3.5.87 \(\int (x+\sqrt {a+x^2})^n \, dx\) [487]
Optimal. Leaf size=52 \[ -\frac {a \left (x+\sqrt {a+x^2}\right )^{-1+n}}{2 (1-n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{1+n}}{2 (1+n)} \]
[Out]
-1/2*a*(x+(x^2+a)^(1/2))^(-1+n)/(1-n)+1/2*(x+(x^2+a)^(1/2))^(1+n)/(1+n)
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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 )^{n+1}}{2 (n+1)}-\frac {a \left (\sqrt {a+x^2}+x\right )^{n-1}}{2 (1-n)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(x + Sqrt[a + x^2])^n,x]
[Out]
-1/2*(a*(x + Sqrt[a + x^2])^(-1 + n))/(1 - n) + (x + Sqrt[a + x^2])^(1 + n)/(2*(1 + n))
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 )^n \, dx &=\frac {1}{2} \text {Subst}\left (\int x^{-2+n} \left (a+x^2\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (a x^{-2+n}+x^n\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=-\frac {a \left (x+\sqrt {a+x^2}\right )^{-1+n}}{2 (1-n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{1+n}}{2 (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 43, normalized size = 0.83 \begin {gather*} \frac {\left (x+\sqrt {a+x^2}\right )^{-1+n} \left (a n+(-1+n) x \left (x+\sqrt {a+x^2}\right )\right )}{-1+n^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(x + Sqrt[a + x^2])^n,x]
[Out]
((x + Sqrt[a + x^2])^(-1 + n)*(a*n + (-1 + n)*x*(x + Sqrt[a + x^2])))/(-1 + n^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 = 0.01, size = 120, normalized size = 2.31
| | |
| method |
result |
size |
| | |
| meijerg |
\(\frac {a^{\frac {1}{2}+\frac {n}{2}} n \left (\frac {8 \sqrt {\pi }\, x^{1+n} a^{-\frac {1}{2}-\frac {n}{2}} \left (\frac {a n}{x^{2}}+n -1\right ) \left (\sqrt {1+\frac {a}{x^{2}}}+1\right )^{-1+n}}{\left (1+n \right ) n \left (-2+2 n \right )}+\frac {4 \sqrt {\pi }\, x^{1+n} a^{-\frac {1}{2}-\frac {n}{2}} \sqrt {1+\frac {a}{x^{2}}}\, \left (\sqrt {1+\frac {a}{x^{2}}}+1\right )^{-1+n}}{\left (1+n \right ) n}\right )}{4 \sqrt {\pi }}\) |
\(120\) |
| | |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x+(x^2+a)^(1/2))^n,x,method=_RETURNVERBOSE)
[Out]
1/4*a^(1/2+1/2*n)/Pi^(1/2)*n*(8*Pi^(1/2)/(1+n)/n*x^(1+n)*a^(-1/2-1/2*n)*(a/x^2*n+n-1)/(-2+2*n)*((1+a/x^2)^(1/2
)+1)^(-1+n)+4*Pi^(1/2)/(1+n)/n*x^(1+n)*a^(-1/2-1/2*n)*(1+a/x^2)^(1/2)*((1+a/x^2)^(1/2)+1)^(-1+n))
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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))^n,x, algorithm="maxima")
[Out]
integrate((x + sqrt(x^2 + a))^n, x)
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Fricas [A]
time = 0.39, size = 32, normalized size = 0.62 \begin {gather*} \frac {{\left (\sqrt {x^{2} + a} n - x\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{2} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x+(x^2+a)^(1/2))^n,x, algorithm="fricas")
[Out]
(sqrt(x^2 + a)*n - x)*(x + sqrt(x^2 + a))^n/(n^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 = 1.65, size = 2147, normalized size = 41.29 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x+(x**2+a)**(1/2))**n,x)
[Out]
Piecewise((-a**(9/2)*a**(n/2)*n**2*x*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*ga
mma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2
)) + a**(9/2)*a**(n/2)*n*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*g
amma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - a**(7/2)*a**(n/2)*n**2
*x**3*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma
(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + a**(7/2)*a**(n/2)*n*x**3*c
osh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n
**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**5*a**(n/2)*n*cosh(n*asinh(x/sqrt(a)) + asinh(
x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*
gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**5*a**(n/2)*n*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 -
n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*
a**4*a**(n/2)*n*x**2*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n
**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1
- n/2)) + 4*a**4*a**(n/2)*n*x**2*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*
gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n
/2)) - 2*a**4*a**(n/2)*n*x**2*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2) + 2*a
**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**4*a**(n/2)*x**2*sqrt(a/x**2 + 1)*sin
h(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 -
n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**4*a**(n/2)*x**2*cosh(n*as
inh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(1 - n/2)
+ 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**3*a**(n/2)*n*x**4*sqrt(a/x**2 +
1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*ga
mma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**3*a**(n/2)*n*x**4*
cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*gamma(
1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) - 2*a**3*a**(n/2)*x**4*sqrt(a
/x**2 + 1)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(
9/2)*gamma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)) + 2*a**3*a**(n/2)*
x**4*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(9/2)*n**2*gamma(1 - n/2) - 2*a**(9/2)*g
amma(1 - n/2) + 2*a**(7/2)*n**2*x**2*gamma(1 - n/2) - 2*a**(7/2)*x**2*gamma(1 - n/2)), Abs(x**2/a) > 1), (-2*a
**(5/2)*a**(n/2)*n*x*sqrt(1 + x**2/a)*sinh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n
**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) + a**(5/2)*a**(n/2)*n*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(
2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) - 2*a**(5/2)*a**(n/2)*x*sqrt(1 + x**2/a)*sinh(n*as
inh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2))
- a**3*a**(n/2)*n**2*sqrt(1 + x**2/a)*sinh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) -
2*a**(5/2)*gamma(1 - n/2)) + 2*a**3*a**(n/2)*n*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a
**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) + 2*a**2*a**(n/2)*n*x**2*cosh(n*asinh(x/sqrt(a)) + as
inh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*gamma(1 - n/2)) + 2*a**2*a**(n/2)*
x**2*cosh(n*asinh(x/sqrt(a)) + asinh(x/sqrt(a)))*gamma(1 - n/2)/(2*a**(5/2)*n**2*gamma(1 - n/2) - 2*a**(5/2)*g
amma(1 - n/2)), True))
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Giac [F]
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))^n,x, algorithm="giac")
[Out]
integrate((x + sqrt(x^2 + a))^n, x)
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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 )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x + (a + x^2)^(1/2))^n,x)
[Out]
int((x + (a + x^2)^(1/2))^n, x)
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