796, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{2 \left (\sqrt{a+x^2}+x\right )^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\frac{\left (x+\sqrt{x^2+a}\right )^2}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
[In] Int[(x + Sqrt[a + x^2])^n/(a + x^2),x]
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Rubi in Sympy [A] time = 12.2769, size = 46, normalized size = 0.78 \[ \frac{2 \left (x + \sqrt{a + x^{2}}\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{n}{2} + \frac{1}{2} \\ \frac{n}{2} + \frac{3}{2} \end{matrix}\middle |{- \frac{\left (x + \sqrt{a + x^{2}}\right )^{2}}{a}} \right )}}{a \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x+(x**2+a)**(1/2))**n/(x**2+a),x)
[Out]
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Mathematica [A] time = 0.0404769, size = 0, normalized size = 0. \[ \int \frac{\left (x+\sqrt{a+x^2}\right )^n}{a+x^2} \, dx \]
Verification is Not applicable to the result.
[In] Integrate[(x + Sqrt[a + x^2])^n/(a + x^2),x]
[Out]
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Maple [F] time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}+a} \left ( x+\sqrt{{x}^{2}+a} \right ) ^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x+(x^2+a)^(1/2))^n/(x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + a))^n/(x^2 + a),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{x^{2} + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + a))^n/(x^2 + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x + \sqrt{a + x^{2}}\right )^{n}}{a + x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x+(x**2+a)**(1/2))**n/(x**2+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{x^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + a))^n/(x^2 + a),x, algorithm="giac")
[Out]