3.30 \(\int \frac{\left (x+\sqrt{b+x^2}\right )^a}{\sqrt{b+x^2}} \, dx\)

Optimal. Leaf size=17 \[ \frac{\left (\sqrt{b+x^2}+x\right )^a}{a} \]

[Out]

(x + Sqrt[b + x^2])^a/a

_______________________________________________________________________________________

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]

[Out]

(x + Sqrt[b + x^2])^a/a

_______________________________________________________________________________________

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]

(x + sqrt(b + x**2))**a/a

_______________________________________________________________________________________

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]

[Out]

(x + Sqrt[b + x^2])^a/a

_______________________________________________________________________________________

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]

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

_______________________________________________________________________________________

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")

[Out]

(x + sqrt(x^2 + b))^a/a

_______________________________________________________________________________________

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]

(x + sqrt(x^2 + b))^a/a

_______________________________________________________________________________________

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]

Piecewise((-sqrt(b)*b**(a/2)*sinh(-a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))/(a*x*s
qrt(b/x**2 + 1)) + b**(a/2)*x*cosh(-a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))/(a*sq
rt(b)) - b**(a/2)*x*sinh(-a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))/(a*sqrt(b)*sqrt
(b/x**2 + 1)) - 2*b**(a/2)*cosh(a*asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(a**2*gamma(
-a/2)), Abs(x**2/b) > 1), (-b**(a/2)*sinh(-a*asinh(x/sqrt(b)) + asinh(x/sqrt(b))
)/(a*sqrt(1 + x**2/b)) - b**(a/2)*x**2*sinh(-a*asinh(x/sqrt(b)) + asinh(x/sqrt(b
)))/(a*b*sqrt(1 + x**2/b)) + b**(a/2)*x*cosh(-a*asinh(x/sqrt(b)) + asinh(x/sqrt(
b)))/(a*sqrt(b)) - 2*b**(a/2)*cosh(a*asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(a**2*gam
ma(-a/2)), True))

_______________________________________________________________________________________

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]

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