3.108 \(\int \frac{a g-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=19 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]

[Out]

(g*x)/Sqrt[a + b*x^2 + c*x^4]

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Rubi [A]  time = 0.0152488, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(a*g - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(g*x)/Sqrt[a + b*x^2 + c*x^4]

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Rubi in Sympy [A]  time = 13.0074, size = 17, normalized size = 0.89 \[ \frac{g x}{\sqrt{a + b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c*g*x**4+a*g)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

g*x/sqrt(a + b*x**2 + c*x**4)

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Mathematica [A]  time = 0.0451867, size = 19, normalized size = 1. \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*g - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

(g*x)/Sqrt[a + b*x^2 + c*x^4]

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Maple [A]  time = 0.007, size = 18, normalized size = 1. \[{gx{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-c*g*x^4+a*g)/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

g*x/(c*x^4+b*x^2+a)^(1/2)

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Maxima [A]  time = 0.768361, size = 23, normalized size = 1.21 \[ \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*g*x^4 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="maxima")

[Out]

g*x/sqrt(c*x^4 + b*x^2 + a)

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Fricas [A]  time = 0.259511, size = 23, normalized size = 1.21 \[ \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*g*x^4 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="fricas")

[Out]

g*x/sqrt(c*x^4 + b*x^2 + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - g \left (\int \left (- \frac{a}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx + \int \frac{c x^{4}}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*g*x**4+a*g)/(c*x**4+b*x**2+a)**(3/2),x)

[Out]

-g*(Integral(-a/(a*sqrt(a + b*x**2 + c*x**4) + b*x**2*sqrt(a + b*x**2 + c*x**4)
+ c*x**4*sqrt(a + b*x**2 + c*x**4)), x) + Integral(c*x**4/(a*sqrt(a + b*x**2 + c
*x**4) + b*x**2*sqrt(a + b*x**2 + c*x**4) + c*x**4*sqrt(a + b*x**2 + c*x**4)), x
))

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GIAC/XCAS [A]  time = 0.329233, size = 95, normalized size = 5. \[ \frac{{\left (b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g\right )} x}{32 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(c*g*x^4 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="giac")

[Out]

1/32*(b^4*g - 8*a*b^2*c*g + 16*a^2*c^2*g)*x/((a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3
*c^4)*sqrt(c*x^4 + b*x^2 + a))