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

Optimal. Leaf size=16 \[ -\frac{1}{b \sqrt{a+b x^2}} \]

[Out]

-(1/(b*Sqrt[a + b*x^2]))

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Rubi [A]  time = 0.0033578, antiderivative size = 16, 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 = {261} \[ -\frac{1}{b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^(3/2),x]

[Out]

-(1/(b*Sqrt[a + b*x^2]))

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0025923, size = 16, normalized size = 1. \[ -\frac{1}{b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^(3/2),x]

[Out]

-(1/(b*Sqrt[a + b*x^2]))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 2.86063, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{\sqrt{b x^{2} + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(b*x^2 + a)*b)

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Fricas [A]  time = 1.30934, size = 46, normalized size = 2.88 \begin{align*} -\frac{\sqrt{b x^{2} + a}}{b^{2} x^{2} + a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b*x^2 + a)/(b^2*x^2 + a*b)

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Sympy [A]  time = 0.540283, size = 24, normalized size = 1.5 \begin{align*} \begin{cases} - \frac{1}{b \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-1/(b*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(3/2)), True))

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Giac [A]  time = 1.99789, size = 19, normalized size = 1.19 \begin{align*} -\frac{1}{\sqrt{b x^{2} + a} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(sqrt(b*x^2 + a)*b)