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

Optimal. Leaf size=43 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{\sqrt{c}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d} \]

[Out]

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

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Rubi [A]  time = 0.12937, antiderivative size = 43, 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 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{\sqrt{c}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.8749, size = 36, normalized size = 0.84 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.063596, size = 76, normalized size = 1.77 \[ \frac{a d x \sqrt{c x^2+d} \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )-b \sqrt{c} \left (c x^2+d\right )}{\sqrt{c} d x^2 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.016, size = 70, normalized size = 1.6 \[ -{\frac{1}{d{x}^{2}}\sqrt{c{x}^{2}+d} \left ( -a\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) xd+b\sqrt{c{x}^{2}+d}\sqrt{c} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227924, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{c} d \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) - 2 \, b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \, c d}, -\frac{a \sqrt{-c} d \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) + b c \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.85985, size = 138, normalized size = 3.21 \[ - a \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{c}} \sqrt{c + \frac{d}{x^{2}}}} \right )}}{c \sqrt{- \frac{1}{c}}} & \text{for}\: - \frac{1}{c} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{c + \frac{d}{x^{2}}} \sqrt{\frac{1}{c}}} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: - \frac{1}{c} < 0 \wedge \frac{1}{c} < \frac{1}{c + \frac{d}{x^{2}}} \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{c + \frac{d}{x^{2}}} \sqrt{\frac{1}{c}}} \right )}}{c \sqrt{\frac{1}{c}}} & \text{for}\: \frac{1}{c} > \frac{1}{c + \frac{d}{x^{2}}} \wedge - \frac{1}{c} < 0 \end{cases}\right ) - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*Piecewise((atan(1/(sqrt(-1/c)*sqrt(c + d/x**2)))/(c*sqrt(-1/c)), -1/c > 0), (
-acoth(1/(sqrt(c + d/x**2)*sqrt(1/c)))/(c*sqrt(1/c)), (-1/c < 0) & (1/c < 1/(c +
d/x**2))), (-atanh(1/(sqrt(c + d/x**2)*sqrt(1/c)))/(c*sqrt(1/c)), (-1/c < 0) &
(1/c > 1/(c + d/x**2)))) - b*sqrt(c + d/x**2)/d

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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