Optimal. Leaf size=17 \[ \frac{\left (\sqrt{b+x^2}+x\right )^a}{a} \]
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Rubi [A] time = 0.0877249, antiderivative size = 17, 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 \[ \frac{\left (\sqrt{b+x^2}+x\right )^a}{a} \]
Antiderivative was successfully verified.
[In] Int[(x + Sqrt[b + x^2])^a/Sqrt[b + x^2],x]
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Rubi in Sympy [A] time = 3.91902, size = 12, normalized size = 0.71 \[ \frac{\left (x + \sqrt{b + x^{2}}\right )^{a}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x+(x**2+b)**(1/2))**a/(x**2+b)**(1/2),x)
[Out]
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Mathematica [A] time = 0.040226, size = 17, normalized size = 1. \[ \frac{\left (\sqrt{b+x^2}+x\right )^a}{a} \]
Antiderivative was successfully verified.
[In] Integrate[(x + Sqrt[b + x^2])^a/Sqrt[b + x^2],x]
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Maple [F] time = 0.037, size = 0, normalized size = 0. \[ \int{1 \left ( x+\sqrt{{x}^{2}+b} \right ) ^{a}{\frac{1}{\sqrt{{x}^{2}+b}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x+(x^2+b)^(1/2))^a/(x^2+b)^(1/2),x)
[Out]
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Maxima [A] time = 1.40199, size = 20, normalized size = 1.18 \[ \frac{{\left (x + \sqrt{x^{2} + b}\right )}^{a}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + b))^a/sqrt(x^2 + b),x, algorithm="maxima")
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Fricas [A] time = 0.241325, size = 20, normalized size = 1.18 \[ \frac{{\left (x + \sqrt{x^{2} + b}\right )}^{a}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + b))^a/sqrt(x^2 + b),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.30149, size = 311, normalized size = 18.29 \[ \begin{cases} - \frac{\sqrt{b} b^{\frac{a}{2}} \sinh{\left (- a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )}}{a x \sqrt{\frac{b}{x^{2}} + 1}} + \frac{b^{\frac{a}{2}} x \cosh{\left (- a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )}}{a \sqrt{b}} - \frac{b^{\frac{a}{2}} x \sinh{\left (- a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )}}{a \sqrt{b} \sqrt{\frac{b}{x^{2}} + 1}} - \frac{2 b^{\frac{a}{2}} \cosh{\left (a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )} \Gamma \left (- \frac{a}{2} + 1\right )}{a^{2} \Gamma \left (- \frac{a}{2}\right )} & \text{for}\: \left |{\frac{x^{2}}{b}}\right | > 1 \\- \frac{b^{\frac{a}{2}} \sinh{\left (- a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )}}{a \sqrt{1 + \frac{x^{2}}{b}}} - \frac{b^{\frac{a}{2}} x^{2} \sinh{\left (- a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )}}{a b \sqrt{1 + \frac{x^{2}}{b}}} + \frac{b^{\frac{a}{2}} x \cosh{\left (- a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} + \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )}}{a \sqrt{b}} - \frac{2 b^{\frac{a}{2}} \cosh{\left (a \operatorname{asinh}{\left (\frac{x}{\sqrt{b}} \right )} \right )} \Gamma \left (- \frac{a}{2} + 1\right )}{a^{2} \Gamma \left (- \frac{a}{2}\right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x+(x**2+b)**(1/2))**a/(x**2+b)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + \sqrt{x^{2} + b}\right )}^{a}}{\sqrt{x^{2} + b}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + b))^a/sqrt(x^2 + b),x, algorithm="giac")
[Out]