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

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 = 0.0906941, 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, Rules used = {1673, 1588, 1247, 636} \[ \frac{g x}{\sqrt{a+b x^2+c x^4}}-\frac{-2 a f+x^2 (2 c e-b f)+b e}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]

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])

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 1247

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

Rule 636

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

Rubi steps

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

[Out]

$Aborted

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Maple [A]  time = 0.004, size = 63, normalized size = 0.9 \begin{align*}{\frac{4\,acgx-{b}^{2}gx-bf{x}^{2}+2\,ce{x}^{2}-2\,af+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+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 = 1.15474, size = 127, normalized size = 1.84 \begin{align*} -\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}} \end{align*}

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 = 1.30622, size = 193, normalized size = 2.8 \begin{align*} \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}} \end{align*}

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

[Out]

Timed out

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Giac [B]  time = 1.18484, size = 262, normalized size = 3.8 \begin{align*} \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}} \end{align*}

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)