3.470 \(\int \sqrt{5+\sqrt{25+x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{2 x^3}{3 \left (\sqrt{x^2+25}+5\right )^{3/2}}+\frac{10 x}{\sqrt{\sqrt{x^2+25}+5}} \]

[Out]

(2*x^3)/(3*(5 + Sqrt[25 + x^2])^(3/2)) + (10*x)/Sqrt[5 + Sqrt[25 + x^2]]

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Rubi [A]  time = 0.0074054, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {2129} \[ \frac{2 x^3}{3 \left (\sqrt{x^2+25}+5\right )^{3/2}}+\frac{10 x}{\sqrt{\sqrt{x^2+25}+5}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[5 + Sqrt[25 + x^2]],x]

[Out]

(2*x^3)/(3*(5 + Sqrt[25 + x^2])^(3/2)) + (10*x)/Sqrt[5 + Sqrt[25 + x^2]]

Rule 2129

Int[Sqrt[(a_) + (b_.)*Sqrt[(c_) + (d_.)*(x_)^2]], x_Symbol] :> Simp[(2*b^2*d*x^3)/(3*(a + b*Sqrt[c + d*x^2])^(
3/2)), x] + Simp[(2*a*x)/Sqrt[a + b*Sqrt[c + d*x^2]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2*c, 0]

Rubi steps

\begin{align*} \int \sqrt{5+\sqrt{25+x^2}} \, dx &=\frac{2 x^3}{3 \left (5+\sqrt{25+x^2}\right )^{3/2}}+\frac{10 x}{\sqrt{5+\sqrt{25+x^2}}}\\ \end{align*}

Mathematica [A]  time = 0.0566115, size = 44, normalized size = 1.07 \[ \frac{2 \left (\sqrt{x^2+25}-5\right ) \sqrt{\sqrt{x^2+25}+5} \left (\sqrt{x^2+25}+10\right )}{3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[5 + Sqrt[25 + x^2]],x]

[Out]

(2*(-5 + Sqrt[25 + x^2])*Sqrt[5 + Sqrt[25 + x^2]]*(10 + Sqrt[25 + x^2]))/(3*x)

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Maple [C]  time = 0.016, size = 64, normalized size = 1.6 \begin{align*} -{\frac{5\,\sqrt{5}}{8\,\sqrt{\pi }} \left ( -{\frac{32\,\sqrt{\pi }\sqrt{2}{x}^{3}}{375}\cosh \left ({\frac{3}{2}{\it Arcsinh} \left ({\frac{x}{5}} \right ) } \right ) }-8\,{\frac{\sqrt{\pi }\sqrt{2}\sinh \left ( 3/2\,{\it Arcsinh} \left ( x/5 \right ) \right ) }{\sqrt{1/25\,{x}^{2}+1}} \left ( -{\frac{4\,{x}^{4}}{1875}}-{\frac{2\,{x}^{2}}{75}}+2/3 \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5+(x^2+25)^(1/2))^(1/2),x)

[Out]

-5/8*5^(1/2)/Pi^(1/2)*(-32/375*Pi^(1/2)*2^(1/2)*x^3*cosh(3/2*arcsinh(1/5*x))-8*Pi^(1/2)*2^(1/2)*(-4/1875*x^4-2
/75*x^2+2/3)*sinh(3/2*arcsinh(1/5*x))/(1/25*x^2+1)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{x^{2} + 25} + 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+(x^2+25)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sqrt(x^2 + 25) + 5), x)

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Fricas [A]  time = 1.18357, size = 84, normalized size = 2.05 \begin{align*} \frac{2 \,{\left (x^{2} + 5 \, \sqrt{x^{2} + 25} - 25\right )} \sqrt{\sqrt{x^{2} + 25} + 5}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+(x^2+25)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

2/3*(x^2 + 5*sqrt(x^2 + 25) - 25)*sqrt(sqrt(x^2 + 25) + 5)/x

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Sympy [B]  time = 1.11623, size = 197, normalized size = 4.8 \begin{align*} - \frac{\sqrt{2} x^{3} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{x^{2} + 25} \sqrt{\sqrt{x^{2} + 25} + 5} + 60 \pi \sqrt{\sqrt{x^{2} + 25} + 5}} - \frac{15 \sqrt{2} x \sqrt{x^{2} + 25} \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{x^{2} + 25} \sqrt{\sqrt{x^{2} + 25} + 5} + 60 \pi \sqrt{\sqrt{x^{2} + 25} + 5}} - \frac{75 \sqrt{2} x \Gamma \left (- \frac{1}{4}\right ) \Gamma \left (\frac{1}{4}\right )}{12 \pi \sqrt{x^{2} + 25} \sqrt{\sqrt{x^{2} + 25} + 5} + 60 \pi \sqrt{\sqrt{x^{2} + 25} + 5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+(x**2+25)**(1/2))**(1/2),x)

[Out]

-sqrt(2)*x**3*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(x**2 + 25)*sqrt(sqrt(x**2 + 25) + 5) + 60*pi*sqrt(sqrt(x**2 +
25) + 5)) - 15*sqrt(2)*x*sqrt(x**2 + 25)*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(x**2 + 25)*sqrt(sqrt(x**2 + 25) +
5) + 60*pi*sqrt(sqrt(x**2 + 25) + 5)) - 75*sqrt(2)*x*gamma(-1/4)*gamma(1/4)/(12*pi*sqrt(x**2 + 25)*sqrt(sqrt(
x**2 + 25) + 5) + 60*pi*sqrt(sqrt(x**2 + 25) + 5))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{x^{2} + 25} + 5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5+(x^2+25)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(sqrt(x^2 + 25) + 5), x)