3.1.31 \(\int (x+\sqrt {b+x^2})^a \, dx\) [31]
Optimal. Leaf size=52 \[ -\frac {b \left (x+\sqrt {b+x^2}\right )^{-1+a}}{2 (1-a)}+\frac {\left (x+\sqrt {b+x^2}\right )^{1+a}}{2 (1+a)} \]
[Out]
-1/2*b*(x+(x^2+b)^(1/2))^(-1+a)/(1-a)+1/2*(x+(x^2+b)^(1/2))^(1+a)/(1+a)
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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 {b+x^2}+x\right )^{a+1}}{2 (a+1)}-\frac {b \left (\sqrt {b+x^2}+x\right )^{a-1}}{2 (1-a)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x + Sqrt[b + x^2])^a,x]
[Out]
-1/2*(b*(x + Sqrt[b + x^2])^(-1 + a))/(1 - a) + (x + Sqrt[b + x^2])^(1 + a)/(2*(1 + a))
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 {b+x^2}\right )^a \, dx &=\frac {1}{2} \text {Subst}\left (\int x^{-2+a} \left (b+x^2\right ) \, dx,x,x+\sqrt {b+x^2}\right )\\ &=\frac {1}{2} \text {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*}
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Mathematica [A]
time = 0.14, size = 46, normalized size = 0.88 \begin {gather*} \frac {1}{2} \left (x+\sqrt {b+x^2}\right )^{-1+a} \left (\frac {b}{-1+a}+\frac {\left (x+\sqrt {b+x^2}\right )^2}{1+a}\right ) \end {gather*}
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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 75.66, size = 738, normalized size = 14.19
result too large to display
Warning: Unable to verify antiderivative.
[In]
mathics('Integrate[(Sqrt[x^2 + b] + x)^a,x]')
[Out]
Piecewise[{{(-a b ^ 2 + a b ^ 2 Cosh[ArcSinh[x / Sqrt[b]] (1 + a)] + a b ^ (3 / 2) x Sqrt[1 + b / x ^ 2] Sinh[
a ArcSinh[x / Sqrt[b]]] - b x ^ 2 Sqrt[1 + b / x ^ 2] Sinh[ArcSinh[x / Sqrt[b]] (1 + a)] + b x ^ 2 Cosh[ArcSin
h[x / Sqrt[b]] (1 + a)] - a b x ^ 2 - a b x ^ 2 Sqrt[1 + b / x ^ 2] Sinh[ArcSinh[x / Sqrt[b]] (1 + a)] + 2 a b
x ^ 2 Cosh[ArcSinh[x / Sqrt[b]] (1 + a)] + a Sqrt[b] x ^ 3 Sqrt[1 + b / x ^ 2] Sinh[a ArcSinh[x / Sqrt[b]]] -
x ^ 4 Sqrt[1 + b / x ^ 2] Sinh[ArcSinh[x / Sqrt[b]] (1 + a)] + x ^ 4 Cosh[ArcSinh[x / Sqrt[b]] (1 + a)] - a x
^ 4 Sqrt[1 + b / x ^ 2] Sinh[ArcSinh[x / Sqrt[b]] (1 + a)] + a x ^ 4 Cosh[ArcSinh[x / Sqrt[b]] (1 + a)] - b ^
(3 / 2) x Cosh[a ArcSinh[x / Sqrt[b]]] - Sqrt[b] x ^ 3 Cosh[a ArcSinh[x / Sqrt[b]]]) b ^ (-1 / 2 + a / 2) / (
-b + a ^ 2 b - x ^ 2 + a ^ 2 x ^ 2), Abs[x ^ 2 / b] > 1}}, 2 a b ^ 3 Cosh[a ArcSinh[x / Sqrt[b]] + ArcSinh[x /
Sqrt[b]]] Gamma[1 - a / 2] b ^ (a / 2) / (-2 b ^ (5 / 2) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma[1 - a /
2]) - a ^ 2 b ^ 3 Gamma[-a / 2] b ^ (a / 2) Sqrt[1 + x ^ 2 / b] Sinh[a ArcSinh[x / Sqrt[b]]] / (-2 b ^ (5 / 2
) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma[1 - a / 2]) + 2 a b ^ 2 x ^ 2 Cosh[a ArcSinh[x / Sqrt[b]] + Arc
Sinh[x / Sqrt[b]]] Gamma[1 - a / 2] b ^ (a / 2) / (-2 b ^ (5 / 2) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma
[1 - a / 2]) - 2 b ^ (5 / 2) x Gamma[1 - a / 2] b ^ (a / 2) Sqrt[1 + x ^ 2 / b] Sinh[a ArcSinh[x / Sqrt[b]] +
ArcSinh[x / Sqrt[b]]] / (-2 b ^ (5 / 2) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma[1 - a / 2]) - 2 a b ^ (5
/ 2) x Gamma[1 - a / 2] b ^ (a / 2) Sqrt[1 + x ^ 2 / b] Sinh[a ArcSinh[x / Sqrt[b]] + ArcSinh[x / Sqrt[b]]] /
(-2 b ^ (5 / 2) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma[1 - a / 2]) + a b ^ (5 / 2) x Cosh[a ArcSinh[x /
Sqrt[b]]] Gamma[-a / 2] b ^ (a / 2) / (-2 b ^ (5 / 2) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma[1 - a / 2])
+ 2 b ^ 2 x ^ 2 Cosh[a ArcSinh[x / Sqrt[b]] + ArcSinh[x / Sqrt[b]]] Gamma[1 - a / 2] b ^ (a / 2) / (-2 b ^ (5
/ 2) Gamma[1 - a / 2] + 2 a ^ 2 b ^ (5 / 2) Gamma[1 - a / 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 |
\(\frac {b^{\frac {a}{2}+\frac {1}{2}} a \left (\frac {8 \sqrt {\pi }\, x^{1+a} b^{-\frac {a}{2}-\frac {1}{2}} \left (\frac {a b}{x^{2}}+a -1\right ) \left (\sqrt {1+\frac {b}{x^{2}}}+1\right )^{a -1}}{\left (1+a \right ) a \left (2 a -2\right )}+\frac {4 \sqrt {\pi }\, x^{1+a} b^{-\frac {a}{2}-\frac {1}{2}} \sqrt {1+\frac {b}{x^{2}}}\, \left (\sqrt {1+\frac {b}{x^{2}}}+1\right )^{a -1}}{\left (1+a \right ) a}\right )}{4 \sqrt {\pi }}\) |
\(120\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x+(x^2+b)^(1/2))^a,x,method=_RETURNVERBOSE)
[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)*(a*b/x^2+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 = 0.00, 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+b)^(1/2))^a,x, algorithm="maxima")
[Out]
integrate((x + sqrt(x^2 + b))^a, x)
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Fricas [A]
time = 0.31, size = 32, normalized size = 0.62 \begin {gather*} \frac {{\left (\sqrt {x^{2} + b} a - x\right )} {\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a^{2} - 1} \end {gather*}
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] Leaf count of result is larger than twice the leaf count of optimal. 2147 vs.
\(2 (37) = 74\)
time = 1.69, 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+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/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)))*ga
mma(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)*gam
ma(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)) + as
inh(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)*g
amma(1 - a/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+b)^(1/2))^a,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+b}\right )}^a \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x + (b + x^2)^(1/2))^a,x)
[Out]
int((x + (b + x^2)^(1/2))^a, x)
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