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

Optimal. Leaf size=14 \[ \frac{g x}{\sqrt{a+b x^4}} \]

[Out]

(g*x)/Sqrt[a + b*x^4]

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Rubi [A]  time = 0.0055367, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {383} \[ \frac{g x}{\sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*x)/Sqrt[a + b*x^4]

Rule 383

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(c*x*(a + b*x^n)^(p + 1))/a, x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rubi steps

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

Mathematica [A]  time = 0.0086365, size = 14, normalized size = 1. \[ \frac{g x}{\sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*x)/Sqrt[a + b*x^4]

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Maple [A]  time = 0.043, size = 13, normalized size = 0.9 \begin{align*}{gx{\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+a*g)/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.05881, size = 16, normalized size = 1.14 \begin{align*} \frac{g x}{\sqrt{b x^{4} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*x/sqrt(b*x^4 + a)

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Fricas [A]  time = 1.3868, size = 28, normalized size = 2. \begin{align*} \frac{g x}{\sqrt{b x^{4} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*x/sqrt(b*x^4 + a)

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Sympy [C]  time = 7.55679, size = 80, normalized size = 5.71 \begin{align*} \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+a*g)/(b*x**4+a)**(3/2),x)

[Out]

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))

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Giac [A]  time = 1.07768, size = 16, normalized size = 1.14 \begin{align*} \frac{g x}{\sqrt{b x^{4} + a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

g*x/sqrt(b*x^4 + a)