3.511 \(\int \frac{x}{(a+b x^2)^{5/2}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{3 b \left (a+b x^2\right )^{3/2}} \]

[Out]

-1/(3*b*(a + b*x^2)^(3/2))

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Rubi [A]  time = 0.0034343, antiderivative size = 18, 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} \[ -\frac{1}{3 b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^(5/2),x]

[Out]

-1/(3*b*(a + b*x^2)^(3/2))

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 )^{5/2}} \, dx &=-\frac{1}{3 b \left (a+b x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0032977, size = 18, normalized size = 1. \[ -\frac{1}{3 b \left (a+b x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^(5/2),x]

[Out]

-1/(3*b*(a + b*x^2)^(3/2))

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Maple [A]  time = 0.002, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2+a)^(5/2),x)

[Out]

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

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Maxima [A]  time = 3.31017, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(5/2),x, algorithm="maxima")

[Out]

-1/3/((b*x^2 + a)^(3/2)*b)

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Fricas [B]  time = 1.34873, size = 73, normalized size = 4.06 \begin{align*} -\frac{\sqrt{b x^{2} + a}}{3 \,{\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.969165, size = 46, normalized size = 2.56 \begin{align*} \begin{cases} - \frac{1}{3 a b \sqrt{a + b x^{2}} + 3 b^{2} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**(5/2),x)

[Out]

Piecewise((-1/(3*a*b*sqrt(a + b*x**2) + 3*b**2*x**2*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(5/2)), True))

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Giac [A]  time = 2.80529, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{3 \,{\left (b x^{2} + a\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(5/2),x, algorithm="giac")

[Out]

-1/3/((b*x^2 + a)^(3/2)*b)