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

Optimal. Leaf size=22 \[ \frac {2 x}{\sqrt {\sqrt {a x^2+b^2}+b}} \]

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Rubi [F]  time = 0.18, 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}}}{\sqrt {b^2+a x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {\sqrt {a x^2+b^2}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]]

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IntegrateAlgebraic [A]  time = 0.09, size = 22, normalized size = 1.00 \begin {gather*} \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]]

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fricas [B]  time = 0.50, size = 38, normalized size = 1.73 \begin {gather*} -\frac {2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}} {\left (b - \sqrt {a x^{2} + b^{2}}\right )}}{a 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)/(a*x^2+b^2)^(1/2),x, algorithm="fricas")

[Out]

-2*sqrt(b + sqrt(a*x^2 + b^2))*(b - sqrt(a*x^2 + b^2))/(a*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}}}}{\sqrt {a x^{2} + b^{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)/(a*x^2+b^2)^(1/2),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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}}}}{\sqrt {a x^{2} + b^{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)/(a*x^2+b^2)^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{\sqrt {b^2+a\,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)/(a*x^2 + b^2)^(1/2),x)

[Out]

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

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sympy [B]  time = 0.72, size = 41, normalized size = 1.86 \begin {gather*} \frac {\sqrt {2} x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{\pi \sqrt {b} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*x*gamma(1/4)*gamma(3/4)/(pi*sqrt(b)*sqrt(sqrt(a*x**2/b**2 + 1) + 1))

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