3.803 \(\int \frac{a+\frac{b}{x^2}}{\sqrt{c+\frac{d}{x^2}} x^4} \, dx\)

Optimal. Leaf size=93 \[ -\frac{c (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{5/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac{b \sqrt{c+\frac{d}{x^2}}}{4 d x^3} \]

[Out]

-(b*Sqrt[c + d/x^2])/(4*d*x^3) + ((3*b*c - 4*a*d)*Sqrt[c + d/x^2])/(8*d^2*x) - (
c*(3*b*c - 4*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(8*d^(5/2))

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Rubi [A]  time = 0.173637, antiderivative size = 93, 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 \[ -\frac{c (3 b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{5/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (3 b c-4 a d)}{8 d^2 x}-\frac{b \sqrt{c+\frac{d}{x^2}}}{4 d x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/(Sqrt[c + d/x^2]*x^4),x]

[Out]

-(b*Sqrt[c + d/x^2])/(4*d*x^3) + ((3*b*c - 4*a*d)*Sqrt[c + d/x^2])/(8*d^2*x) - (
c*(3*b*c - 4*a*d)*ArcTanh[Sqrt[d]/(Sqrt[c + d/x^2]*x)])/(8*d^(5/2))

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Rubi in Sympy [A]  time = 14.415, size = 82, normalized size = 0.88 \[ - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{4 d x^{3}} + \frac{c \left (4 a d - 3 b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 d^{\frac{5}{2}}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (4 a d - 3 b c\right )}{8 d^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/x**4/(c+d/x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.181506, size = 129, normalized size = 1.39 \[ \frac{-\sqrt{d} \left (c x^2+d\right ) \left (4 a d x^2-3 b c x^2+2 b d\right )+c x^4 \log (x) \sqrt{c x^2+d} (3 b c-4 a d)+c x^4 \sqrt{c x^2+d} (4 a d-3 b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{8 d^{5/2} x^5 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/(Sqrt[c + d/x^2]*x^4),x]

[Out]

(-(Sqrt[d]*(d + c*x^2)*(2*b*d - 3*b*c*x^2 + 4*a*d*x^2)) + c*(3*b*c - 4*a*d)*x^4*
Sqrt[d + c*x^2]*Log[x] + c*(-3*b*c + 4*a*d)*x^4*Sqrt[d + c*x^2]*Log[d + Sqrt[d]*
Sqrt[d + c*x^2]])/(8*d^(5/2)*Sqrt[c + d/x^2]*x^5)

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Maple [A]  time = 0.032, size = 148, normalized size = 1.6 \[ -{\frac{1}{8\,{x}^{5}}\sqrt{c{x}^{2}+d} \left ( -4\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{3}+4\,a\sqrt{c{x}^{2}+d}{x}^{2}{d}^{7/2}-3\,bc\sqrt{c{x}^{2}+d}{x}^{2}{d}^{5/2}+3\,b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{2}+2\,b\sqrt{c{x}^{2}+d}{d}^{7/2} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{d}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/x^4/(c+d/x^2)^(1/2),x)

[Out]

-1/8*(c*x^2+d)^(1/2)*(-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)^(1/2)*x^2*d^(7/2)-3*b*c*(c*x^2+d)^(1/2)*x^2*d^(5/2)+3*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)^(1/2)*d^(7/2))/((c*x^2+d)/x^2)^(1/2
)/x^5/d^(9/2)

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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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235741, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, b c^{2} - 4 \, a c d\right )} \sqrt{d} x^{3} \log \left (-\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (2 \, b d^{2} -{\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, d^{3} x^{3}}, \frac{{\left (3 \, b c^{2} - 4 \, a c d\right )} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) -{\left (2 \, b d^{2} -{\left (3 \, b c d - 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, d^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 12.8797, size = 150, normalized size = 1.61 \[ - \frac{a \sqrt{c} \sqrt{1 + \frac{d}{c x^{2}}}}{2 d x} + \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 d^{\frac{3}{2}}} + \frac{3 b c^{\frac{3}{2}}}{8 d^{2} x \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b \sqrt{c}}{8 d x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{5}{2}}} - \frac{b}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/x**4/(c+d/x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{\sqrt{c + \frac{d}{x^{2}}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^4),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/(sqrt(c + d/x^2)*x^4), x)