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

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {1}{b \sqrt {a+b x^2}} \]

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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fricas [A]  time = 0.70, size = 24, normalized size = 1.50 \[ -\frac {\sqrt {b x^{2} + a}}{b^{2} x^{2} + a b} \]

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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giac [A]  time = 1.05, size = 14, normalized size = 0.88 \[ -\frac {1}{\sqrt {b x^{2} + a} b} \]

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)

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ -\frac {1}{\sqrt {b \,x^{2}+a}\, b} \]

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 = 1.32, size = 14, normalized size = 0.88 \[ -\frac {1}{\sqrt {b x^{2} + a} b} \]

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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mupad [B]  time = 0.04, size = 14, normalized size = 0.88 \[ -\frac {1}{b\,\sqrt {b\,x^2+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2)^(3/2),x)

[Out]

-1/(b*(a + b*x^2)^(1/2))

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sympy [A]  time = 0.54, size = 24, normalized size = 1.50 \[ \begin {cases} - \frac {1}{b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

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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