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

Optimal. Leaf size=149 \[ -\frac {2 b x}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 x \sqrt {a x^2+b^2}}{3 \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 \sqrt {2} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

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Rubi [F]  time = 0.21, 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^2+a x^2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.43, size = 212, normalized size = 1.42 \begin {gather*} \frac {6 b^2 \left (\sqrt {a x^2+b^2}+b\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )-6 b^2 \sqrt {a x^2+b^2}+4 a x^2 \sqrt {a x^2+b^2}+3 \sqrt {2} b^{3/2} \sqrt {\sqrt {a x^2+b^2}-b} \left (\sqrt {a x^2+b^2}+b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a x^2+b^2}-b}}{\sqrt {2} \sqrt {b}}\right )-4 a b x^2-6 b^3}{6 a x \sqrt {\sqrt {a x^2+b^2}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*b^3 - 4*a*b*x^2 - 6*b^2*Sqrt[b^2 + a*x^2] + 4*a*x^2*Sqrt[b^2 + a*x^2] + 3*Sqrt[2]*b^(3/2)*Sqrt[-b + Sqrt[b
^2 + a*x^2]]*(b + Sqrt[b^2 + a*x^2])*ArcTan[Sqrt[-b + Sqrt[b^2 + a*x^2]]/(Sqrt[2]*Sqrt[b])] + 6*b^2*(b + Sqrt[
b^2 + a*x^2])*Hypergeometric2F1[-1/2, 1, 1/2, (b - Sqrt[b^2 + a*x^2])/(2*b)])/(6*a*x*Sqrt[b + Sqrt[b^2 + a*x^2
]])

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IntegrateAlgebraic [A]  time = 0.22, size = 81, normalized size = 0.54 \begin {gather*} \frac {2 a x^3}{3 \left (b+\sqrt {b^2+a x^2}\right )^{3/2}}+\frac {\sqrt {2} b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*x^3)/(3*(b + Sqrt[b^2 + a*x^2])^(3/2)) + (Sqrt[2]*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sq
rt[b^2 + a*x^2]])])/Sqrt[a]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a \,x^{2}+b^{2}}}{\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*x**2 + b**2)/sqrt(b + sqrt(a*x**2 + b**2)), x)

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