3.25.46 \(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{x^6} \, dx\) [2446]

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

[Out]

1/1920*(b+(a*x^2+b^2)^(1/2))^(1/2)*(105*a^(5/2)*x^4+a^(1/2)*(-432*b^4+48*b^3*(a*x^2+b^2)^(1/2))+a^(3/2)*(14*b^
2*x^2-70*b*x^2*(a*x^2+b^2)^(1/2)))/a^(1/2)/b^4/x^5+7/128*a^(5/2)*arctan(1/2*a^(1/2)*x*2^(1/2)/b^(1/2)/(b+(a*x^
2+b^2)^(1/2))^(1/2)-1/2*(b+(a*x^2+b^2)^(1/2))^(1/2)*2^(1/2)/b^(1/2))*2^(1/2)/b^(9/2)

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Rubi [F]
time = 0.10, 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 [A]
time = 0.35, size = 147, normalized size = 0.74 \begin {gather*} \frac {\sqrt {b+\sqrt {b^2+a x^2}} \left (-432 b^4+14 a b^2 x^2+105 a^2 x^4+48 b^3 \sqrt {b^2+a x^2}-70 a b x^2 \sqrt {b^2+a x^2}\right )}{1920 b^4 x^5}+\frac {7 a^{5/2} \text {ArcTan}\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]

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

[Out]

(Sqrt[b + Sqrt[b^2 + a*x^2]]*(-432*b^4 + 14*a*b^2*x^2 + 105*a^2*x^4 + 48*b^3*Sqrt[b^2 + a*x^2] - 70*a*b*x^2*Sq
rt[b^2 + a*x^2]))/(1920*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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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 3.
time = 0.04, 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*} \text {Failed to integrate} \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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Fricas [A]
time = 173.28, size = 330, normalized size = 1.67 \begin {gather*} \left [\frac {105 \, \sqrt {\frac {1}{2}} a^{2} x^{5} \sqrt {-\frac {a}{b}} \log \left (-\frac {a^{2} x^{3} + 4 \, a b^{2} x - 4 \, \sqrt {a x^{2} + b^{2}} a b x - 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a x^{2} + b^{2}} b^{2} \sqrt {-\frac {a}{b}} - \sqrt {\frac {1}{2}} {\left (a b x^{2} + 2 \, b^{3}\right )} \sqrt {-\frac {a}{b}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{x^{3}}\right ) + 2 \, {\left (105 \, a^{2} x^{4} + 14 \, a b^{2} x^{2} - 432 \, b^{4} - 2 \, {\left (35 \, a b x^{2} - 24 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{3840 \, b^{4} x^{5}}, -\frac {105 \, \sqrt {\frac {1}{2}} a^{2} x^{5} \sqrt {\frac {a}{b}} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {b + \sqrt {a x^{2} + b^{2}}} b \sqrt {\frac {a}{b}}}{a x}\right ) - {\left (105 \, a^{2} x^{4} + 14 \, a b^{2} x^{2} - 432 \, b^{4} - 2 \, {\left (35 \, a b x^{2} - 24 \, b^{3}\right )} \sqrt {a x^{2} + b^{2}}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{1920 \, b^{4} x^{5}}\right ] \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]

[1/3840*(105*sqrt(1/2)*a^2*x^5*sqrt(-a/b)*log(-(a^2*x^3 + 4*a*b^2*x - 4*sqrt(a*x^2 + b^2)*a*b*x - 4*(2*sqrt(1/
2)*sqrt(a*x^2 + b^2)*b^2*sqrt(-a/b) - sqrt(1/2)*(a*b*x^2 + 2*b^3)*sqrt(-a/b))*sqrt(b + sqrt(a*x^2 + b^2)))/x^3
) + 2*(105*a^2*x^4 + 14*a*b^2*x^2 - 432*b^4 - 2*(35*a*b*x^2 - 24*b^3)*sqrt(a*x^2 + b^2))*sqrt(b + sqrt(a*x^2 +
b^2)))/(b^4*x^5), -1/1920*(105*sqrt(1/2)*a^2*x^5*sqrt(a/b)*arctan(2*sqrt(1/2)*sqrt(b + sqrt(a*x^2 + b^2))*b*s
qrt(a/b)/(a*x)) - (105*a^2*x^4 + 14*a*b^2*x^2 - 432*b^4 - 2*(35*a*b*x^2 - 24*b^3)*sqrt(a*x^2 + b^2))*sqrt(b +
sqrt(a*x^2 + b^2)))/(b^4*x^5)]

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Sympy [C] Result contains complex when optimal does not.
time = 0.95, 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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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