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]
-1/2*a*(x+(x^2+a)^(1/2))^(-1+b)/(1-b)+1/2*(x+(x^2+a)^(1/2))^(1+b)/(1+b)
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Rubi [A] time = 0.02, antiderivative size = 52, normalized size of antiderivative = 1.00,
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 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]]
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]
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 1.38, size = 32, normalized size = 0.62 \[ \frac {{\left (\sqrt {x^{2} + a} b - x\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{b}}{b^{2} - 1} \]
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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x + \sqrt {x^{2} + a}\right )}^{b}\,{d x} \]
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)
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maple [B] time = 0.03, size = 120, normalized size = 2.31 \[ \frac {\left (\frac {8 \sqrt {\pi }\, \left (b +\frac {a b}{x^{2}}-1\right ) a^{-\frac {b}{2}-\frac {1}{2}} x^{b +1} \left (\sqrt {\frac {a}{x^{2}}+1}+1\right )^{b -1}}{\left (b +1\right ) \left (2 b -2\right ) b}+\frac {4 \sqrt {\pi }\, \sqrt {\frac {a}{x^{2}}+1}\, a^{-\frac {b}{2}-\frac {1}{2}} x^{b +1} \left (\sqrt {\frac {a}{x^{2}}+1}+1\right )^{b -1}}{\left (b +1\right ) b}\right ) b \,a^{\frac {b}{2}+\frac {1}{2}}}{4 \sqrt {\pi }} \]
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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x + \sqrt {x^{2} + a}\right )}^{b}\,{d x} \]
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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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (x+\sqrt {x^2+a}\right )}^b \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x + (a + x^2)^(1/2))^b,x)
[Out]
int((x + (a + x^2)^(1/2))^b, x)
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sympy [B] time = 3.53, size = 2147, normalized size = 41.29 \[ \text {result too large to display} \]
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/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*as
inh(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)) + as
inh(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)*g
amma(1 - b/2)), True))
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