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

Optimal. Leaf size=91 \[ \frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{3/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-4 a d)}{8 d x}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{4 d x} \]

[Out]

((b*c - 4*a*d)*Sqrt[c + d/x^2])/(8*d*x) - (b*(c + d/x^2)^(3/2))/(4*d*x) + (c*(b*
c - 4*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(8*d^(3/2))

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Rubi [A]  time = 0.159678, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{3/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-4 a d)}{8 d x}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{4 d x} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - 4*a*d)*Sqrt[c + d/x^2])/(8*d*x) - (b*(c + d/x^2)^(3/2))/(4*d*x) + (c*(b*
c - 4*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(8*d^(3/2))

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Rubi in Sympy [A]  time = 13.0123, size = 76, normalized size = 0.84 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{4 d x} - \frac{c \left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 d^{\frac{3}{2}}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (4 a d - b c\right )}{8 d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(c + d/x**2)**(3/2)/(4*d*x) - c*(4*a*d - b*c)*atanh(sqrt(d)/(x*sqrt(c + d/x**
2)))/(8*d**(3/2)) - sqrt(c + d/x**2)*(4*a*d - b*c)/(8*d*x)

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Mathematica [A]  time = 0.157091, size = 119, normalized size = 1.31 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^4 \log (x) (b c-4 a d)+\sqrt{d} \sqrt{c x^2+d} \left (4 a d x^2+b c x^2+2 b d\right )+c x^4 (4 a d-b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )\right )}{8 d^{3/2} x^3 \sqrt{c x^2+d}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c + d/x^2]*(Sqrt[d]*Sqrt[d + c*x^2]*(2*b*d + b*c*x^2 + 4*a*d*x^2) + c*(b*
c - 4*a*d)*x^4*Log[x] + c*(-(b*c) + 4*a*d)*x^4*Log[d + Sqrt[d]*Sqrt[d + c*x^2]])
)/(8*d^(3/2)*x^3*Sqrt[d + c*x^2])

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Maple [B]  time = 0.018, size = 187, normalized size = 2.1 \[ -{\frac{1}{8\,{x}^{3}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( -4\,ac\sqrt{c{x}^{2}+d}{x}^{4}{d}^{5/2}+b{c}^{2}\sqrt{c{x}^{2}+d}{x}^{4}{d}^{{\frac{3}{2}}}+4\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{3}+4\,a \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}{d}^{5/2}-bc \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}{x}^{2}{d}^{{\frac{3}{2}}}-b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{2}+2\,b \left ( c{x}^{2}+d \right ) ^{3/2}{d}^{5/2} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{d}^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*((c*x^2+d)/x^2)^(1/2)/x^3*(-4*a*c*(c*x^2+d)^(1/2)*x^4*d^(5/2)+b*c^2*(c*x^2+
d)^(1/2)*x^4*d^(3/2)+4*a*c*ln(2*(d^(1/2)*(c*x^2+d)^(1/2)+d)/x)*x^4*d^3+4*a*(c*x^
2+d)^(3/2)*x^2*d^(5/2)-b*c*(c*x^2+d)^(3/2)*x^2*d^(3/2)-b*c^2*ln(2*(d^(1/2)*(c*x^
2+d)^(1/2)+d)/x)*x^4*d^2+2*b*(c*x^2+d)^(3/2)*d^(5/2))/(c*x^2+d)^(1/2)/d^(7/2)

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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 = 0.238281, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c^{2} - 4 \, a c d\right )} \sqrt{d} x^{3} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (2 \, b d^{2} +{\left (b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, d^{2} x^{3}}, -\frac{{\left (b c^{2} - 4 \, a c d\right )} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (2 \, b d^{2} +{\left (b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, d^{2} x^{3}}\right ] \]

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]

[-1/16*((b*c^2 - 4*a*c*d)*sqrt(d)*x^3*log((2*d*x*sqrt((c*x^2 + d)/x^2) - (c*x^2
+ 2*d)*sqrt(d))/x^2) + 2*(2*b*d^2 + (b*c*d + 4*a*d^2)*x^2)*sqrt((c*x^2 + d)/x^2)
)/(d^2*x^3), -1/8*((b*c^2 - 4*a*c*d)*sqrt(-d)*x^3*arctan(sqrt(-d)/(x*sqrt((c*x^2
+ d)/x^2))) + (2*b*d^2 + (b*c*d + 4*a*d^2)*x^2)*sqrt((c*x^2 + d)/x^2))/(d^2*x^3
)]

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Sympy [A]  time = 9.85855, size = 144, normalized size = 1.58 \[ - \frac{a \sqrt{c} \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} - \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 \sqrt{d}} - \frac{b c^{\frac{3}{2}}}{8 d x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b \sqrt{c}}{8 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{3}{2}}} - \frac{b d}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*sqrt(c)*sqrt(1 + d/(c*x**2))/(2*x) - a*c*asinh(sqrt(d)/(sqrt(c)*x))/(2*sqrt(d
)) - b*c**(3/2)/(8*d*x*sqrt(1 + d/(c*x**2))) - 3*b*sqrt(c)/(8*x**3*sqrt(1 + d/(c
*x**2))) + b*c**2*asinh(sqrt(d)/(sqrt(c)*x))/(8*d**(3/2)) - b*d/(4*sqrt(c)*x**5*
sqrt(1 + d/(c*x**2)))

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GIAC/XCAS [A]  time = 0.249531, size = 176, normalized size = 1.93 \[ -\frac{\frac{{\left (b c^{3}{\rm sign}\left (x\right ) - 4 \, a c^{2} d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d} d} + \frac{{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{3}{\rm sign}\left (x\right ) + 4 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{2} d{\rm sign}\left (x\right ) + \sqrt{c x^{2} + d} b c^{3} d{\rm sign}\left (x\right ) - 4 \, \sqrt{c x^{2} + d} a c^{2} d^{2}{\rm sign}\left (x\right )}{c^{2} d x^{4}}}{8 \, c} \]

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]

-1/8*((b*c^3*sign(x) - 4*a*c^2*d*sign(x))*arctan(sqrt(c*x^2 + d)/sqrt(-d))/(sqrt
(-d)*d) + ((c*x^2 + d)^(3/2)*b*c^3*sign(x) + 4*(c*x^2 + d)^(3/2)*a*c^2*d*sign(x)
+ sqrt(c*x^2 + d)*b*c^3*d*sign(x) - 4*sqrt(c*x^2 + d)*a*c^2*d^2*sign(x))/(c^2*d
*x^4))/c