3.31 \(\int (x+\sqrt{b+x^2})^a \, dx\)
Optimal. Leaf size=52 \[ \frac{\left (\sqrt{b+x^2}+x\right )^{a+1}}{2 (a+1)}-\frac{b \left (\sqrt{b+x^2}+x\right )^{a-1}}{2 (1-a)} \]
[Out]
-(b*(x + Sqrt[b + x^2])^(-1 + a))/(2*(1 - a)) + (x + Sqrt[b + x^2])^(1 + a)/(2*(1 + a))
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Rubi [A] time = 0.0239267, 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} \[ \frac{\left (\sqrt{b+x^2}+x\right )^{a+1}}{2 (a+1)}-\frac{b \left (\sqrt{b+x^2}+x\right )^{a-1}}{2 (1-a)} \]
Antiderivative was successfully verified.
[In]
Int[(x + Sqrt[b + x^2])^a,x]
[Out]
-(b*(x + Sqrt[b + x^2])^(-1 + a))/(2*(1 - a)) + (x + Sqrt[b + x^2])^(1 + a)/(2*(1 + a))
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{b+x^2}\right )^a \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^{-2+a} \left (b+x^2\right ) \, dx,x,x+\sqrt{b+x^2}\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b x^{-2+a}+x^a\right ) \, dx,x,x+\sqrt{b+x^2}\right )\\ &=-\frac{b \left (x+\sqrt{b+x^2}\right )^{-1+a}}{2 (1-a)}+\frac{\left (x+\sqrt{b+x^2}\right )^{1+a}}{2 (1+a)}\\ \end{align*}
Mathematica [A] time = 0.0595495, size = 46, normalized size = 0.88 \[ \frac{1}{2} \left (\sqrt{b+x^2}+x\right )^{a-1} \left (\frac{\left (\sqrt{b+x^2}+x\right )^2}{a+1}+\frac{b}{a-1}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(x + Sqrt[b + x^2])^a,x]
[Out]
((x + Sqrt[b + x^2])^(-1 + a)*(b/(-1 + a) + (x + Sqrt[b + x^2])^2/(1 + a)))/2
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Maple [B] time = 0.022, size = 120, normalized size = 2.3 \begin{align*}{\frac{a}{4\,\sqrt{\pi }}{b}^{{\frac{a}{2}}+{\frac{1}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+a}{b}^{-a/2-1/2}}{ \left ( 1+a \right ) a \left ( 2\,a-2 \right ) } \left ({\frac{ab}{{x}^{2}}}+a-1 \right ) \left ( \sqrt{1+{\frac{b}{{x}^{2}}}}+1 \right ) ^{a-1}}+4\,{\frac{\sqrt{\pi }{x}^{1+a}{b}^{-a/2-1/2}}{ \left ( 1+a \right ) a}\sqrt{1+{\frac{b}{{x}^{2}}}} \left ( \sqrt{1+{\frac{b}{{x}^{2}}}}+1 \right ) ^{a-1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x+(x^2+b)^(1/2))^a,x)
[Out]
1/4*b^(1/2*a+1/2)/Pi^(1/2)*a*(8*Pi^(1/2)/(1+a)/a*x^(1+a)*b^(-1/2*a-1/2)*(1/x^2*a*b+a-1)/(2*a-2)*((1+1/x^2*b)^(
1/2)+1)^(a-1)+4*Pi^(1/2)/(1+a)/a*x^(1+a)*b^(-1/2*a-1/2)*(1+1/x^2*b)^(1/2)*((1+1/x^2*b)^(1/2)+1)^(a-1))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + \sqrt{x^{2} + b}\right )}^{a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x+(x^2+b)^(1/2))^a,x, algorithm="maxima")
[Out]
integrate((x + sqrt(x^2 + b))^a, x)
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Fricas [A] time = 2.24924, size = 74, normalized size = 1.42 \begin{align*} \frac{{\left (\sqrt{x^{2} + b} a - x\right )}{\left (x + \sqrt{x^{2} + b}\right )}^{a}}{a^{2} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x+(x^2+b)^(1/2))^a,x, algorithm="fricas")
[Out]
(sqrt(x^2 + b)*a - x)*(x + sqrt(x^2 + b))^a/(a^2 - 1)
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Sympy [B] time = 2.65297, 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+b)**(1/2))**a,x)
[Out]
Piecewise((-a**2*b**(9/2)*b**(a/2)*x*sqrt(b/x**2 + 1)*sinh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(9/2)*ga
mma(1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2
)) - a**2*b**(7/2)*b**(a/2)*x**3*sqrt(b/x**2 + 1)*sinh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(9/2)*gamma(
