3.496 \(\int \frac{x^5}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^6}{6 a \left (a+b x^2\right )^3} \]

[Out]

x^6/(6*a*(a + b*x^2)^3)

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Rubi [A]  time = 0.006774, 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} \[ \frac{x^6}{6 a \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

x^6/(6*a*(a + b*x^2)^3)

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

Mathematica [A]  time = 0.0132677, size = 35, normalized size = 1.84 \[ -\frac{a^2+3 a b x^2+3 b^2 x^4}{6 b^3 \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

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

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Maple [B]  time = 0.049, size = 48, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{6\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{a}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{1}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

-1/6*a^2/b^3/(b*x^2+a)^3+1/2*a/b^3/(b*x^2+a)^2-1/2/b^3/(b*x^2+a)

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Maxima [B]  time = 0.979301, size = 78, normalized size = 4.11 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.58219, size = 116, normalized size = 6.11 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \,{\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.58234, size = 60, normalized size = 3.16 \begin{align*} - \frac{a^{2} + 3 a b x^{2} + 3 b^{2} x^{4}}{6 a^{3} b^{3} + 18 a^{2} b^{4} x^{2} + 18 a b^{5} x^{4} + 6 b^{6} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

-(a**2 + 3*a*b*x**2 + 3*b**2*x**4)/(6*a**3*b**3 + 18*a**2*b**4*x**2 + 18*a*b**5*x**4 + 6*b**6*x**6)

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Giac [A]  time = 1.13551, size = 45, normalized size = 2.37 \begin{align*} -\frac{3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \,{\left (b x^{2} + a\right )}^{3} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")

[Out]

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