3.1.19 \(\int \frac {(x+\sqrt {a+x^2})^b}{\sqrt {a+x^2}} \, dx\) [19]

Optimal. Leaf size=17 \[ \frac {\left (x+\sqrt {a+x^2}\right )^b}{b} \]

[Out]

(x+(x^2+a)^(1/2))^b/b

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Rubi [A]
time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2147, 30} \begin {gather*} \frac {\left (\sqrt {a+x^2}+x\right )^b}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[a + x^2])^b/Sqrt[a + x^2],x]

[Out]

(x + Sqrt[a + x^2])^b/b

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 \frac {\left (x+\sqrt {a+x^2}\right )^b}{\sqrt {a+x^2}} \, dx &=\text {Subst}\left (\int x^{-1+b} \, dx,x,x+\sqrt {a+x^2}\right )\\ &=\frac {\left (x+\sqrt {a+x^2}\right )^b}{b}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} \frac {\left (x+\sqrt {a+x^2}\right )^b}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + Sqrt[a + x^2])^b/Sqrt[a + x^2],x]

[Out]

(x + Sqrt[a + x^2])^b/b

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 7.68, size = 269, normalized size = 15.82 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x a^{\frac {1}{2}+\frac {b}{2}} \sqrt {\frac {a+x^2}{x^2}} \text {Sinh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ] \left (-1+b\right )\right ]+x^3 a^{-\frac {1}{2}+\frac {b}{2}} \sqrt {\frac {a+x^2}{x^2}} \text {Sinh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ] \left (-1+b\right )\right ]+\left (a+x^2\right ) \left (\frac {x \text {Cosh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ] \left (-1+b\right )\right ]}{\sqrt {a}}+\text {Cosh}\left [b \text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]\right ]\right ) a^{\frac {b}{2}}}{b \left (a+x^2\right )},\text {Abs}\left [\frac {x^2}{a}\right ]>1\right \}\right \},\frac {-2 \text {Cosh}\left [b \text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]\right ] \text {Gamma}\left [1-\frac {b}{2}\right ] a^{\frac {b}{2}}}{b^2 \text {Gamma}\left [-\frac {b}{2}\right ]}+\frac {a^{\frac {b}{2}} \text {Sinh}\left [b \text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]\right ]}{b \sqrt {1+\frac {x^2}{a}}}+\frac {x \text {Cosh}\left [b \text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]\right ] a^{\frac {b}{2}}}{\sqrt {a} b}+\frac {x^2 a^{\frac {b}{2}} \text {Sinh}\left [b \text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {a}}\right ]\right ]}{a b \sqrt {1+\frac {x^2}{a}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(x + Sqrt[x^2 + a])^b/Sqrt[x^2 + a],x]')

[Out]

Piecewise[{{(x a ^ (1 / 2 + b / 2) Sqrt[(a + x ^ 2) / x ^ 2] Sinh[ArcSinh[x / Sqrt[a]] (-1 + b)] + x ^ 3 a ^ (
-1 / 2 + b / 2) Sqrt[(a + x ^ 2) / x ^ 2] Sinh[ArcSinh[x / Sqrt[a]] (-1 + b)] + (a + x ^ 2) (x Cosh[ArcSinh[x
/ Sqrt[a]] (-1 + b)] / Sqrt[a] + Cosh[b ArcSinh[x / Sqrt[a]]]) a ^ (b / 2)) / (b (a + x ^ 2)), Abs[x ^ 2 / a]
> 1}}, -2 Cosh[b ArcSinh[x / Sqrt[a]]] Gamma[1 - b / 2] a ^ (b / 2) / (b ^ 2 Gamma[-b / 2]) + a ^ (b / 2) Sinh
[b ArcSinh[x / Sqrt[a]] - ArcSinh[x / Sqrt[a]]] / (b Sqrt[1 + x ^ 2 / a]) + x Cosh[b ArcSinh[x / Sqrt[a]] - Ar
cSinh[x / Sqrt[a]]] a ^ (b / 2) / (Sqrt[a] b) + x ^ 2 a ^ (b / 2) Sinh[b ArcSinh[x / Sqrt[a]] - ArcSinh[x / Sq
rt[a]]] / (a b Sqrt[1 + x ^ 2 / a])]

