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

Optimal. Leaf size=69 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

[Out]

(g*x)/Sqrt[a + b*x^2 + c*x^4] - (b*e - 2*a*f + (2*c*e - b*f)*x^2)/((b^2 - 4*a*c)
*Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.201119, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 22.647, size = 54, normalized size = 0.78 \[ \frac{2 a f - b e + g x \left (- 4 a c + b^{2}\right ) + x^{2} \left (b f - 2 c e\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a*f - b*e + g*x*(-4*a*c + b**2) + x**2*(b*f - 2*c*e))/((-4*a*c + b**2)*sqrt(a
+ b*x**2 + c*x**4))

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Mathematica [A]  time = 0.0915398, size = 61, normalized size = 0.88 \[ \frac{-4 a c g x+2 a f+b^2 g x-b e+b f x^2-2 c e x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-(b*e) + 2*a*f + b^2*g*x - 4*a*c*g*x - 2*c*e*x^2 + b*f*x^2)/((b^2 - 4*a*c)*Sqrt
[a + b*x^2 + c*x^4])

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Maple [A]  time = 0.008, size = 63, normalized size = 0.9 \[{\frac{4\,acgx-{b}^{2}gx-bf{x}^{2}+2\,ce{x}^{2}-2\,fa+be}{4\,ac-{b}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.780754, size = 127, normalized size = 1.84 \[ -\frac{\sqrt{c x^{4} + b x^{2} + a}{\left ({\left (2 \, c e - b f\right )} x^{2} + b e - 2 \, a f -{\left (b^{2} g - 4 \, a c g\right )} x\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*x^4 + b*x^2 + a)*((2*c*e - b*f)*x^2 + b*e - 2*a*f - (b^2*g - 4*a*c*g)*x)
/((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)

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Fricas [A]  time = 0.266905, size = 124, normalized size = 1.8 \[ \frac{\sqrt{c x^{4} + b x^{2} + a}{\left ({\left (b^{2} - 4 \, a c\right )} g x -{\left (2 \, c e - b f\right )} x^{2} - b e + 2 \, a f\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(c*x^4 + b*x^2 + a)*((b^2 - 4*a*c)*g*x - (2*c*e - b*f)*x^2 - b*e + 2*a*f)/((
b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{a g}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac{e x}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac{f x^{3}}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac{c g x^{4}}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-a*g/(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(-e*x/(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(-f*x**3/(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*g*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 = 0.313919, size = 262, normalized size = 3.8 \[ \frac{{\left (\frac{{\left (b^{3} f - 4 \, a b c f - 2 \, b^{2} c e + 8 \, a c^{2} e\right )} x}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}} + \frac{b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}\right )} x + \frac{2 \, a b^{2} f - 8 \, a^{2} c f - b^{3} e + 4 \, a b c e}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}}{8 \, \sqrt{c x^{4} + b x^{2} + a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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