3.17.29 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2} \, dx\)

Optimal. Leaf size=110 \[ \frac {\sqrt {2} \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{x} \]

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Rubi [F]  time = 0.16, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/x^2, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 99, normalized size = 0.90 \begin {gather*} \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (\sqrt {2} \sqrt {\sqrt {a x^2+b^2}-b} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )-2 \sqrt {b}\right )}{2 \sqrt {b} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-2*Sqrt[b] + Sqrt[2]*Sqrt[-b + Sqrt[b^2 + a*x^2]]*ArcTan[Sqrt[-b + Sqrt[b^2 + a*
x^2]]/(Sqrt[2]*Sqrt[b])]))/(2*Sqrt[b]*x)

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IntegrateAlgebraic [A]  time = 0.15, size = 78, normalized size = 0.71 \begin {gather*} -\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x}+\frac {\sqrt {a} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {2} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/x^2,x]

[Out]

-(Sqrt[b + Sqrt[b^2 + a*x^2]]/x) + (Sqrt[a]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])
/(Sqrt[2]*Sqrt[b])

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fricas [A]  time = 39.45, size = 213, normalized size = 1.94 \begin {gather*} \left [\frac {\sqrt {2} x \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x - 2 \, {\left (2 \, \sqrt {2} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {2} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) - 4 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{4 \, x}, -\frac {\sqrt {2} x \sqrt {\frac {a}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) + 2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{2 \, x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(sqrt(2)*x*sqrt(-a/b)*log(-(a^2*x^3 + 4*a*b^2*x - 4*sqrt(a*x^2 + b^2)*a*b*x - 2*(2*sqrt(2)*sqrt(a*x^2 + b
^2)*b^2*sqrt(-a/b) - sqrt(2)*(a*b*x^2 + 2*b^3)*sqrt(-a/b))*sqrt(b + sqrt(a*x^2 + b^2)))/x^3) - 4*sqrt(b + sqrt
(a*x^2 + b^2)))/x, -1/2*(sqrt(2)*x*sqrt(a/b)*arctan(sqrt(2)*sqrt(b + sqrt(a*x^2 + b^2))*b*sqrt(a/b)/(a*x)) + 2
*sqrt(b + sqrt(a*x^2 + b^2)))/x]

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.04, size = 31, normalized size = 0.28

method	result	size
meijerg	\(-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \hypergeom \left (\left [-\frac {1}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {1}{2}, \frac {1}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{x}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-(b^2)^(1/4)*2^(1/2)/x*hypergeom([-1/2,-1/4,1/4],[1/2,1/2],-x^2*a/b^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 0.91, size = 48, normalized size = 0.44 \begin {gather*} \frac {\sqrt {b} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, \frac {1}{4} \\ \frac {1}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{4 \pi x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*gamma(-1/4)*gamma(1/4)*hyper((-1/2, -1/4, 1/4), (1/2, 1/2), a*x**2*exp_polar(I*pi)/b**2)/(4*pi*x)

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