3.5.86 \(\int (a+x^2) (x+\sqrt {a+x^2})^n \, dx\) [486]
Optimal. Leaf size=108 \[ -\frac {a^3 \left (x+\sqrt {a+x^2}\right )^{-3+n}}{8 (3-n)}-\frac {3 a^2 \left (x+\sqrt {a+x^2}\right )^{-1+n}}{8 (1-n)}+\frac {3 a \left (x+\sqrt {a+x^2}\right )^{1+n}}{8 (1+n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{3+n}}{8 (3+n)} \]
[Out]
-1/8*a^3*(x+(x^2+a)^(1/2))^(-3+n)/(3-n)-3/8*a^2*(x+(x^2+a)^(1/2))^(-1+n)/(1-n)+3/8*a*(x+(x^2+a)^(1/2))^(1+n)/(
1+n)+1/8*(x+(x^2+a)^(1/2))^(3+n)/(3+n)
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Rubi [A]
time = 0.04, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2147, 276}
\begin {gather*} -\frac {a^3 \left (\sqrt {a+x^2}+x\right )^{n-3}}{8 (3-n)}-\frac {3 a^2 \left (\sqrt {a+x^2}+x\right )^{n-1}}{8 (1-n)}+\frac {3 a \left (\sqrt {a+x^2}+x\right )^{n+1}}{8 (n+1)}+\frac {\left (\sqrt {a+x^2}+x\right )^{n+3}}{8 (n+3)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + x^2)*(x + Sqrt[a + x^2])^n,x]
[Out]
-1/8*(a^3*(x + Sqrt[a + x^2])^(-3 + n))/(3 - n) - (3*a^2*(x + Sqrt[a + x^2])^(-1 + n))/(8*(1 - n)) + (3*a*(x +
Sqrt[a + x^2])^(1 + n))/(8*(1 + n)) + (x + Sqrt[a + x^2])^(3 + n)/(8*(3 + n))
Rule 276
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rule 2147
Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^m, Subst[Int[x^n*((d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1)/(-d + x)^(2*(m + 1
))), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])
Rubi steps
\begin {align*} \int \left (a+x^2\right ) \left (x+\sqrt {a+x^2}\right )^n \, dx &=\frac {1}{8} \text {Subst}\left (\int x^{-4+n} \left (a+x^2\right )^3 \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {1}{8} \text {Subst}\left (\int \left (a^3 x^{-4+n}+3 a^2 x^{-2+n}+3 a x^n+x^{2+n}\right ) \, dx,x,x+\sqrt {a+x^2}\right )\\ &=-\frac {a^3 \left (x+\sqrt {a+x^2}\right )^{-3+n}}{8 (3-n)}-\frac {3 a^2 \left (x+\sqrt {a+x^2}\right )^{-1+n}}{8 (1-n)}+\frac {3 a \left (x+\sqrt {a+x^2}\right )^{1+n}}{8 (1+n)}+\frac {\left (x+\sqrt {a+x^2}\right )^{3+n}}{8 (3+n)}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 92, normalized size = 0.85 \begin {gather*} \frac {1}{8} \left (x+\sqrt {a+x^2}\right )^{-3+n} \left (\frac {a^3}{-3+n}+\frac {3 a^2 \left (x+\sqrt {a+x^2}\right )^2}{-1+n}+\frac {3 a \left (x+\sqrt {a+x^2}\right )^4}{1+n}+\frac {\left (x+\sqrt {a+x^2}\right )^6}{3+n}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + x^2)*(x + Sqrt[a + x^2])^n,x]
[Out]
((x + Sqrt[a + x^2])^(-3 + n)*(a^3/(-3 + n) + (3*a^2*(x + Sqrt[a + x^2])^2)/(-1 + n) + (3*a*(x + Sqrt[a + x^2]
)^4)/(1 + n) + (x + Sqrt[a + x^2])^6/(3 + n)))/8
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 0.02, size = 167, normalized size = 1.55
| | |
| method |
result |
size |
| | |
| meijerg |
\(\frac {2^{n} x^{3+n} \hypergeom \left (\left [-\frac {n}{2}, -\frac {3}{2}-\frac {n}{2}, \frac {1}{2}-\frac {n}{2}\right ], \left [1-n , -\frac {1}{2}-\frac {n}{2}\right ], -\frac {a}{x^{2}}\right )}{3+n}+\frac {a^{\frac {3}{2}+\frac {n}{2}} n \left (\frac {8 \sqrt {\pi }\, x^{1+n} a^{-\frac {1}{2}-\frac {n}{2}} \left (\frac {a n}{x^{2}}+n -1\right ) \left (\sqrt {1+\frac {a}{x^{2}}}+1\right )^{-1+n}}{\left (1+n \right ) n \left (-2+2 n \right )}+\frac {4 \sqrt {\pi }\, x^{1+n} a^{-\frac {1}{2}-\frac {n}{2}} \sqrt {1+\frac {a}{x^{2}}}\, \left (\sqrt {1+\frac {a}{x^{2}}}+1\right )^{-1+n}}{\left (1+n \right ) n}\right )}{4 \sqrt {\pi }}\) |
\(167\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^2+a)*(x+(x^2+a)^(1/2))^n,x,method=_RETURNVERBOSE)
[Out]
