3.773 \(\int \left (a+\frac{b}{x^2}\right ) \sqrt{c+\frac{d}{x^2}} \, dx\)

Optimal. Leaf size=85 \[ -\frac{\sqrt{c+\frac{d}{x^2}} (2 a d+b c)}{2 c x}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{2 \sqrt{d}}+\frac{a x \left (c+\frac{d}{x^2}\right )^{3/2}}{c} \]

[Out]

-((b*c + 2*a*d)*Sqrt[c + d/x^2])/(2*c*x) + (a*(c + d/x^2)^(3/2)*x)/c - ((b*c + 2
*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*Sqrt[d])

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Rubi [A]  time = 0.146613, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{\sqrt{c+\frac{d}{x^2}} (2 a d+b c)}{2 c x}-\frac{(2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{2 \sqrt{d}}+\frac{a x \left (c+\frac{d}{x^2}\right )^{3/2}}{c} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)*Sqrt[c + d/x^2],x]

[Out]

-((b*c + 2*a*d)*Sqrt[c + d/x^2])/(2*c*x) + (a*(c + d/x^2)^(3/2)*x)/c - ((b*c + 2
*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(2*Sqrt[d])

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Rubi in Sympy [A]  time = 13.4391, size = 71, normalized size = 0.84 \[ \frac{a x \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{c} - \frac{\left (2 a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{2 \sqrt{d}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (2 a d + b c\right )}{2 c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2),x)

[Out]

a*x*(c + d/x**2)**(3/2)/c - (2*a*d + b*c)*atanh(sqrt(d)/(x*sqrt(c + d/x**2)))/(2
*sqrt(d)) - sqrt(c + d/x**2)*(2*a*d + b*c)/(2*c*x)

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Mathematica [A]  time = 0.118758, size = 109, normalized size = 1.28 \[ \frac{\sqrt{c+\frac{d}{x^2}} \left (\sqrt{d} \left (2 a x^2-b\right ) \sqrt{c x^2+d}+x^2 \log (x) (2 a d+b c)-x^2 (2 a d+b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )\right )}{2 \sqrt{d} x \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)*Sqrt[c + d/x^2],x]

[Out]

(Sqrt[c + d/x^2]*(Sqrt[d]*(-b + 2*a*x^2)*Sqrt[d + c*x^2] + (b*c + 2*a*d)*x^2*Log
[x] - (b*c + 2*a*d)*x^2*Log[d + Sqrt[d]*Sqrt[d + c*x^2]]))/(2*Sqrt[d]*x*Sqrt[d +
c*x^2])

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Maple [A]  time = 0.016, size = 142, normalized size = 1.7 \[{\frac{1}{2\,x}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 2\,\sqrt{c{x}^{2}+d}a{x}^{2}{d}^{3/2}+bc\sqrt{c{x}^{2}+d}{x}^{2}\sqrt{d}-2\,{d}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) a{x}^{2}-bc\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{2}d-b \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}\sqrt{d} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{d}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)*(c+d/x^2)^(1/2),x)

[Out]

1/2*((c*x^2+d)/x^2)^(1/2)*(2*(c*x^2+d)^(1/2)*a*x^2*d^(3/2)+b*c*(c*x^2+d)^(1/2)*x
^2*d^(1/2)-2*d^2*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*a*x^2-b*c*ln(2*(d^(1/2)*(c*
x^2+d)^(1/2)+d)/x)*x^2*d-b*(c*x^2+d)^(3/2)*d^(1/2))/x/(c*x^2+d)^(1/2)/d^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234256, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (b c + 2 \, a d\right )} \sqrt{d} x \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (2 \, a d x^{2} - b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \, d x}, \frac{{\left (b c + 2 \, a d\right )} \sqrt{-d} x \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (2 \, a d x^{2} - b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \, d x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2),x, algorithm="fricas")

[Out]

[1/4*((b*c + 2*a*d)*sqrt(d)*x*log((2*d*x*sqrt((c*x^2 + d)/x^2) - (c*x^2 + 2*d)*s
qrt(d))/x^2) + 2*(2*a*d*x^2 - b*d)*sqrt((c*x^2 + d)/x^2))/(d*x), 1/2*((b*c + 2*a
*d)*sqrt(-d)*x*arctan(sqrt(-d)/(x*sqrt((c*x^2 + d)/x^2))) + (2*a*d*x^2 - b*d)*sq
rt((c*x^2 + d)/x^2))/(d*x)]

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Sympy [A]  time = 6.00896, size = 107, normalized size = 1.26 \[ \frac{a \sqrt{c} x}{\sqrt{1 + \frac{d}{c x^{2}}}} - a \sqrt{d} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )} + \frac{a d}{\sqrt{c} x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{b \sqrt{c} \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} - \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)*(c+d/x**2)**(1/2),x)

[Out]

a*sqrt(c)*x/sqrt(1 + d/(c*x**2)) - a*sqrt(d)*asinh(sqrt(d)/(sqrt(c)*x)) + a*d/(s
qrt(c)*x*sqrt(1 + d/(c*x**2))) - b*sqrt(c)*sqrt(1 + d/(c*x**2))/(2*x) - b*c*asin
h(sqrt(d)/(sqrt(c)*x))/(2*sqrt(d))

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GIAC/XCAS [A]  time = 0.239893, size = 103, normalized size = 1.21 \[ \frac{2 \, \sqrt{c x^{2} + d} a c{\rm sign}\left (x\right ) + \frac{{\left (b c^{2}{\rm sign}\left (x\right ) + 2 \, a c d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d}} - \frac{\sqrt{c x^{2} + d} b c{\rm sign}\left (x\right )}{x^{2}}}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)*sqrt(c + d/x^2),x, algorithm="giac")

[Out]

1/2*(2*sqrt(c*x^2 + d)*a*c*sign(x) + (b*c^2*sign(x) + 2*a*c*d*sign(x))*arctan(sq
rt(c*x^2 + d)/sqrt(-d))/sqrt(-d) - sqrt(c*x^2 + d)*b*c*sign(x)/x^2)/c