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

Optimal. Leaf size=59 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.139945, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{(2 b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 c^{3/2}}+\frac{a x^2 \sqrt{c+\frac{d}{x^2}}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.3052, size = 48, normalized size = 0.81 \[ \frac{a x^{2} \sqrt{c + \frac{d}{x^{2}}}}{2 c} - \frac{\left (\frac{a d}{2} - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{c^{\frac{3}{2}}} \]

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

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Mathematica [A]  time = 0.0749944, size = 82, normalized size = 1.39 \[ \frac{\sqrt{c x^2+d} (2 b c-a d) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )+a \sqrt{c} x \left (c x^2+d\right )}{2 c^{3/2} x \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 90, normalized size = 1.5 \[{\frac{1}{2\,x}\sqrt{c{x}^{2}+d} \left ( ax\sqrt{c{x}^{2}+d}{c}^{{\frac{3}{2}}}+2\,b\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ){c}^{2}-ad\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) c \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{c}^{-{\frac{5}{2}}}} \]

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]

1/2*(c*x^2+d)^(1/2)*(a*x*(c*x^2+d)^(1/2)*c^(3/2)+2*b*ln(c^(1/2)*x+(c*x^2+d)^(1/2
))*c^2-a*d*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*c)/((c*x^2+d)/x^2)^(1/2)/x/c^(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)*x/sqrt(c + d/x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233514, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, b c - a d\right )} \sqrt{c} \log \left (2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right )}{4 \, c^{2}}, \frac{a c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, b c - a d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{2 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.4467, size = 66, normalized size = 1.12 \[ \frac{a \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2 c} - \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 c^{\frac{3}{2}}} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{\sqrt{c}} \]

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

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GIAC/XCAS [A]  time = 0.246198, size = 107, normalized size = 1.81 \[ -\frac{1}{2} \, d{\left (\frac{a \sqrt{\frac{c x^{2} + d}{x^{2}}}}{{\left (c - \frac{c x^{2} + d}{x^{2}}\right )} c} + \frac{{\left (2 \, b c - a d\right )} \arctan \left (\frac{\sqrt{\frac{c x^{2} + d}{x^{2}}}}{\sqrt{-c}}\right )}{\sqrt{-c} c d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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