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

Optimal. Leaf size=29 \[ \frac{2 a g x+e x^2}{2 a \sqrt{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0230818, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {1856} \[ \frac{2 a g x+e x^2}{2 a \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0368414, size = 27, normalized size = 0.93 \[ \frac{x (2 a g+e x)}{2 a \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.042, size = 24, normalized size = 0.8 \begin{align*}{\frac{x \left ( 2\,ag+ex \right ) }{2\,a}{\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+e*x)/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.07298, size = 34, normalized size = 1.17 \begin{align*} \frac{2 \, a g x + e x^{2}}{2 \, \sqrt{b x^{4} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.34993, size = 74, normalized size = 2.55 \begin{align*} \frac{\sqrt{b x^{4} + a}{\left (2 \, a g x + e x^{2}\right )}}{2 \,{\left (a b x^{4} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 11.0955, size = 104, normalized size = 3.59 \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 )} + \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+a*g+e*x)/(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)) + e*x**2/(2*a**(3/2)*sqrt(1 + b*
x**4/a))

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Giac [A]  time = 1.0737, size = 43, normalized size = 1.48 \begin{align*} -\frac{x{\left (\frac{2 \, g}{a^{2} b^{4}} + \frac{x e}{a^{3} b^{4}}\right )}}{64 \, \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+e*x)/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

-1/64*x*(2*g/(a^2*b^4) + x*e/(a^3*b^4))/sqrt(b*x^4 + a)