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

Optimal. Leaf size=38 \[ -\frac{-2 a b g x+a f-b e x^2}{2 a b \sqrt{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0298362, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {1856} \[ -\frac{-2 a b g x+a f-b e x^2}{2 a b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1856

Int[(P4_)/((a_) + (b_.)*(x_)^4)^(3/2), x_Symbol] :> With[{d = Coeff[P4, x, 0], e = Coeff[P4, x, 1], f = Coeff[
P4, x, 3], g = Coeff[P4, x, 4]}, -Simp[(a*f + 2*a*g*x - b*e*x^2)/(2*a*b*Sqrt[a + b*x^4]), x] /; EqQ[b*d + a*g,
0]] /; FreeQ[{a, b}, x] && PolyQ[P4, x, 4] && EqQ[Coeff[P4, x, 2], 0]

Rubi steps

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

Mathematica [A]  time = 0.0415902, size = 38, normalized size = 1. \[ \frac{2 a b g x-a f+b e x^2}{2 a b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 35, normalized size = 0.9 \begin{align*}{\frac{2\,abgx+be{x}^{2}-af}{2\,ab}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07834, size = 59, normalized size = 1.55 \begin{align*} \frac{\sqrt{b x^{4} + a}{\left (2 \, a b g x + b e x^{2} - a f\right )}}{2 \,{\left (a b^{2} x^{4} + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.30205, size = 93, normalized size = 2.45 \begin{align*} \frac{\sqrt{b x^{4} + a}{\left (2 \, a b g x + b e x^{2} - a f\right )}}{2 \,{\left (a b^{2} x^{4} + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 13.7758, size = 133, normalized size = 3.5 \begin{align*} f \left (\begin{cases} - \frac{1}{2 b \sqrt{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases}\right ) + \frac{g x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{5}{4}\right )} - \frac{b g x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} + \frac{e x^{2}}{2 a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{4}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

f*Piecewise((-1/(2*b*sqrt(a + b*x**4)), Ne(b, 0)), (x**4/(4*a**(3/2)), True)) + g*x*gamma(1/4)*hyper((1/4, 3/2
), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(5/4)) - b*g*x**5*gamma(5/4)*hyper((5/4, 3/2), (9/4,), b*
x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(9/4)) + e*x**2/(2*a**(3/2)*sqrt(1 + b*x**4/a))

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Giac [A]  time = 1.09686, size = 42, normalized size = 1.11 \begin{align*} \frac{{\left (2 \, g + \frac{x e}{a}\right )} x - \frac{f}{b}}{2 \, \sqrt{b x^{4} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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