1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) +
a*b**(9/2)*b**(a/2)*x*cosh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*
x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) + a*b**(7/2)*b**(a/2)*x**3*c
osh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*
b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) + 2*a*b**5*b**(a/2)*cosh(a*asinh(x/sqrt(b)) + asinh(
x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*
gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) - 2*a*b**5*b**(a/2)*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 -
a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) - 2*
a*b**4*b**(a/2)*x**2*sqrt(b/x**2 + 1)*sinh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9
/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1
- a/2)) + 4*a*b**4*b**(a/2)*x**2*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*
gamma(1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a
/2)) - 2*a*b**4*b**(a/2)*x**2*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x**2*gamma(1 -
a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) - 2*a*b**3*b**(a/2)*x**4*sqrt(b/x**2 + 1)*s
inh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x*
*2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) + 2*a*b**3*b**(a/2)*x**4*cosh(
a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x**2*g
amma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) - 2*b**4*b**(a/2)*x**2*sqrt(b/x**2
+ 1)*sinh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(
7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) + 2*b**4*b**(a/2)*x**2*
cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7/2)*x
**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) - 2*b**3*b**(a/2)*x**4*sqrt(b
/x**2 + 1)*sinh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2
*b**(7/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)) + 2*b**3*b**(a/2)*
x**4*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(9/2)*gamma(1 - a/2) + 2*a**2*b**(7
/2)*x**2*gamma(1 - a/2) - 2*b**(9/2)*gamma(1 - a/2) - 2*b**(7/2)*x**2*gamma(1 - a/2)), Abs(x**2)/Abs(b) > 1),
(-a**2*b**3*b**(a/2)*sqrt(1 + x**2/b)*sinh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(5/2)*gamma(1 - a/2) - 2
*b**(5/2)*gamma(1 - a/2)) - 2*a*b**(5/2)*b**(a/2)*x*sqrt(1 + x**2/b)*sinh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)
))*gamma(1 - a/2)/(2*a**2*b**(5/2)*gamma(1 - a/2) - 2*b**(5/2)*gamma(1 - a/2)) + a*b**(5/2)*b**(a/2)*x*cosh(a*
asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(5/2)*gamma(1 - a/2) - 2*b**(5/2)*gamma(1 - a/2)) + 2*a*b**3*b**(a/2)
*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(5/2)*gamma(1 - a/2) - 2*b**(5/2)*gamma
(1 - a/2)) + 2*a*b**2*b**(a/2)*x**2*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(5/2
)*gamma(1 - a/2) - 2*b**(5/2)*gamma(1 - a/2)) - 2*b**(5/2)*b**(a/2)*x*sqrt(1 + x**2/b)*sinh(a*asinh(x/sqrt(b))
+ asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(5/2)*gamma(1 - a/2) - 2*b**(5/2)*gamma(1 - a/2)) + 2*b**2*b**(
a/2)*x**2*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(1 - a/2)/(2*a**2*b**(5/2)*gamma(1 - a/2) - 2*b**(5
/2)*gamma(1 - a/2)), True))
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x + \sqrt{x^{2} + b}\right )}^{a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x+(x^2+b)^(1/2))^a,x, algorithm="giac")
[Out]
integrate((x + sqrt(x^2 + b))^a, x)