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

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]

________________________________________________________________________________________

Rubi [A]  time = 0.0181643, 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, Rules used = {1588} \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]

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]

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{a g-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\frac{g x}{\sqrt{a+b x^2+c x^4}}\\ \end{align*}

Mathematica [F]  time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 18, normalized size = 1. \begin{align*}{gx{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.12618, size = 23, normalized size = 1.21 \begin{align*} \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.21719, size = 39, normalized size = 2.05 \begin{align*} \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.21693, size = 95, normalized size = 5. \begin{align*} \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}} \end{align*}

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))