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

Optimal. Leaf size=21 \[ \frac{x^2}{2 a \sqrt{a+b x^4}} \]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0036308, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {264} \[ \frac{x^2}{2 a \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^4)^(3/2),x]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

Rule 264

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

Rubi steps

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

Mathematica [A]  time = 0.0031739, size = 21, normalized size = 1. \[ \frac{x^2}{2 a \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^4)^(3/2),x]

[Out]

x^2/(2*a*Sqrt[a + b*x^4])

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Maple [A]  time = 0.003, size = 18, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,a}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.951382, size = 23, normalized size = 1.1 \begin{align*} \frac{x^{2}}{2 \, \sqrt{b x^{4} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/(sqrt(b*x^4 + a)*a)

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Fricas [A]  time = 1.49794, size = 55, normalized size = 2.62 \begin{align*} \frac{\sqrt{b x^{4} + a} x^{2}}{2 \,{\left (a b x^{4} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.570206, size = 20, normalized size = 0.95 \begin{align*} \frac{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(x/(b*x**4+a)**(3/2),x)

[Out]

x**2/(2*a**(3/2)*sqrt(1 + b*x**4/a))

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Giac [A]  time = 1.12439, size = 23, normalized size = 1.1 \begin{align*} \frac{x^{2}}{2 \, \sqrt{b x^{4} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2/(sqrt(b*x^4 + a)*a)