3.1.17 \(\int (x+\sqrt {a+x^2})^b \, dx\) [17]
Optimal. Leaf size=52 \[ -\frac {a \left (x+\sqrt {a+x^2}\right )^{-1+b}}{2 (1-b)}+\frac {\left (x+\sqrt {a+x^2}\right )^{1+b}}{2 (1+b)} \]
[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 = {2142, 14}
\begin {gather*} \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)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x + Sqrt[a + x^2])^b,x]
[Out]
-1/2*(a*(x + Sqrt[a + x^2])^(-1 + b))/(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 2142
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} \text {Subst}\left (\int x^{-2+b} \left (a+x^2\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {1}{2} \text {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 = 0.12, size = 43, normalized size = 0.83 \begin {gather*} \frac {\left (x+\sqrt {a+x^2}\right )^{-1+b} \left (a b+(-1+b) x \left (x+\sqrt {a+x^2}\right )\right )}{-1+b^2} \end {gather*}
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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 94.03, size = 738, normalized size = 14.19
result too large to display
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[(x + Sqrt[x^2 + a])^b,x]')
[Out]
Piecewise[{{(-a ^ 2 b + a ^ 2 b Cosh[ArcSinh[x / Sqrt[a]] (1 + b)] - a ^ (3 / 2) x Cosh[b ArcSinh[x / Sqrt[a]]
] + a ^ (3 / 2) b x Sqrt[1 + a / x ^ 2] Sinh[b ArcSinh[x / Sqrt[a]]] - a x ^ 2 Sqrt[1 + a / x ^ 2] Sinh[ArcSin
h[x / Sqrt[a]] (1 + b)] + a x ^ 2 Cosh[ArcSinh[x / Sqrt[a]] (1 + b)] - a b x ^ 2 - a b x ^ 2 Sqrt[1 + a / x ^
2] Sinh[ArcSinh[x / Sqrt[a]] (1 + b)] + 2 a b x ^ 2 Cosh[ArcSinh[x / Sqrt[a]] (1 + b)] - Sqrt[a] x ^ 3 Cosh[b
ArcSinh[x / Sqrt[a]]] + Sqrt[a] b x ^ 3 Sqrt[1 + a / x ^ 2] Sinh[b ArcSinh[x / Sqrt[a]]] - x ^ 4 Sqrt[1 + a /
x ^ 2] Sinh[ArcSinh[x / Sqrt[a]] (1 + b)] + x ^ 4 Cosh[ArcSinh[x / Sqrt[a]] (1 + b)] - b x ^ 4 Sqrt[1 + a / x
^ 2] Sinh[ArcSinh[x / Sqrt[a]] (1 + b)] + b x ^ 4 Cosh[ArcSinh[x / Sqrt[a]] (1 + b)]) a ^ (-1 / 2 + b / 2) / (
-a + a b ^ 2 - x ^ 2 + b ^ 2 x ^ 2), Abs[x ^ 2 / a] > 1}}, 2 a ^ 3 b Cosh[b ArcSinh[x / Sqrt[a]] + ArcSinh[x /
Sqrt[a]]] Gamma[1 - b / 2] a ^ (b / 2) / (-2 a ^ (5 / 2) Gamma[1 - b / 2] + 2 a ^ (5 / 2) b ^ 2 Gamma[1 - b /
2]) - a ^ 3 b ^ 2 Gamma[-b / 2] a ^ (b / 2) Sqrt[1 + x ^ 2 / a] Sinh[b ArcSinh[x / Sqrt[a]]] / (-2 a ^ (5 / 2
) Gamma[1 - b / 2] + 2 a ^ (5 / 2) b ^ 2 Gamma[1 - b / 2]) - 2 a ^ (5 / 2) x Gamma[1 - b / 2] a ^ (b / 2) Sqrt
[1 + x ^ 2 / a] Sinh[b ArcSinh[x / Sqrt[a]] + ArcSinh[x / Sqrt[a]]] / (-2 a ^ (5 / 2) Gamma[1 - b / 2] + 2 a ^
(5 / 2) b ^ 2 Gamma[1 - b / 2]) - 2 a ^ (5 / 2) b x Gamma[1 - b / 2] a ^ (b / 2) Sqrt[1 + x ^ 2 / a] Sinh[b A
rcSinh[x / Sqrt[a]] + ArcSinh[x / Sqrt[a]]] / (-2 a ^ (5 / 2) Gamma[1 - b / 2] + 2 a ^ (5 / 2) b ^ 2 Gamma[1 -
b / 2]) + a ^ (5 / 2) b x Cosh[b ArcSinh[x / Sqrt[a]]] Gamma[-b / 2] a ^ (b / 2) / (-2 a ^ (5 / 2) Gamma[1 -
b / 2] + 2 a ^ (5 / 2) b ^ 2 Gamma[1 - b / 2]) + 2 a ^ 2 x ^ 2 Cosh[b ArcSinh[x / Sqrt[a]] + ArcSinh[x / Sqrt[
a]]] Gamma[1 - b / 2] a ^ (b / 2) / (-2 a ^ (5 / 2) Gamma[1 - b / 2] + 2 a ^ (5 / 2) b ^ 2 Gamma[1 - b / 2]) +
2 a ^ 2 b x ^ 2 Cosh[b ArcSinh[x / Sqrt[a]] + ArcSinh[x / Sqrt[a]]] Gamma[1 - b / 2] a ^ (b / 2) / (-2 a ^ (5
/ 2) Gamma[1 - b / 2] + 2 a ^ (5 / 2) b ^ 2 Gamma[1 - b / 2])]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs.
\(2(44)=88\).
time = 0.03, size = 120, normalized size = 2.31
| | |
| method |
result |
size |
| | |
| meijerg |
\(\frac {a^{\frac {b}{2}+\frac {1}{2}} b \left (\frac {8 \sqrt {\pi }\, x^{1+b} a^{-\frac {b}{2}-\frac {1}{2}} \left (\frac {a b}{x^{2}}+b -1\right ) \left (\sqrt {1+\frac {a}{x^{2}}}+1\right )^{-1+b}}{\left (1+b \right ) b \left (-2+2 b \right )}+\frac {4 \sqrt {\pi }\, x^{1+b} a^{-\frac {b}{2}-\frac {1}{2}} \sqrt {1+\frac {a}{x^{2}}}\, \left (\sqrt {1+\frac {a}{x^{2}}}+1\right )^{-1+b}}{\left (1+b \right ) b}\right )}{4 \sqrt {\pi }}\) |
\(120\) |
| | |
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|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x+(x^2+a)^(1/2))^b,x,method=_RETURNVERBOSE)
[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)*(a*b/x^2+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 \begin {gather*} \text {Failed to integrate} \end {gather*}
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 = 0.32, size = 32, normalized size = 0.62 \begin {gather*} \frac {{\left (\sqrt {x^{2} + a} b - x\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{b}}{b^{2} - 1} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 2147 vs.
\(2 (37) = 74\)
time = 1.73, size = 2147, normalized size = 41.29
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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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x+(x^2+a)^(1/2))^b,x)
[Out]
Could not integrate
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (x+\sqrt {x^2+a}\right )}^b \,d x \end {gather*}
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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