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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 18, 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} \[ -\frac {1}{7 b \left (a+b x^2\right )^{7/2}} \]

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 = 0.00, size = 18, normalized size = 1.00 \[ -\frac {1}{7 b \left (a+b x^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^(9/2),x]

[Out]

-1/7*1/(b*(a + b*x^2)^(7/2))

________________________________________________________________________________________

fricas [B]  time = 0.87, size = 57, normalized size = 3.17 \[ -\frac {\sqrt {b x^{2} + a}}{7 \, {\left (b^{5} x^{8} + 4 \, a b^{4} x^{6} + 6 \, a^{2} b^{3} x^{4} + 4 \, a^{3} b^{2} x^{2} + a^{4} b\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.97, size = 14, normalized size = 0.78 \[ -\frac {1}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ -\frac {1}{7 \left (b \,x^{2}+a \right )^{\frac {7}{2}} b} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 14, normalized size = 0.78 \[ -\frac {1}{7 \, {\left (b x^{2} + a\right )}^{\frac {7}{2}} b} \]

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)

________________________________________________________________________________________

mupad [B]  time = 4.60, size = 14, normalized size = 0.78 \[ -\frac {1}{7\,b\,{\left (b\,x^2+a\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2)^(9/2),x)

[Out]

-1/(7*b*(a + b*x^2)^(7/2))

________________________________________________________________________________________

sympy [A]  time = 5.73, size = 90, normalized size = 5.00 \[ \begin {cases} - \frac {1}{7 a^{3} b \sqrt {a + b x^{2}} + 21 a^{2} b^{2} x^{2} \sqrt {a + b x^{2}} + 21 a b^{3} x^{4} \sqrt {a + b x^{2}} + 7 b^{4} x^{6} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________