3.31 \(\int \left (x+\sqrt{b+x^2}\right )^a \, dx\)
Optimal. Leaf size=52 \[ \frac{\left (\sqrt{b+x^2}+x\right )^{a+1}}{2 (a+1)}-\frac{b \left (\sqrt{b+x^2}+x\right )^{a-1}}{2 (1-a)} \]
[Out]
-(b*(x + Sqrt[b + x^2])^(-1 + a))/(2*(1 - a)) + (x + Sqrt[b + x^2])^(1 + a)/(2*(
1 + a))
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Rubi [A] time = 0.0464369, antiderivative size = 52, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154
\[ \frac{\left (\sqrt{b+x^2}+x\right )^{a+1}}{2 (a+1)}-\frac{b \left (\sqrt{b+x^2}+x\right )^{a-1}}{2 (1-a)} \]
Antiderivative was successfully verified.
[In] Int[(x + Sqrt[b + x^2])^a,x]
[Out]
-(b*(x + Sqrt[b + x^2])^(-1 + a))/(2*(1 - a)) + (x + Sqrt[b + x^2])^(1 + a)/(2*(
1 + a))
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x + \sqrt{b + x^{2}}\right )^{a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x+(x**2+b)**(1/2))**a,x)
[Out]
Integral((x + sqrt(b + x**2))**a, x)
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Mathematica [A] time = 0.0233956, size = 36, normalized size = 0.69 \[ \frac{\left (\sqrt{b+x^2}+x\right )^a \left (a \sqrt{b+x^2}-x\right )}{a^2-1} \]
Antiderivative was successfully verified.
[In] Integrate[(x + Sqrt[b + x^2])^a,x]
[Out]
((x + Sqrt[b + x^2])^a*(-x + a*Sqrt[b + x^2]))/(-1 + a^2)
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Maple [B] time = 0.033, size = 120, normalized size = 2.3 \[{\frac{a}{4\,\sqrt{\pi }}{b}^{{\frac{a}{2}}+{\frac{1}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+a}{b}^{-a/2-1/2}}{ \left ( 1+a \right ) a \left ( 2\,a-2 \right ) } \left ({\frac{ab}{{x}^{2}}}+a-1 \right ) \left ( \sqrt{{\frac{b}{{x}^{2}}}+1}+1 \right ) ^{a-1}}+4\,{\frac{\sqrt{\pi }{x}^{1+a}{b}^{-a/2-1/2}}{ \left ( 1+a \right ) a}\sqrt{{\frac{b}{{x}^{2}}}+1} \left ( \sqrt{{\frac{b}{{x}^{2}}}+1}+1 \right ) ^{a-1}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x+(x^2+b)^(1/2))^a,x)
[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)*(1/x^2*a
*b+a-1)/(2*a-2)*((1/x^2*b+1)^(1/2)+1)^(a-1)+4*Pi^(1/2)/(1+a)/a*x^(1+a)*b^(-1/2*a
-1/2)*(1/x^2*b+1)^(1/2)*((1/x^2*b+1)^(1/2)+1)^(a-1))
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x + \sqrt{x^{2} + b}\right )}^{a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + b))^a,x, algorithm="maxima")
[Out]
integrate((x + sqrt(x^2 + b))^a, x)
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Fricas [A] time = 0.244311, size = 43, normalized size = 0.83 \[ \frac{{\left (\sqrt{x^{2} + b} a - x\right )}{\left (x + \sqrt{x^{2} + b}\right )}^{a}}{a^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + b))^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 [A] time = 6.69675, 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+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)))*g
amma(-a/2)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 +
1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 + 1)) - 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(-a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamm
a(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 + 1)) + a*b**(9/2)*b**(a/2)*x*cosh(a*as
inh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2)*x
**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 +
1)) + a*b**(7/2)*b**(a/2)*x**3*cosh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(
9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-
