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

Optimal. Leaf size=47 \[ \frac{a x \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{\sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.0975331, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{a x \sqrt{c+\frac{d}{x^2}}}{c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.3511, size = 39, normalized size = 0.83 \[ \frac{a x \sqrt{c + \frac{d}{x^{2}}}}{c} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{\sqrt{d}} \]

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

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Mathematica [A]  time = 0.0920009, size = 89, normalized size = 1.89 \[ \frac{a \sqrt{d} \left (c x^2+d\right )+b c \log (x) \sqrt{c x^2+d}-b c \sqrt{c x^2+d} \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{c \sqrt{d} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 73, normalized size = 1.6 \[{\frac{1}{cx}\sqrt{c{x}^{2}+d} \left ( a\sqrt{c{x}^{2}+d}\sqrt{d}-b\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) c \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{\frac{1}{\sqrt{d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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 = 0.229864, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a d x \sqrt{\frac{c x^{2} + d}{x^{2}}} + b c \sqrt{d} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right )}{2 \, c d}, \frac{a d x \sqrt{\frac{c x^{2} + d}{x^{2}}} + b c \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{c d}\right ] \]

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]

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

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Sympy [A]  time = 3.51324, size = 39, normalized size = 0.83 \[ \frac{a \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}}{c} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{\sqrt{d}} \]

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(d)*sqrt(c*x**2/d + 1)/c - b*asinh(sqrt(d)/(sqrt(c)*x))/sqrt(d)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{\sqrt{c + \frac{d}{x^{2}}}}\,{d x} \]

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]

integrate((a + b/x^2)/sqrt(c + d/x^2), x)