3.21.69 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{(b^2+a x^2)^3} \, dx\)

Optimal. Leaf size=149 \[ \frac {35 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{64 \sqrt {a} b^{9/2}}+\frac {7 x \left (15 a x^2+23 b^2\right )}{192 b^4 \left (a x^2+b^2\right )^{3/2} \sqrt {\sqrt {a x^2+b^2}+b}}+\frac {x \left (35 a x^2+59 b^2\right )}{96 b^3 \left (a x^2+b^2\right )^2 \sqrt {\sqrt {a x^2+b^2}+b}} \]

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Rubi [F]  time = 1.58, 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 )^3} \, 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)^3,x]

[Out]

(3*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(b - Sqrt[-a]*x), x])/(16*b^5) + (3*Defer[Int][Sqrt[b + Sqrt[b^2 + a
*x^2]]/(b + Sqrt[-a]*x), x])/(16*b^5) + ((-a)^(3/2)*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b - a*x)^
3, x])/(8*b^3) - (3*a*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b - a*x)^2, x])/(16*b^4) + ((-a)^(3/2)*
Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b + a*x)^3, x])/(8*b^3) - (3*a*Defer[Int][Sqrt[b + Sqrt[b^2 +
a*x^2]]/(Sqrt[-a]*b + a*x)^2, x])/(16*b^4)

Rubi steps

\begin {align*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx &=\int \left (-\frac {a^3 \sqrt {b+\sqrt {b^2+a x^2}}}{8 (-a)^{3/2} b^3 \left (\sqrt {-a} b-a x\right )^3}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{16 b^4 \left (\sqrt {-a} b-a x\right )^2}-\frac {a^3 \sqrt {b+\sqrt {b^2+a x^2}}}{8 (-a)^{3/2} b^3 \left (\sqrt {-a} b+a x\right )^3}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{16 b^4 \left (\sqrt {-a} b+a x\right )^2}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{8 b^4 \left (-a b^2-a^2 x^2\right )}\right ) \, dx\\ &=-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{-a b^2-a^2 x^2} \, dx}{8 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3}\\ &=-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \left (-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b-\sqrt {-a} x\right )}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b+\sqrt {-a} x\right )}\right ) \, dx}{8 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3}\\ &=\frac {3 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {-a} x} \, dx}{16 b^5}+\frac {3 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {-a} x} \, dx}{16 b^5}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3}\\ \end {align*}

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Mathematica [F]  time = 0.54, 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 )^3} \, 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)^3,x]

[Out]

Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^3, x]

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IntegrateAlgebraic [A]  time = 0.34, size = 149, normalized size = 1.00 \begin {gather*} \frac {7 x \left (23 b^2+15 a x^2\right )}{192 b^4 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {x \left (59 b^2+35 a x^2\right )}{96 b^3 \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {35 \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{64 \sqrt {a} b^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^3,x]

[Out]

(7*x*(23*b^2 + 15*a*x^2))/(192*b^4*(b^2 + a*x^2)^(3/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (x*(59*b^2 + 35*a*x^2))/
(96*b^3*(b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (35*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x
^2]])])/(64*Sqrt[a]*b^(9/2))

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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)^3,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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^3, x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{3}}\, dx\]

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)^3,x)

[Out]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^3,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 )}^{3}}\,{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)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^3, 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 )}^3} \,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)^3,x)

[Out]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(a*x^2 + b^2)^3, 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 )^{3}}\, 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)**3,x)

[Out]

Integral(sqrt(b + sqrt(a*x**2 + b**2))/(a*x**2 + b**2)**3, x)

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