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

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

[Out]

2/3*x^3/(5+(x^2+25)^(1/2))^(3/2)+10*x/(5+(x^2+25)^(1/2))^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.48, size = 30, normalized size = 0.73 \[ \frac {2 \, {\left (x^{2} + 5 \, \sqrt {x^{2} + 25} - 25\right )} \sqrt {\sqrt {x^{2} + 25} + 5}}{3 \, x} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {x^{2} + 25} + 5}\,{d x} \]

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)

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maple [C]  time = 0.02, size = 64, normalized size = 1.56 \[ -\frac {5 \sqrt {5}\, \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \left (\frac {x}{5}\right )}{2}\right )}{375}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{1875} x^{4}-\frac {2}{75} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {x}{5}\right )}{2}\right )}{\sqrt {\frac {x^{2}}{25}+1}}\right )}{8 \sqrt {\pi }} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sqrt {x^{2} + 25} + 5}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\sqrt {x^2+25}+5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 + 25)^(1/2) + 5)^(1/2),x)

[Out]

int(((x^2 + 25)^(1/2) + 5)^(1/2), x)

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sympy [B]  time = 1.22, size = 197, normalized size = 4.80 \[ - \frac {\sqrt {2} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 25} \sqrt {\sqrt {x^{2} + 25} + 5} + 60 \pi \sqrt {\sqrt {x^{2} + 25} + 5}} - \frac {15 \sqrt {2} x \sqrt {x^{2} + 25} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 25} \sqrt {\sqrt {x^{2} + 25} + 5} + 60 \pi \sqrt {\sqrt {x^{2} + 25} + 5}} - \frac {75 \sqrt {2} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 25} \sqrt {\sqrt {x^{2} + 25} + 5} + 60 \pi \sqrt {\sqrt {x^{2} + 25} + 5}} \]

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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