3.493 \(\int \frac{x^{11}}{(a^2+2 a b x^2+b^2 x^4)^2} \, dx\)

Optimal. Leaf size=91 \[ \frac{a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac{5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac{5 a^3}{b^6 \left (a+b x^2\right )}+\frac{5 a^2 \log \left (a+b x^2\right )}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4} \]

[Out]

(-2*a*x^2)/b^5 + x^4/(4*b^4) + a^5/(6*b^6*(a + b*x^2)^3) - (5*a^4)/(4*b^6*(a + b*x^2)^2) + (5*a^3)/(b^6*(a + b
*x^2)) + (5*a^2*Log[a + b*x^2])/b^6

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Rubi [A]  time = 0.0926364, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ \frac{a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac{5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac{5 a^3}{b^6 \left (a+b x^2\right )}+\frac{5 a^2 \log \left (a+b x^2\right )}{b^6}-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(-2*a*x^2)/b^5 + x^4/(4*b^4) + a^5/(6*b^6*(a + b*x^2)^3) - (5*a^4)/(4*b^6*(a + b*x^2)^2) + (5*a^3)/(b^6*(a + b
*x^2)) + (5*a^2*Log[a + b*x^2])/b^6

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^{11}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{x^{11}}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \frac{x^5}{\left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \left (-\frac{4 a}{b^9}+\frac{x}{b^8}-\frac{a^5}{b^9 (a+b x)^4}+\frac{5 a^4}{b^9 (a+b x)^3}-\frac{10 a^3}{b^9 (a+b x)^2}+\frac{10 a^2}{b^9 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{2 a x^2}{b^5}+\frac{x^4}{4 b^4}+\frac{a^5}{6 b^6 \left (a+b x^2\right )^3}-\frac{5 a^4}{4 b^6 \left (a+b x^2\right )^2}+\frac{5 a^3}{b^6 \left (a+b x^2\right )}+\frac{5 a^2 \log \left (a+b x^2\right )}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0305118, size = 78, normalized size = 0.86 \[ \frac{\frac{2 a^5}{\left (a+b x^2\right )^3}-\frac{15 a^4}{\left (a+b x^2\right )^2}+\frac{60 a^3}{a+b x^2}+60 a^2 \log \left (a+b x^2\right )-24 a b x^2+3 b^2 x^4}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(-24*a*b*x^2 + 3*b^2*x^4 + (2*a^5)/(a + b*x^2)^3 - (15*a^4)/(a + b*x^2)^2 + (60*a^3)/(a + b*x^2) + 60*a^2*Log[
a + b*x^2])/(12*b^6)

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Maple [A]  time = 0.052, size = 86, normalized size = 1. \begin{align*} -2\,{\frac{a{x}^{2}}{{b}^{5}}}+{\frac{{x}^{4}}{4\,{b}^{4}}}+{\frac{{a}^{5}}{6\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{5\,{a}^{4}}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+5\,{\frac{{a}^{3}}{{b}^{6} \left ( b{x}^{2}+a \right ) }}+5\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.00571, size = 134, normalized size = 1.47 \begin{align*} \frac{60 \, a^{3} b^{2} x^{4} + 105 \, a^{4} b x^{2} + 47 \, a^{5}}{12 \,{\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} + \frac{5 \, a^{2} \log \left (b x^{2} + a\right )}{b^{6}} + \frac{b x^{4} - 8 \, a x^{2}}{4 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")

[Out]

1/12*(60*a^3*b^2*x^4 + 105*a^4*b*x^2 + 47*a^5)/(b^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6) + 5*a^2*log(b
*x^2 + a)/b^6 + 1/4*(b*x^4 - 8*a*x^2)/b^5

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Fricas [A]  time = 1.63722, size = 285, normalized size = 3.13 \begin{align*} \frac{3 \, b^{5} x^{10} - 15 \, a b^{4} x^{8} - 63 \, a^{2} b^{3} x^{6} - 9 \, a^{3} b^{2} x^{4} + 81 \, a^{4} b x^{2} + 47 \, a^{5} + 60 \,{\left (a^{2} b^{3} x^{6} + 3 \, a^{3} b^{2} x^{4} + 3 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{9} x^{6} + 3 \, a b^{8} x^{4} + 3 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^10 - 15*a*b^4*x^8 - 63*a^2*b^3*x^6 - 9*a^3*b^2*x^4 + 81*a^4*b*x^2 + 47*a^5 + 60*(a^2*b^3*x^6 + 3
*a^3*b^2*x^4 + 3*a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^9*x^6 + 3*a*b^8*x^4 + 3*a^2*b^7*x^2 + a^3*b^6)

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Sympy [A]  time = 0.784291, size = 100, normalized size = 1.1 \begin{align*} \frac{5 a^{2} \log{\left (a + b x^{2} \right )}}{b^{6}} - \frac{2 a x^{2}}{b^{5}} + \frac{47 a^{5} + 105 a^{4} b x^{2} + 60 a^{3} b^{2} x^{4}}{12 a^{3} b^{6} + 36 a^{2} b^{7} x^{2} + 36 a b^{8} x^{4} + 12 b^{9} x^{6}} + \frac{x^{4}}{4 b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

5*a**2*log(a + b*x**2)/b**6 - 2*a*x**2/b**5 + (47*a**5 + 105*a**4*b*x**2 + 60*a**3*b**2*x**4)/(12*a**3*b**6 +
36*a**2*b**7*x**2 + 36*a*b**8*x**4 + 12*b**9*x**6) + x**4/(4*b**4)

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Giac [A]  time = 1.13509, size = 123, normalized size = 1.35 \begin{align*} \frac{5 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac{b^{4} x^{4} - 8 \, a b^{3} x^{2}}{4 \, b^{8}} - \frac{110 \, a^{2} b^{3} x^{6} + 270 \, a^{3} b^{2} x^{4} + 225 \, a^{4} b x^{2} + 63 \, a^{5}}{12 \,{\left (b x^{2} + a\right )}^{3} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")

[Out]

5*a^2*log(abs(b*x^2 + a))/b^6 + 1/4*(b^4*x^4 - 8*a*b^3*x^2)/b^8 - 1/12*(110*a^2*b^3*x^6 + 270*a^3*b^2*x^4 + 22
5*a^4*b*x^2 + 63*a^5)/((b*x^2 + a)^3*b^6)