3.25.47 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx\)

Optimal. Leaf size=198 \[ \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (a^{3/2} \left (210 b x^2 \sqrt {a x^2+b^2}-42 b^2 x^2\right )-315 a^{5/2} x^4+\sqrt {a} \left (16 b^4-144 b^3 \sqrt {a x^2+b^2}\right )\right )}{640 \sqrt {a} b^5 x^5}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{64 \sqrt {2} b^{11/2}} \]

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Rubi [F]  time = 0.68, 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^6 \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]]/(x^6*Sqrt[b^2 + a*x^2]),x]

[Out]

Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(x^6*Sqrt[b^2 + a*x^2]), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx &=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6 \sqrt {b^2+a x^2}} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.25, size = 81, normalized size = 0.41 \begin {gather*} -\frac {a^3 x \, _2F_1\left (-\frac {5}{2},3;-\frac {3}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )}{20 b^3 \left (\sqrt {a x^2+b^2}-b\right )^3 \sqrt {\sqrt {a x^2+b^2}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(x^6*Sqrt[b^2 + a*x^2]),x]

[Out]

-1/20*(a^3*x*Hypergeometric2F1[-5/2, 3, -3/2, (b - Sqrt[b^2 + a*x^2])/(2*b)])/(b^3*(-b + Sqrt[b^2 + a*x^2])^3*
Sqrt[b + Sqrt[b^2 + a*x^2]])

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IntegrateAlgebraic [A]  time = 0.37, size = 166, normalized size = 0.84 \begin {gather*} \frac {\sqrt {b+\sqrt {b^2+a x^2}} \left (-315 a^{5/2} x^4+\sqrt {a} \left (16 b^4-144 b^3 \sqrt {b^2+a x^2}\right )+a^{3/2} \left (-42 b^2 x^2+210 b x^2 \sqrt {b^2+a x^2}\right )\right )}{640 \sqrt {a} b^5 x^5}-\frac {63 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{128 \sqrt {2} b^{11/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[b + Sqrt[b^2 + a*x^2]]/(x^6*Sqrt[b^2 + a*x^2]),x]

[Out]

(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-315*a^(5/2)*x^4 + Sqrt[a]*(16*b^4 - 144*b^3*Sqrt[b^2 + a*x^2]) + a^(3/2)*(-42*b
^2*x^2 + 210*b*x^2*Sqrt[b^2 + a*x^2])))/(640*Sqrt[a]*b^5*x^5) - (63*a^(5/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b
]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(128*Sqrt[2]*b^(11/2))

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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)/x^6/(a*x^2+b^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 {b + \sqrt {a x^{2} + b^{2}}}}{\sqrt {a x^{2} + b^{2}} x^{6}}\,{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^6/(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^6), x)

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{x^{6} \sqrt {a \,x^{2}+b^{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/x^6/(a*x^2+b^2)^(1/2),x)

[Out]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/x^6/(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}} x^{6}}\,{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^6/(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^6), 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^6\,\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)/(x^6*(a*x^2 + b^2)^(1/2)),x)

[Out]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(x^6*(a*x^2 + b^2)^(1/2)), x)

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sympy [C]  time = 1.99, size = 49, normalized size = 0.25 \begin {gather*} - \frac {\Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {5}{2}, \frac {1}{4}, \frac {3}{4} \\ - \frac {3}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{5 \pi \sqrt {b} x^{5}} \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**6/(a*x**2+b**2)**(1/2),x)

[Out]

-gamma(1/4)*gamma(3/4)*hyper((-5/2, 1/4, 3/4), (-3/2, 1/2), a*x**2*exp_polar(I*pi)/b**2)/(5*pi*sqrt(b)*x**5)

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