2^n/(3+n)*x^(3+n)*hypergeom([-1/2*n,-3/2-1/2*n,1/2-1/2*n],[1-n,-1/2-1/2*n],-a/x^2)+1/4*a^(3/2+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^2+a)*(x+(x^2+a)^(1/2))^n,x, algorithm="maxima")
[Out]
integrate((x^2 + a)*(x + sqrt(x^2 + a))^n, x)
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Fricas [A]
time = 0.35, size = 78, normalized size = 0.72 \begin {gather*} -\frac {{\left (3 \, {\left (n^{2} - 1\right )} x^{3} + 3 \, {\left (a n^{2} - 3 \, a\right )} x - {\left (a n^{3} + {\left (n^{3} - n\right )} x^{2} - 7 \, a n\right )} \sqrt {x^{2} + a}\right )} {\left (x + \sqrt {x^{2} + a}\right )}^{n}}{n^{4} - 10 \, n^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^2+a)*(x+(x^2+a)^(1/2))^n,x, algorithm="fricas")
[Out]
-(3*(n^2 - 1)*x^3 + 3*(a*n^2 - 3*a)*x - (a*n^3 + (n^3 - n)*x^2 - 7*a*n)*sqrt(x^2 + a))*(x + sqrt(x^2 + a))^n/(
n^4 - 10*n^2 + 9)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2149 vs.
\(2 (85) = 170\).
time = 15.84, size = 14884, normalized size = 137.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**2+a)*(x+(x**2+a)**(1/2))**n,x)
[Out]
a*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*
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**(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)
*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*
*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)*gam
ma(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
*cosh(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)) + asin
h(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)*s
inh(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*
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)*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)*
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*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)*gamm
a(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)
*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)), 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*
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**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)) +
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
)*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)
*gamma(1 - n/2)), True)) + Piecewise((2*a**(73/2)*a**(n/2)*n**3*x*cosh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(
71/2)*n**4*gamma(1 - n/2) - 20*a**(71/2)*n**2*gamma(1 - n/2) + 18*a**(71/2)*gamma(1 - n/2) + 2*a**(69/2)*n**4*
x**2*gamma(1 - n/2) - 20*a**(69/2)*n**2*x**2*gamma(1 - n/2) + 18*a**(69/2)*x**2*gamma(1 - n/2)) - 2*a**(73/2)*
a**(n/2)*n**2*x*sqrt(a/x**2 + 1)*sinh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(71/2)*n**4*gamma(1 - n/2) - 20*a*
*(71/2)*n**2*gamma(1 - n/2) + 18*a**(71/2)*gamma(1 - n/2) + 2*a**(69/2)*n**4*x**2*gamma(1 - n/2) - 20*a**(69/2
)*n**2*x**2*gamma(1 - n/2) + 18*a**(69/2)*x**2*gamma(1 - n/2)) - a**(71/2)*a**(n/2)*n**4*x**3*sqrt(a/x**2 + 1)
*sinh(n*asinh(x/sqrt(a)))*gamma(-n/2)/(2*a**(71/2)*n**4*gamma(1 - n/2) - 20*a**(71/2)*n**2*gamma(1 - n/2) + 18
*a**(71/2)*gamma(1 - n/2) + 2*a**(69/2)*n**4*x*...
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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^2+a)*(x+(x^2+a)^(1/2))^n,x, algorithm="giac")
[Out]
integrate((x^2 + a)*(x + sqrt(x^2 + a))^n, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (x^2+a\right )\,{\left (x+\sqrt {x^2+a}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + x^2)*(x + (a + x^2)^(1/2))^n,x)
[Out]
int((a + x^2)*(x + (a + x^2)^(1/2))^n, x)
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