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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^6}{6 a \left (a+b x^2\right )^3} \end {gather*}

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 = 0.01, size = 35, normalized size = 1.84 \begin {gather*} -\frac {a^2+3 a b x^2+3 b^2 x^4}{6 b^3 \left (a+b x^2\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.49, size = 58, normalized size = 3.05 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 33, normalized size = 1.74 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, {\left (b x^{2} + a\right )}^{3} b^{3}} \end {gather*}

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)

________________________________________________________________________________________

maple [B]  time = 0.01, size = 48, normalized size = 2.53 \begin {gather*} -\frac {a^{2}}{6 \left (b \,x^{2}+a \right )^{3} b^{3}}+\frac {a}{2 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {1}{2 \left (b \,x^{2}+a \right ) b^{3}} \end {gather*}

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)

________________________________________________________________________________________

maxima [B]  time = 1.38, size = 58, normalized size = 3.05 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 4.29, size = 60, normalized size = 3.16 \begin {gather*} -\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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 0.40, size = 60, normalized size = 3.16 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________