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

Optimal. Leaf size=25 \[ -\frac{f-2 b g x}{2 b \sqrt{a+b x^4}} \]

[Out]

-(f - 2*b*g*x)/(2*b*Sqrt[a + b*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0282155, antiderivative size = 25, 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 = {1856} \[ -\frac{f-2 b g x}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0366762, size = 27, normalized size = 1.08 \[ \frac{2 b g x-f}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-f + 2*b*g*x)/(2*b*Sqrt[a + b*x^4])

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 24, normalized size = 1. \begin{align*}{\frac{2\,bgx-f}{2\,b}{\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)/(b*x^4+a)^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.08228, size = 31, normalized size = 1.24 \begin{align*} \frac{2 \, b g x - f}{2 \, \sqrt{b x^{4} + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*g*x - f)/(sqrt(b*x^4 + a)*b)

________________________________________________________________________________________

Fricas [A]  time = 1.50489, size = 69, normalized size = 2.76 \begin{align*} \frac{\sqrt{b x^{4} + a}{\left (2 \, b g x - f\right )}}{2 \,{\left (b^{2} x^{4} + a 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)/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 10.4613, size = 109, normalized size = 4.36 \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Giac [A]  time = 1.08497, size = 30, normalized size = 1.2 \begin{align*} \frac{2 \, g 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)/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

1/2*(2*g*x - f/b)/sqrt(b*x^4 + a)