3.1.30 \(\int \frac {(x+\sqrt {b+x^2})^a}{\sqrt {b+x^2}} \, dx\) [30]

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

[Out]

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

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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 {b+x^2}+x\right )^a}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x + Sqrt[b + x^2])^a/a

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

(x + Sqrt[b + x^2])^a/a

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 7.54, size = 269, normalized size = 15.82 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {x b^{\frac {1}{2}+\frac {a}{2}} \sqrt {\frac {b+x^2}{x^2}} \text {Sinh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ] \left (-1+a\right )\right ]+x^3 b^{-\frac {1}{2}+\frac {a}{2}} \sqrt {\frac {b+x^2}{x^2}} \text {Sinh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ] \left (-1+a\right )\right ]+\left (b+x^2\right ) \left (\frac {x \text {Cosh}\left [\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ] \left (-1+a\right )\right ]}{\sqrt {b}}+\text {Cosh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ]\right ) b^{\frac {a}{2}}}{a \left (b+x^2\right )},\text {Abs}\left [\frac {x^2}{b}\right ]>1\right \}\right \},\frac {-2 \text {Cosh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ] \text {Gamma}\left [1-\frac {a}{2}\right ] b^{\frac {a}{2}}}{a^2 \text {Gamma}\left [-\frac {a}{2}\right ]}+\frac {b^{\frac {a}{2}} \text {Sinh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ]}{a \sqrt {1+\frac {x^2}{b}}}+\frac {x \text {Cosh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ] b^{\frac {a}{2}}}{a \sqrt {b}}+\frac {x^2 b^{\frac {a}{2}} \text {Sinh}\left [a \text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]-\text {ArcSinh}\left [\frac {x}{\sqrt {b}}\right ]\right ]}{a b \sqrt {1+\frac {x^2}{b}}}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((x+(x^2+b)^(1/2))^a/(x^2+b)^(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+b)^(1/2))^a/(x^2+b)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.60, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\left (x + \sqrt {x^{2} + b}\right )}^{a}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + sqrt(x^2 + b))^a/a

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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 = 1.46, size = 311, normalized size = 18.29 \begin {gather*} \begin {cases} \frac {\sqrt {b} b^{\frac {a}{2}} \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a x \sqrt {\frac {b}{x^{2}} + 1}} + \frac {b^{\frac {a}{2}} x \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b}} + \frac {b^{\frac {a}{2}} x \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b} \sqrt {\frac {b}{x^{2}} + 1}} - \frac {2 b^{\frac {a}{2}} \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )} \Gamma \left (1 - \frac {a}{2}\right )}{a^{2} \Gamma \left (- \frac {a}{2}\right )} & \text {for}\: \left |{\frac {x^{2}}{b}}\right | > 1 \\\frac {b^{\frac {a}{2}} \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {1 + \frac {x^{2}}{b}}} + \frac {b^{\frac {a}{2}} x^{2} \sinh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a b \sqrt {1 + \frac {x^{2}}{b}}} + \frac {b^{\frac {a}{2}} x \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} - \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )}}{a \sqrt {b}} - \frac {2 b^{\frac {a}{2}} \cosh {\left (a \operatorname {asinh}{\left (\frac {x}{\sqrt {b}} \right )} \right )} \Gamma \left (1 - \frac {a}{2}\right )}{a^{2} \Gamma \left (- \frac {a}{2}\right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + sqrt(x^2 + b))^a/a

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Mupad [B]
time = 0.31, size = 15, normalized size = 0.88 \begin {gather*} \frac {{\left (x+\sqrt {x^2+b}\right )}^a}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + (b + x^2)^(1/2))^a/a

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