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x +\sqrt {x^{2}+a}\right )^{b}}{\sqrt {x^{2}+a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+(x^2+a)^(1/2))^b/(x^2+a)^(1/2),x)

[Out]

int((x+(x^2+a)^(1/2))^b/(x^2+a)^(1/2),x)

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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^2+a)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + sqrt(x^2 + a))^b/sqrt(x^2 + a), x)

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Fricas [A]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\left (x + \sqrt {x^{2} + a}\right )}^{b}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^b/(x^2+a)^(1/2),x, algorithm="fricas")

[Out]

(x + sqrt(x^2 + a))^b/b

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (12) = 24\)
time = 1.25, size = 311, normalized size = 18.29 \begin {gather*} \begin {cases} \frac {\sqrt {a} a^{\frac {b}{2}} \sinh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{b x \sqrt {\frac {a}{x^{2}} + 1}} - \frac {2 a^{\frac {b}{2}} \cosh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )} \Gamma \left (1 - \frac {b}{2}\right )}{b^{2} \Gamma \left (- \frac {b}{2}\right )} + \frac {a^{\frac {b}{2}} x \cosh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} b} + \frac {a^{\frac {b}{2}} x \sinh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} b \sqrt {\frac {a}{x^{2}} + 1}} & \text {for}\: \left |{\frac {x^{2}}{a}}\right | > 1 \\\frac {a^{\frac {b}{2}} \sinh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{b \sqrt {1 + \frac {x^{2}}{a}}} - \frac {2 a^{\frac {b}{2}} \cosh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )} \Gamma \left (1 - \frac {b}{2}\right )}{b^{2} \Gamma \left (- \frac {b}{2}\right )} + \frac {a^{\frac {b}{2}} x^{2} \sinh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{a b \sqrt {1 + \frac {x^{2}}{a}}} + \frac {a^{\frac {b}{2}} x \cosh {\left (b \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {a}} \right )} \right )}}{\sqrt {a} b} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x**2+a)**(1/2))**b/(x**2+a)**(1/2),x)

[Out]

Piecewise((sqrt(a)*a**(b/2)*sinh(b*asinh(x/sqrt(a)) - asinh(x/sqrt(a)))/(b*x*sqrt(a/x**2 + 1)) - 2*a**(b/2)*co
sh(b*asinh(x/sqrt(a)))*gamma(1 - b/2)/(b**2*gamma(-b/2)) + a**(b/2)*x*cosh(b*asinh(x/sqrt(a)) - asinh(x/sqrt(a
)))/(sqrt(a)*b) + a**(b/2)*x*sinh(b*asinh(x/sqrt(a)) - asinh(x/sqrt(a)))/(sqrt(a)*b*sqrt(a/x**2 + 1)), Abs(x**
2/a) > 1), (a**(b/2)*sinh(b*asinh(x/sqrt(a)) - asinh(x/sqrt(a)))/(b*sqrt(1 + x**2/a)) - 2*a**(b/2)*cosh(b*asin
h(x/sqrt(a)))*gamma(1 - b/2)/(b**2*gamma(-b/2)) + a**(b/2)*x**2*sinh(b*asinh(x/sqrt(a)) - asinh(x/sqrt(a)))/(a
*b*sqrt(1 + x**2/a)) + a**(b/2)*x*cosh(b*asinh(x/sqrt(a)) - asinh(x/sqrt(a)))/(sqrt(a)*b), True))

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Giac [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} \frac {\left (x+\sqrt {a+x^{2}}\right )^{b}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+(x^2+a)^(1/2))^b/(x^2+a)^(1/2),x)

[Out]

(x + sqrt(x^2 + a))^b/b

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Mupad [B]
time = 0.30, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\left (x+\sqrt {x^2+a}\right )}^b}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + (a + x^2)^(1/2))^b/(a + x^2)^(1/2),x)

[Out]

(x + (a + x^2)^(1/2))^b/b

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