3.501 \(\int \frac{1}{(a+b x^2)^{3/2}} \, dx\)

Optimal. Leaf size=16 \[ \frac{x}{a \sqrt{a+b x^2}} \]

[Out]

x/(a*Sqrt[a + b*x^2])

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Rubi [A]  time = 0.0018622, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {191} \[ \frac{x}{a \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

Rule 191

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{x}{a \sqrt{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0033032, size = 16, normalized size = 1. \[ \frac{x}{a \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(-3/2),x]

[Out]

x/(a*Sqrt[a + b*x^2])

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Maple [A]  time = 0.002, size = 15, normalized size = 0.9 \begin{align*}{\frac{x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 3.00858, size = 19, normalized size = 1.19 \begin{align*} \frac{x}{\sqrt{b x^{2} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(3/2),x, algorithm="maxima")

[Out]

x/(sqrt(b*x^2 + a)*a)

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Fricas [A]  time = 1.17569, size = 47, normalized size = 2.94 \begin{align*} \frac{\sqrt{b x^{2} + a} x}{a b x^{2} + a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(3/2),x, algorithm="fricas")

[Out]

sqrt(b*x^2 + a)*x/(a*b*x^2 + a^2)

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Sympy [A]  time = 0.55124, size = 17, normalized size = 1.06 \begin{align*} \frac{x}{a^{\frac{3}{2}} \sqrt{1 + \frac{b x^{2}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+a)**(3/2),x)

[Out]

x/(a**(3/2)*sqrt(1 + b*x**2/a))

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Giac [A]  time = 3.08707, size = 19, normalized size = 1.19 \begin{align*} \frac{x}{\sqrt{b x^{2} + a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+a)^(3/2),x, algorithm="giac")

[Out]

x/(sqrt(b*x^2 + a)*a)