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

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Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, 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*}

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Mathematica [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

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

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fricas [A]  time = 0.71, size = 17, normalized size = 0.89 \[ \frac {g x}{\sqrt {c x^{4} + b x^{2} + a}} \]

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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giac [B]  time = 1.91, size = 60, normalized size = 3.16 \[ \frac {{\left (b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g\right )} x}{\sqrt {c x^{4} + b x^{2} + a} {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]

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, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 18, normalized size = 0.95 \[ \frac {g x}{\sqrt {c \,x^{4}+b \,x^{2}+a}} \]

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 = 0.63, size = 17, normalized size = 0.89 \[ \frac {g x}{\sqrt {c x^{4} + b x^{2} + a}} \]

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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mupad [B]  time = 0.99, size = 17, normalized size = 0.89 \[ \frac {g\,x}{\sqrt {c\,x^4+b\,x^2+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(g*x)/(a + b*x^2 + c*x^4)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - g \left (\int \left (- \frac {a}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx + \int \frac {c x^{4}}{a \sqrt {a + b x^{2} + c x^{4}} + b x^{2} \sqrt {a + b x^{2} + c x^{4}} + c x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx\right ) \]

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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