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

Optimal. Leaf size=84 \[ -\frac{\sqrt{c+\frac{d}{x^2}} (a d+2 b c)}{2 c}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{a x^2 \left (c+\frac{d}{x^2}\right )^{3/2}}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.179291, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{\sqrt{c+\frac{d}{x^2}} (a d+2 b c)}{2 c}+\frac{(a d+2 b c) \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{2 \sqrt{c}}+\frac{a x^2 \left (c+\frac{d}{x^2}\right )^{3/2}}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.9538, size = 68, normalized size = 0.81 \[ \frac{a x^{2} \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{2 c} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (\frac{a d}{2} + b c\right )}{c} + \frac{\left (\frac{a d}{2} + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{\sqrt{c}} \]

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

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Mathematica [A]  time = 0.11842, size = 72, normalized size = 0.86 \[ \frac{1}{2} \sqrt{c+\frac{d}{x^2}} \left (\frac{x (a d+2 b c) \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{\sqrt{c} \sqrt{c x^2+d}}+a x^2-2 b\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.019, size = 127, normalized size = 1.5 \[{\frac{1}{2\,d}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( 2\,bc\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) xd+a{x}^{2}\sqrt{c{x}^{2}+d}\sqrt{c}d+2\,b{c}^{3/2}{x}^{2}\sqrt{c{x}^{2}+d}-2\,b \left ( c{x}^{2}+d \right ) ^{3/2}\sqrt{c}+a{d}^{2}\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) x \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{\frac{1}{\sqrt{c}}}} \]

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]

1/2*((c*x^2+d)/x^2)^(1/2)*(2*b*c*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x*d+a*x^2*(c*x^2+
d)^(1/2)*c^(1/2)*d+2*b*c^(3/2)*x^2*(c*x^2+d)^(1/2)-2*b*(c*x^2+d)^(3/2)*c^(1/2)+a
*d^2*ln(c^(1/2)*x+(c*x^2+d)^(1/2))*x)/(c*x^2+d)^(1/2)/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 = 0.233111, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b c + a d\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right ) + 2 \,{\left (a c x^{2} - 2 \, b c\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{4 \, c}, -\frac{{\left (2 \, b c + a d\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (a c x^{2} - 2 \, b c\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{2 \, c}\right ] \]

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]

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

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Sympy [A]  time = 11.2965, size = 107, normalized size = 1.27 \[ \frac{a \sqrt{d} x \sqrt{\frac{c x^{2}}{d} + 1}}{2} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )}}{2 \sqrt{c}} + b \sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{d}} \right )} - \frac{b c x}{\sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} - \frac{b \sqrt{d}}{x \sqrt{\frac{c x^{2}}{d} + 1}} \]

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

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GIAC/XCAS [A]  time = 0.259688, size = 124, normalized size = 1.48 \[ \frac{1}{2} \, \sqrt{c x^{2} + d} a x{\rm sign}\left (x\right ) + \frac{2 \, b \sqrt{c} d{\rm sign}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d} - \frac{{\left (2 \, b c^{\frac{3}{2}}{\rm sign}\left (x\right ) + a \sqrt{c} d{\rm sign}\left (x\right )\right )}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2}\right )}{4 \, c} \]

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]

1/2*sqrt(c*x^2 + d)*a*x*sign(x) + 2*b*sqrt(c)*d*sign(x)/((sqrt(c)*x - sqrt(c*x^2
+ d))^2 - d) - 1/4*(2*b*c^(3/2)*sign(x) + a*sqrt(c)*d*sign(x))*ln((sqrt(c)*x -
sqrt(c*x^2 + d))^2)/c