3.526 \(\int \frac{x}{(a+b x^2)^{9/2}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{7 b \left (a+b x^2\right )^{7/2}} \]

[Out]

-1/(7*b*(a + b*x^2)^(7/2))

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Rubi [A]  time = 0.0037261, 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} \[ -\frac{1}{7 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^(9/2),x]

[Out]

-1/(7*b*(a + b*x^2)^(7/2))

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 )^{9/2}} \, dx &=-\frac{1}{7 b \left (a+b x^2\right )^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0044308, size = 18, normalized size = 1. \[ -\frac{1}{7 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^(9/2),x]

[Out]

-1/(7*b*(a + b*x^2)^(7/2))

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*} -{\frac{1}{7\,b} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2+a)^(9/2),x)

[Out]

-1/7/b/(b*x^2+a)^(7/2)

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Maxima [A]  time = 2.29914, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

-1/7/((b*x^2 + a)^(7/2)*b)

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Fricas [B]  time = 1.37224, size = 116, normalized size = 6.44 \begin{align*} -\frac{\sqrt{b x^{2} + a}}{7 \,{\left (b^{5} x^{8} + 4 \, a b^{4} x^{6} + 6 \, a^{2} b^{3} x^{4} + 4 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

-1/7*sqrt(b*x^2 + a)/(b^5*x^8 + 4*a*b^4*x^6 + 6*a^2*b^3*x^4 + 4*a^3*b^2*x^2 + a^4*b)

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Sympy [A]  time = 5.12714, size = 90, normalized size = 5. \begin{align*} \begin{cases} - \frac{1}{7 a^{3} b \sqrt{a + b x^{2}} + 21 a^{2} b^{2} x^{2} \sqrt{a + b x^{2}} + 21 a b^{3} x^{4} \sqrt{a + b x^{2}} + 7 b^{4} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**(9/2),x)

[Out]

Piecewise((-1/(7*a**3*b*sqrt(a + b*x**2) + 21*a**2*b**2*x**2*sqrt(a + b*x**2) + 21*a*b**3*x**4*sqrt(a + b*x**2
) + 7*b**4*x**6*sqrt(a + b*x**2)), Ne(b, 0)), (x**2/(2*a**(9/2)), True))

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Giac [A]  time = 1.51784, size = 19, normalized size = 1.06 \begin{align*} -\frac{1}{7 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

-1/7/((b*x^2 + a)^(7/2)*b)