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

Optimal. Leaf size=61 \[ \frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{2 d^{3/2}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{2 d x} \]

[Out]

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

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Rubi [A]  time = 0.12548, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{(b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{2 d^{3/2}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{2 d x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.4797, size = 51, normalized size = 0.84 \[ - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{2 d x} - \frac{\left (2 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{2 d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.131212, size = 108, normalized size = 1.77 \[ \frac{x^2 \log (x) \sqrt{c x^2+d} (2 a d-b c)+x^2 \sqrt{c x^2+d} (b c-2 a d) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )-b \sqrt{d} \left (c x^2+d\right )}{2 d^{3/2} x^3 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 105, normalized size = 1.7 \[ -{\frac{1}{2\,{x}^{3}}\sqrt{c{x}^{2}+d} \left ( 2\,{d}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) a{x}^{2}-bc\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{2}d+b\sqrt{c{x}^{2}+d}{d}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{d}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(c*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)^(1/2)*d^(3/2))/((c*x^2+d)/x^2)^(
1/2)/x^3/d^(5/2)

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229325, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (b c - 2 \, a d\right )} \sqrt{d} x \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \, b d \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \, d^{2} x}, -\frac{{\left (b c - 2 \, a d\right )} \sqrt{-d} x \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) + b d \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \, d^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 6.75813, size = 66, normalized size = 1.08 \[ - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{\sqrt{d}} - \frac{b \sqrt{c} \sqrt{1 + \frac{d}{c x^{2}}}}{2 d x} + \frac{b c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 d^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*asinh(sqrt(d)/(sqrt(c)*x))/sqrt(d) - b*sqrt(c)*sqrt(1 + d/(c*x**2))/(2*d*x) +
b*c*asinh(sqrt(d)/(sqrt(c)*x))/(2*d**(3/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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