3.204 \(\int \frac {x}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=16 \[ -\frac {1}{18 b \left (a+b x^2\right )^9} \]

[Out]

-1/18/b/(b*x^2+a)^9

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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {261} \[ -\frac {1}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b*x^2)^10,x]

[Out]

-1/(18*b*(a + b*x^2)^9)

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 )^{10}} \, dx &=-\frac {1}{18 b \left (a+b x^2\right )^9}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.00 \[ -\frac {1}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b*x^2)^10,x]

[Out]

-1/18*1/(b*(a + b*x^2)^9)

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fricas [B]  time = 0.90, size = 103, normalized size = 6.44 \[ -\frac {1}{18 \, {\left (b^{10} x^{18} + 9 \, a b^{9} x^{16} + 36 \, a^{2} b^{8} x^{14} + 84 \, a^{3} b^{7} x^{12} + 126 \, a^{4} b^{6} x^{10} + 126 \, a^{5} b^{5} x^{8} + 84 \, a^{6} b^{4} x^{6} + 36 \, a^{7} b^{3} x^{4} + 9 \, a^{8} b^{2} x^{2} + a^{9} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/18/(b^10*x^18 + 9*a*b^9*x^16 + 36*a^2*b^8*x^14 + 84*a^3*b^7*x^12 + 126*a^4*b^6*x^10 + 126*a^5*b^5*x^8 + 84*
a^6*b^4*x^6 + 36*a^7*b^3*x^4 + 9*a^8*b^2*x^2 + a^9*b)

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giac [A]  time = 0.59, size = 14, normalized size = 0.88 \[ -\frac {1}{18 \, {\left (b x^{2} + a\right )}^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/18/((b*x^2 + a)^9*b)

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maple [A]  time = 0.00, size = 15, normalized size = 0.94 \[ -\frac {1}{18 \left (b \,x^{2}+a \right )^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(b*x^2+a)^10,x)

[Out]

-1/18/b/(b*x^2+a)^9

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maxima [A]  time = 1.30, size = 14, normalized size = 0.88 \[ -\frac {1}{18 \, {\left (b x^{2} + a\right )}^{9} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/18/((b*x^2 + a)^9*b)

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mupad [B]  time = 0.13, size = 14, normalized size = 0.88 \[ -\frac {1}{18\,b\,{\left (b\,x^2+a\right )}^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(a + b*x^2)^10,x)

[Out]

-1/(18*b*(a + b*x^2)^9)

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sympy [B]  time = 0.89, size = 110, normalized size = 6.88 \[ - \frac {1}{18 a^{9} b + 162 a^{8} b^{2} x^{2} + 648 a^{7} b^{3} x^{4} + 1512 a^{6} b^{4} x^{6} + 2268 a^{5} b^{5} x^{8} + 2268 a^{4} b^{6} x^{10} + 1512 a^{3} b^{7} x^{12} + 648 a^{2} b^{8} x^{14} + 162 a b^{9} x^{16} + 18 b^{10} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(b*x**2+a)**10,x)

[Out]

-1/(18*a**9*b + 162*a**8*b**2*x**2 + 648*a**7*b**3*x**4 + 1512*a**6*b**4*x**6 + 2268*a**5*b**5*x**8 + 2268*a**
4*b**6*x**10 + 1512*a**3*b**7*x**12 + 648*a**2*b**8*x**14 + 162*a*b**9*x**16 + 18*b**10*x**18)

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