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

Optimal. Leaf size=57 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0666024, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {1673, 1588, 12, 1107, 613} \[ \frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

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

Rule 1673

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 613

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[(-2*(b + 2*c*x))/((b^2 - 4*a*c)*Sqrt[a + b*x
+ c*x^2]), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \frac{a g+e x-c g x^4}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx &=\int \frac{e x}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx+\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}}+e \int \frac{x}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\\ &=\frac{g x}{\sqrt{a+b x^2+c x^4}}+\frac{1}{2} e \operatorname{Subst}\left (\int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{e \left (b+2 c x^2\right )}{\left (b^2-4 a c\right ) \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 + e*x - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]

[Out]

$Aborted

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Maple [A]  time = 0.005, size = 52, normalized size = 0.9 \begin{align*}{\frac{4\,acgx-{b}^{2}gx+2\,ce{x}^{2}+be}{4\,ac-{b}^{2}}{\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+e*x)/(c*x^4+b*x^2+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.23641, size = 69, normalized size = 1.21 \begin{align*} -\frac{2 \, c e x^{2} + b e -{\left (b^{2} g - 4 \, a c g\right )} x}{\sqrt{c x^{4} + b x^{2} + a}{\left (b^{2} - 4 \, a c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.33697, size = 173, normalized size = 3.04 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2} + a}{\left (2 \, c e x^{2} -{\left (b^{2} - 4 \, a c\right )} g x + b e\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.17025, size = 228, normalized size = 4. \begin{align*} -\frac{{\left (\frac{2 \,{\left (b^{2} c e - 4 \, 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{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}}}{16 \, \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+e*x)/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

-1/16*((2*(b^2*c*e - 4*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 + (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)