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

Optimal. Leaf size=198 \[ \frac {7 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{64 \sqrt {2} b^{9/2}}+\frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (a^{3/2} \left (14 b^2 x^2-70 b x^2 \sqrt {a x^2+b^2}\right )+105 a^{5/2} x^4+\sqrt {a} \left (48 b^3 \sqrt {a x^2+b^2}-432 b^4\right )\right )}{1920 \sqrt {a} b^4 x^5} \]

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Rubi [F]  time = 0.15, 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} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 0.25, size = 95, normalized size = 0.48 \begin {gather*} \frac {\sqrt {\sqrt {a x^2+b^2}+b} \left (\left (2 b \sqrt {a x^2+b^2}+a x^2+2 b^2\right ) \, _2F_1\left (-\frac {5}{2},2;-\frac {3}{2};\frac {b-\sqrt {b^2+a x^2}}{2 b}\right )-20 b^2\right )}{80 b^2 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-20*b^2 + (2*b^2 + a*x^2 + 2*b*Sqrt[b^2 + a*x^2])*Hypergeometric2F1[-5/2, 2, -3/
2, (b - Sqrt[b^2 + a*x^2])/(2*b)]))/(80*b^2*x^5)

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IntegrateAlgebraic [A]  time = 0.40, size = 166, normalized size = 0.84 \begin {gather*} \frac {\sqrt {b+\sqrt {b^2+a x^2}} \left (105 a^{5/2} x^4+\sqrt {a} \left (-432 b^4+48 b^3 \sqrt {b^2+a x^2}\right )+a^{3/2} \left (14 b^2 x^2-70 b x^2 \sqrt {b^2+a x^2}\right )\right )}{1920 \sqrt {a} b^4 x^5}+\frac {7 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^{9/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b + Sqrt[b^2 + a*x^2]]*(105*a^(5/2)*x^4 + Sqrt[a]*(-432*b^4 + 48*b^3*Sqrt[b^2 + a*x^2]) + a^(3/2)*(14*b^
2*x^2 - 70*b*x^2*Sqrt[b^2 + a*x^2])))/(1920*Sqrt[a]*b^4*x^5) + (7*a^(5/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*
Sqrt[b + Sqrt[b^2 + a*x^2]])])/(128*Sqrt[2]*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)/x^6,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}}}}{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,x, algorithm="giac")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/x^6, x)

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maple [C]  time = 0.05, size = 31, normalized size = 0.16

method	result	size
meijerg	\(-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \hypergeom \left (\left [-\frac {5}{2}, -\frac {1}{4}, \frac {1}{4}\right ], \left [-\frac {3}{2}, \frac {1}{2}\right ], -\frac {x^{2} a}{b^{2}}\right )}{5 x^{5}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*(b^2)^(1/4)*2^(1/2)/x^5*hypergeom([-5/2,-1/4,1/4],[-3/2,1/2],-x^2*a/b^2)

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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}}}}{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,x, algorithm="maxima")

[Out]

integrate(sqrt(b + 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} \,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,x)

[Out]

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

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sympy [C]  time = 1.58, size = 51, normalized size = 0.26 \begin {gather*} \frac {\sqrt {b} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} - \frac {5}{2}, - \frac {1}{4}, \frac {1}{4} \\ - \frac {3}{2}, \frac {1}{2} \end {matrix}\middle | {\frac {a x^{2} e^{i \pi }}{b^{2}}} \right )}}{20 \pi 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,x)

[Out]

sqrt(b)*gamma(-1/4)*gamma(1/4)*hyper((-5/2, -1/4, 1/4), (-3/2, 1/2), a*x**2*exp_polar(I*pi)/b**2)/(20*pi*x**5)

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