3.807 \(\int \frac{a+\frac{b}{x^2}}{\left (c+\frac{d}{x^2}\right )^{3/2} x^3} \, dx\)

Optimal. Leaf size=42 \[ -\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.122619, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{b c-a d}{d^2 \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{c+\frac{d}{x^2}}}{d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.1969, size = 34, normalized size = 0.81 \[ - \frac{b \sqrt{c + \frac{d}{x^{2}}}}{d^{2}} + \frac{a d - b c}{d^{2} \sqrt{c + \frac{d}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.0532218, size = 36, normalized size = 0.86 \[ \frac{a d x^2-b \left (2 c x^2+d\right )}{d^2 x^2 \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x]

[Out]

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

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Maple [A]  time = 0.01, size = 46, normalized size = 1.1 \[{\frac{ \left ( ad{x}^{2}-2\,bc{x}^{2}-bd \right ) \left ( c{x}^{2}+d \right ) }{{d}^{2}{x}^{4}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^2)/(c+d/x^2)^(3/2)/x^3,x)

[Out]

(a*d*x^2-2*b*c*x^2-b*d)*(c*x^2+d)/((c*x^2+d)/x^2)^(3/2)/d^2/x^4

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Maxima [A]  time = 1.38262, size = 62, normalized size = 1.48 \[ -b{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{d^{2}} + \frac{c}{\sqrt{c + \frac{d}{x^{2}}} d^{2}}\right )} + \frac{a}{\sqrt{c + \frac{d}{x^{2}}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.216663, size = 62, normalized size = 1.48 \[ -\frac{{\left ({\left (2 \, b c - a d\right )} x^{2} + b d\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c d^{2} x^{2} + d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.5744, size = 68, normalized size = 1.62 \[ \begin{cases} \frac{a}{d \sqrt{c + \frac{d}{x^{2}}}} - \frac{2 b c}{d^{2} \sqrt{c + \frac{d}{x^{2}}}} - \frac{b}{d x^{2} \sqrt{c + \frac{d}{x^{2}}}} & \text{for}\: d \neq 0 \\\frac{- \frac{a}{2 x^{2}} - \frac{b}{4 x^{4}}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**3,x)

[Out]

Piecewise((a/(d*sqrt(c + d/x**2)) - 2*b*c/(d**2*sqrt(c + d/x**2)) - b/(d*x**2*sq
rt(c + d/x**2)), Ne(d, 0)), ((-a/(2*x**2) - b/(4*x**4))/c**(3/2), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^3),x, algorithm="giac")

[Out]

integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^3), x)