3.515 \(\int \frac{x^9}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^{10}}{10 a \left (a+b x^2\right )^5} \]

[Out]

x^10/(10*a*(a + b*x^2)^5)

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Rubi [A]  time = 0.0067681, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {28, 264} \[ \frac{x^{10}}{10 a \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

x^10/(10*a*(a + b*x^2)^5)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^9}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac{x^{10}}{10 a \left (a+b x^2\right )^5}\\ \end{align*}

Mathematica [B]  time = 0.0158142, size = 57, normalized size = 3. \[ -\frac{10 a^2 b^2 x^4+5 a^3 b x^2+a^4+10 a b^3 x^6+5 b^4 x^8}{10 b^5 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(a^4 + 5*a^3*b*x^2 + 10*a^2*b^2*x^4 + 10*a*b^3*x^6 + 5*b^4*x^8)/(10*b^5*(a + b*x^2)^5)

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Maple [B]  time = 0.049, size = 81, normalized size = 4.3 \begin{align*}{\frac{{a}^{3}}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{a}{{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{{a}^{2}}{{b}^{5} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{{a}^{4}}{10\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/2*a^3/b^5/(b*x^2+a)^4+1/b^5*a/(b*x^2+a)^2-a^2/b^5/(b*x^2+a)^3-1/10*a^4/b^5/(b*x^2+a)^5-1/2/b^5/(b*x^2+a)

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Maxima [B]  time = 1.25286, size = 138, normalized size = 7.26 \begin{align*} -\frac{5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/10*(5*b^4*x^8 + 10*a*b^3*x^6 + 10*a^2*b^2*x^4 + 5*a^3*b*x^2 + a^4)/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^
6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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Fricas [B]  time = 1.67945, size = 212, normalized size = 11.16 \begin{align*} -\frac{5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \,{\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/10*(5*b^4*x^8 + 10*a*b^3*x^6 + 10*a^2*b^2*x^4 + 5*a^3*b*x^2 + a^4)/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^
6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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Sympy [B]  time = 1.18708, size = 107, normalized size = 5.63 \begin{align*} - \frac{a^{4} + 5 a^{3} b x^{2} + 10 a^{2} b^{2} x^{4} + 10 a b^{3} x^{6} + 5 b^{4} x^{8}}{10 a^{5} b^{5} + 50 a^{4} b^{6} x^{2} + 100 a^{3} b^{7} x^{4} + 100 a^{2} b^{8} x^{6} + 50 a b^{9} x^{8} + 10 b^{10} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-(a**4 + 5*a**3*b*x**2 + 10*a**2*b**2*x**4 + 10*a*b**3*x**6 + 5*b**4*x**8)/(10*a**5*b**5 + 50*a**4*b**6*x**2 +
100*a**3*b**7*x**4 + 100*a**2*b**8*x**6 + 50*a*b**9*x**8 + 10*b**10*x**10)

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Giac [B]  time = 1.16209, size = 74, normalized size = 3.89 \begin{align*} -\frac{5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \,{\left (b x^{2} + a\right )}^{5} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-1/10*(5*b^4*x^8 + 10*a*b^3*x^6 + 10*a^2*b^2*x^4 + 5*a^3*b*x^2 + a^4)/((b*x^2 + a)^5*b^5)