3.20.53 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{(b^2+a x^2)^{5/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac {5 x}{12 b^2 \left (a x^2+b^2\right ) \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{8 \sqrt {a} b^{7/2}}+\frac {x \left (15 a x^2+23 b^2\right )}{24 b^3 \left (a x^2+b^2\right )^{3/2} \sqrt {\sqrt {a x^2+b^2}+b}} \]

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Rubi [F]  time = 0.20, 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}}}{\left (b^2+a x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]

[Out]

Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2), x]

Rubi steps

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

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Mathematica [F]  time = 0.09, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]

[Out]

Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2), x]

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IntegrateAlgebraic [A]  time = 0.28, size = 137, normalized size = 1.00 \begin {gather*} \frac {5 x}{12 b^2 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {x \left (23 b^2+15 a x^2\right )}{24 b^3 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{8 \sqrt {a} b^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]

[Out]

(5*x)/(12*b^2*(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (x*(23*b^2 + 15*a*x^2))/(24*b^3*(b^2 + a*x^2)^(3/2)
*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (5*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(8*Sqrt[a]*b^(7/
2))

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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((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),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 {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{\frac {5}{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)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^(5/2), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{\frac {5}{2}}}\, dx\]

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)^(5/2),x)

[Out]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),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}}}}{{\left (a x^{2} + b^{2}\right )}^{\frac {5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^(5/2), 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}}}{{\left (b^2+a\,x^2\right )}^{5/2}} \,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)^(5/2),x)

[Out]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(a*x^2 + b^2)^(5/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\left (a x^{2} + b^{2}\right )^{\frac {5}{2}}}\, dx \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)**(5/2),x)

[Out]

Integral(sqrt(b + sqrt(a*x**2 + b**2))/(a*x**2 + b**2)**(5/2), x)

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