3.17 \(\int (x+\sqrt{a+x^2})^b \, dx\)

Optimal. Leaf size=52 \[ \frac{\left (\sqrt{a+x^2}+x\right )^{b+1}}{2 (b+1)}-\frac{a \left (\sqrt{a+x^2}+x\right )^{b-1}}{2 (1-b)} \]

[Out]

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

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Rubi [A]  time = 0.0232867, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2117, 14} \[ \frac{\left (\sqrt{a+x^2}+x\right )^{b+1}}{2 (b+1)}-\frac{a \left (\sqrt{a+x^2}+x\right )^{b-1}}{2 (1-b)} \]

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[a + x^2])^b,x]

[Out]

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

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \left (x+\sqrt{a+x^2}\right )^b \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^{-2+b} \left (a+x^2\right ) \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a x^{-2+b}+x^b\right ) \, dx,x,x+\sqrt{a+x^2}\right )\\ &=-\frac{a \left (x+\sqrt{a+x^2}\right )^{-1+b}}{2 (1-b)}+\frac{\left (x+\sqrt{a+x^2}\right )^{1+b}}{2 (1+b)}\\ \end{align*}

Mathematica [A]  time = 0.0386577, size = 43, normalized size = 0.83 \[ \frac{\left (\sqrt{a+x^2}+x\right )^{b-1} \left ((b-1) x \left (\sqrt{a+x^2}+x\right )+a b\right )}{b^2-1} \]

Antiderivative was successfully verified.

[In]

Integrate[(x + Sqrt[a + x^2])^b,x]

[Out]

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

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Maple [B]  time = 0.026, size = 120, normalized size = 2.3 \begin{align*}{\frac{b}{4\,\sqrt{\pi }}{a}^{{\frac{b}{2}}+{\frac{1}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+b}{a}^{-b/2-1/2}}{ \left ( 1+b \right ) b \left ( -2+2\,b \right ) } \left ({\frac{ab}{{x}^{2}}}+b-1 \right ) \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+b}}+4\,{\frac{\sqrt{\pi }{x}^{1+b}{a}^{-b/2-1/2}}{ \left ( 1+b \right ) b}\sqrt{1+{\frac{a}{{x}^{2}}}} \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+b}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a^(1/2*b+1/2)/Pi^(1/2)*b*(8*Pi^(1/2)/(1+b)/b*x^(1+b)*a^(-1/2*b-1/2)*(1/x^2*a*b+b-1)/(-2+2*b)*((1+1/x^2*a)^
(1/2)+1)^(-1+b)+4*Pi^(1/2)/(1+b)/b*x^(1+b)*a^(-1/2*b-1/2)*(1+1/x^2*a)^(1/2)*((1+1/x^2*a)^(1/2)+1)^(-1+b))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + \sqrt{x^{2} + a}\right )}^{b}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^b,x, algorithm="maxima")

[Out]

integrate((x + sqrt(x^2 + a))^b, x)

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Fricas [A]  time = 2.2043, size = 74, normalized size = 1.42 \begin{align*} \frac{{\left (\sqrt{x^{2} + a} b - x\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{b}}{b^{2} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^b,x, algorithm="fricas")

[Out]

(sqrt(x^2 + a)*b - x)*(x + sqrt(x^2 + a))^b/(b^2 - 1)

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Sympy [B]  time = 2.73252, size = 2149, normalized size = 41.33 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**(9/2)*a**(b/2)*b**2*x*sqrt(a/x**2 + 1)*sinh(b*asinh(x/sqrt(a)))*gamma(-b/2)/(2*a**(9/2)*b**2*ga
mma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2
)) + a**(9/2)*a**(b/2)*b*x*cosh(b*asinh(x/sqrt(a)))*gamma(-b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*g
amma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) - a**(7/2)*a**(b/2)*b**2
*x**3*sqrt(a/x**2 + 1)*sinh(b*asinh(x/sqrt(a)))*gamma(-b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma
(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) + a**(7/2)*a**(b/2)*b*x**3*c
osh(b*asinh(x/sqrt(a)))*gamma(-b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a**(7/2)*b
**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) + 2*a**5*a**(b/2)*b*cosh(b*asinh(x/sqrt(a)) + asinh(
x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a**(7/2)*b**2*x**2*
gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) - 2*a**5*a**(b/2)*b*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 -
b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) - 2*
a**4*a**(b/2)*b*x**2*sqrt(a/x**2 + 1)*sinh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b
**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1
- b/2)) + 4*a**4*a**(b/2)*b*x**2*cosh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*
gamma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b
/2)) - 2*a**4*a**(b/2)*b*x**2*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2) + 2*a
**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) - 2*a**4*a**(b/2)*x**2*sqrt(a/x**2 + 1)*sin
h(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(1 -
b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) + 2*a**4*a**(b/2)*x**2*cosh(b*as
inh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(1 - b/2)
+ 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) - 2*a**3*a**(b/2)*b*x**4*sqrt(a/x**2 +
1)*sinh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*ga
mma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) + 2*a**3*a**(b/2)*b*x**4*
cosh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*gamma(
1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) - 2*a**3*a**(b/2)*x**4*sqrt(a
/x**2 + 1)*sinh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(
9/2)*gamma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)) + 2*a**3*a**(b/2)*
x**4*cosh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(9/2)*b**2*gamma(1 - b/2) - 2*a**(9/2)*g
amma(1 - b/2) + 2*a**(7/2)*b**2*x**2*gamma(1 - b/2) - 2*a**(7/2)*x**2*gamma(1 - b/2)), Abs(x**2)/Abs(a) > 1),
(-2*a**(5/2)*a**(b/2)*b*x*sqrt(1 + x**2/a)*sinh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(5
/2)*b**2*gamma(1 - b/2) - 2*a**(5/2)*gamma(1 - b/2)) + a**(5/2)*a**(b/2)*b*x*cosh(b*asinh(x/sqrt(a)))*gamma(-b
/2)/(2*a**(5/2)*b**2*gamma(1 - b/2) - 2*a**(5/2)*gamma(1 - b/2)) - 2*a**(5/2)*a**(b/2)*x*sqrt(1 + x**2/a)*sinh
(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(5/2)*b**2*gamma(1 - b/2) - 2*a**(5/2)*gamma(1 -
b/2)) - a**3*a**(b/2)*b**2*sqrt(1 + x**2/a)*sinh(b*asinh(x/sqrt(a)))*gamma(-b/2)/(2*a**(5/2)*b**2*gamma(1 - b/
2) - 2*a**(5/2)*gamma(1 - b/2)) + 2*a**3*a**(b/2)*b*cosh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)
/(2*a**(5/2)*b**2*gamma(1 - b/2) - 2*a**(5/2)*gamma(1 - b/2)) + 2*a**2*a**(b/2)*b*x**2*cosh(b*asinh(x/sqrt(a))
+ asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(5/2)*b**2*gamma(1 - b/2) - 2*a**(5/2)*gamma(1 - b/2)) + 2*a**2*a**(
b/2)*x**2*cosh(b*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - b/2)/(2*a**(5/2)*b**2*gamma(1 - b/2) - 2*a**(5
/2)*gamma(1 - b/2)), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + \sqrt{x^{2} + a}\right )}^{b}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^b,x, algorithm="giac")

[Out]

integrate((x + sqrt(x^2 + a))^b, x)