3.514 \(\int \frac{x^{11}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=109 \[ \frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6} \]

[Out]

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

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Rubi [A]  time = 0.102303, antiderivative size = 109, 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} \[ \frac{a^5}{10 b^6 \left (a+b x^2\right )^5}-\frac{5 a^4}{8 b^6 \left (a+b x^2\right )^4}+\frac{5 a^3}{3 b^6 \left (a+b x^2\right )^3}-\frac{5 a^2}{2 b^6 \left (a+b x^2\right )^2}+\frac{5 a}{2 b^6 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^6} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

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

Mathematica [A]  time = 0.0232354, size = 72, normalized size = 0.66 \[ \frac{\frac{a \left (1100 a^2 b^2 x^4+625 a^3 b x^2+137 a^4+900 a b^3 x^6+300 b^4 x^8\right )}{\left (a+b x^2\right )^5}+60 \log \left (a+b x^2\right )}{120 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((a*(137*a^4 + 625*a^3*b*x^2 + 1100*a^2*b^2*x^4 + 900*a*b^3*x^6 + 300*b^4*x^8))/(a + b*x^2)^5 + 60*Log[a + b*x
^2])/(120*b^6)

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Maple [A]  time = 0.056, size = 98, normalized size = 0.9 \begin{align*}{\frac{{a}^{5}}{10\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{5\,{a}^{4}}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{5\,{a}^{3}}{3\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{5\,{a}^{2}}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,a}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{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)^3,x)

[Out]

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

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Maxima [A]  time = 1.20553, size = 163, normalized size = 1.5 \begin{align*} \frac{300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5}}{120 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} b^{6}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, 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)^3,x, algorithm="maxima")

[Out]

1/120*(300*a*b^4*x^8 + 900*a^2*b^3*x^6 + 1100*a^3*b^2*x^4 + 625*a^4*b*x^2 + 137*a^5)/(b^11*x^10 + 5*a*b^10*x^8
+ 10*a^2*b^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6) + 1/2*log(b*x^2 + a)/b^6

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Fricas [A]  time = 1.69063, size = 367, normalized size = 3.37 \begin{align*} \frac{300 \, a b^{4} x^{8} + 900 \, a^{2} b^{3} x^{6} + 1100 \, a^{3} b^{2} x^{4} + 625 \, a^{4} b x^{2} + 137 \, a^{5} + 60 \,{\left (b^{5} x^{10} + 5 \, a b^{4} x^{8} + 10 \, a^{2} b^{3} x^{6} + 10 \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x^{2} + a^{5}\right )} \log \left (b x^{2} + a\right )}{120 \,{\left (b^{11} x^{10} + 5 \, a b^{10} x^{8} + 10 \, a^{2} b^{9} x^{6} + 10 \, a^{3} b^{8} x^{4} + 5 \, a^{4} b^{7} x^{2} + a^{5} 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)^3,x, algorithm="fricas")

[Out]

1/120*(300*a*b^4*x^8 + 900*a^2*b^3*x^6 + 1100*a^3*b^2*x^4 + 625*a^4*b*x^2 + 137*a^5 + 60*(b^5*x^10 + 5*a*b^4*x
^8 + 10*a^2*b^3*x^6 + 10*a^3*b^2*x^4 + 5*a^4*b*x^2 + a^5)*log(b*x^2 + a))/(b^11*x^10 + 5*a*b^10*x^8 + 10*a^2*b
^9*x^6 + 10*a^3*b^8*x^4 + 5*a^4*b^7*x^2 + a^5*b^6)

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Sympy [A]  time = 1.26721, size = 124, normalized size = 1.14 \begin{align*} \frac{137 a^{5} + 625 a^{4} b x^{2} + 1100 a^{3} b^{2} x^{4} + 900 a^{2} b^{3} x^{6} + 300 a b^{4} x^{8}}{120 a^{5} b^{6} + 600 a^{4} b^{7} x^{2} + 1200 a^{3} b^{8} x^{4} + 1200 a^{2} b^{9} x^{6} + 600 a b^{10} x^{8} + 120 b^{11} x^{10}} + \frac{\log{\left (a + b x^{2} \right )}}{2 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)**3,x)

[Out]

(137*a**5 + 625*a**4*b*x**2 + 1100*a**3*b**2*x**4 + 900*a**2*b**3*x**6 + 300*a*b**4*x**8)/(120*a**5*b**6 + 600
*a**4*b**7*x**2 + 1200*a**3*b**8*x**4 + 1200*a**2*b**9*x**6 + 600*a*b**10*x**8 + 120*b**11*x**10) + log(a + b*
x**2)/(2*b**6)

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Giac [A]  time = 1.13556, size = 101, normalized size = 0.93 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac{137 \, b^{4} x^{10} + 385 \, a b^{3} x^{8} + 470 \, a^{2} b^{2} x^{6} + 270 \, a^{3} b x^{4} + 60 \, a^{4} x^{2}}{120 \,{\left (b x^{2} + a\right )}^{5} 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)^3,x, algorithm="giac")

[Out]

1/2*log(abs(b*x^2 + a))/b^6 - 1/120*(137*b^4*x^10 + 385*a*b^3*x^8 + 470*a^2*b^2*x^6 + 270*a^3*b*x^4 + 60*a^4*x
^2)/((b*x^2 + a)^5*b^5)