a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 + 1)) + 2*a*b**5*b**(a/2)*cosh(a*asinh(x/s
qrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2
*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x*
*2*gamma(-a/2 + 1)) - 2*a*b**5*b**(a/2)*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-
a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2
*b**(7/2)*x**2*gamma(-a/2 + 1)) - 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(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/
2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b
**(7/2)*x**2*gamma(-a/2 + 1)) + 4*a*b**4*b**(a/2)*x**2*cosh(a*asinh(x/sqrt(b)) +
asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**
(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(
-a/2 + 1)) - 2*a*b**4*b**(a/2)*x**2*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/2
+ 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**
(7/2)*x**2*gamma(-a/2 + 1)) - 2*a*b**3*b**(a/2)*x**4*sqrt(b/x**2 + 1)*sinh(a*asi
nh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/2 +
1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7
/2)*x**2*gamma(-a/2 + 1)) + 2*a*b**3*b**(a/2)*x**4*cosh(a*asinh(x/sqrt(b)) + asi
nh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2
)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2
+ 1)) - 2*b**4*b**(a/2)*x**2*sqrt(b/x**2 + 1)*sinh(a*asinh(x/sqrt(b)) + asinh(x
/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2)*x*
*2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 + 1
)) + 2*b**4*b**(a/2)*x**2*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2
+ 1)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) -
2*b**(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 + 1)) - 2*b**3*b**(a/2)*
x**4*sqrt(b/x**2 + 1)*sinh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1
)/(2*a**2*b**(9/2)*gamma(-a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b*
*(9/2)*gamma(-a/2 + 1) - 2*b**(7/2)*x**2*gamma(-a/2 + 1)) + 2*b**3*b**(a/2)*x**4
*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(9/2)*ga
mma(-a/2 + 1) + 2*a**2*b**(7/2)*x**2*gamma(-a/2 + 1) - 2*b**(9/2)*gamma(-a/2 + 1
) - 2*b**(7/2)*x**2*gamma(-a/2 + 1)), Abs(x**2/b) > 1), (-a**2*b**3*b**(a/2)*sqr
t(1 + x**2/b)*sinh(a*asinh(x/sqrt(b)))*gamma(-a/2)/(2*a**2*b**(5/2)*gamma(-a/2 +
1) - 2*b**(5/2)*gamma(-a/2 + 1)) - 2*a*b**(5/2)*b**(a/2)*x*sqrt(1 + x**2/b)*sin
h(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(5/2)*gamma(
-a/2 + 1) - 2*b**(5/2)*gamma(-a/2 + 1)) + a*b**(5/2)*b**(a/2)*x*cosh(a*asinh(x/s
qrt(b)))*gamma(-a/2)/(2*a**2*b**(5/2)*gamma(-a/2 + 1) - 2*b**(5/2)*gamma(-a/2 +
1)) + 2*a*b**3*b**(a/2)*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 +
1)/(2*a**2*b**(5/2)*gamma(-a/2 + 1) - 2*b**(5/2)*gamma(-a/2 + 1)) + 2*a*b**2*b*
*(a/2)*x**2*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*
b**(5/2)*gamma(-a/2 + 1) - 2*b**(5/2)*gamma(-a/2 + 1)) - 2*b**(5/2)*b**(a/2)*x*s
qrt(1 + x**2/b)*sinh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a
**2*b**(5/2)*gamma(-a/2 + 1) - 2*b**(5/2)*gamma(-a/2 + 1)) + 2*b**2*b**(a/2)*x**
2*cosh(a*asinh(x/sqrt(b)) + asinh(x/sqrt(b)))*gamma(-a/2 + 1)/(2*a**2*b**(5/2)*g
amma(-a/2 + 1) - 2*b**(5/2)*gamma(-a/2 + 1)), True))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (x + \sqrt{x^{2} + b}\right )}^{a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(x^2 + b))^a,x, algorithm="giac")
[Out]
integrate((x + sqrt(x^2 + b))^a, x)