3.526 \(\int \frac{x^{10}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=121 \[ -\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac{21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac{21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac{63 x}{256 b^5 \left (a+b x^2\right )}+\frac{63 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 \sqrt{a} b^{11/2}}-\frac{x^9}{10 b \left (a+b x^2\right )^5} \]

[Out]

-x^9/(10*b*(a + b*x^2)^5) - (9*x^7)/(80*b^2*(a + b*x^2)^4) - (21*x^5)/(160*b^3*(a + b*x^2)^3) - (21*x^3)/(128*
b^4*(a + b*x^2)^2) - (63*x)/(256*b^5*(a + b*x^2)) + (63*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*Sqrt[a]*b^(11/2))

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Rubi [A]  time = 0.0688564, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 288, 205} \[ -\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac{21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac{21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac{63 x}{256 b^5 \left (a+b x^2\right )}+\frac{63 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 \sqrt{a} b^{11/2}}-\frac{x^9}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-x^9/(10*b*(a + b*x^2)^5) - (9*x^7)/(80*b^2*(a + b*x^2)^4) - (21*x^5)/(160*b^3*(a + b*x^2)^3) - (21*x^3)/(128*
b^4*(a + b*x^2)^2) - (63*x)/(256*b^5*(a + b*x^2)) + (63*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*Sqrt[a]*b^(11/2))

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 288

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{10}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^{10}}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{x^9}{10 b \left (a+b x^2\right )^5}+\frac{1}{10} \left (9 b^4\right ) \int \frac{x^8}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{x^9}{10 b \left (a+b x^2\right )^5}-\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}+\frac{1}{80} \left (63 b^2\right ) \int \frac{x^6}{\left (a b+b^2 x^2\right )^4} \, dx\\ &=-\frac{x^9}{10 b \left (a+b x^2\right )^5}-\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac{21 x^5}{160 b^3 \left (a+b x^2\right )^3}+\frac{21}{32} \int \frac{x^4}{\left (a b+b^2 x^2\right )^3} \, dx\\ &=-\frac{x^9}{10 b \left (a+b x^2\right )^5}-\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac{21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac{21 x^3}{128 b^4 \left (a+b x^2\right )^2}+\frac{63 \int \frac{x^2}{\left (a b+b^2 x^2\right )^2} \, dx}{128 b^2}\\ &=-\frac{x^9}{10 b \left (a+b x^2\right )^5}-\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac{21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac{21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac{63 x}{256 b^5 \left (a+b x^2\right )}+\frac{63 \int \frac{1}{a b+b^2 x^2} \, dx}{256 b^4}\\ &=-\frac{x^9}{10 b \left (a+b x^2\right )^5}-\frac{9 x^7}{80 b^2 \left (a+b x^2\right )^4}-\frac{21 x^5}{160 b^3 \left (a+b x^2\right )^3}-\frac{21 x^3}{128 b^4 \left (a+b x^2\right )^2}-\frac{63 x}{256 b^5 \left (a+b x^2\right )}+\frac{63 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 \sqrt{a} b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0509354, size = 88, normalized size = 0.73 \[ \frac{63 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 \sqrt{a} b^{11/2}}-\frac{x \left (2688 a^2 b^2 x^4+1470 a^3 b x^2+315 a^4+2370 a b^3 x^6+965 b^4 x^8\right )}{1280 b^5 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-(x*(315*a^4 + 1470*a^3*b*x^2 + 2688*a^2*b^2*x^4 + 2370*a*b^3*x^6 + 965*b^4*x^8))/(1280*b^5*(a + b*x^2)^5) + (
63*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*Sqrt[a]*b^(11/2))

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Maple [A]  time = 0.056, size = 80, normalized size = 0.7 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ( -{\frac{193\,{x}^{9}}{256\,b}}-{\frac{237\,a{x}^{7}}{128\,{b}^{2}}}-{\frac{21\,{a}^{2}{x}^{5}}{10\,{b}^{3}}}-{\frac{147\,{a}^{3}{x}^{3}}{128\,{b}^{4}}}-{\frac{63\,{a}^{4}x}{256\,{b}^{5}}} \right ) }+{\frac{63}{256\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

(-193/256/b*x^9-237/128/b^2*a*x^7-21/10*a^2/b^3*x^5-147/128*a^3/b^4*x^3-63/256*a^4/b^5*x)/(b*x^2+a)^5+63/256/b
^5/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.513, size = 861, normalized size = 7.12 \begin{align*} \left [-\frac{1930 \, a b^{5} x^{9} + 4740 \, a^{2} b^{4} x^{7} + 5376 \, a^{3} b^{3} x^{5} + 2940 \, a^{4} b^{2} x^{3} + 630 \, a^{5} b x + 315 \,{\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 )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{2560 \,{\left (a b^{11} x^{10} + 5 \, a^{2} b^{10} x^{8} + 10 \, a^{3} b^{9} x^{6} + 10 \, a^{4} b^{8} x^{4} + 5 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}, -\frac{965 \, a b^{5} x^{9} + 2370 \, a^{2} b^{4} x^{7} + 2688 \, a^{3} b^{3} x^{5} + 1470 \, a^{4} b^{2} x^{3} + 315 \, a^{5} b x - 315 \,{\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 )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{1280 \,{\left (a b^{11} x^{10} + 5 \, a^{2} b^{10} x^{8} + 10 \, a^{3} b^{9} x^{6} + 10 \, a^{4} b^{8} x^{4} + 5 \, a^{5} b^{7} x^{2} + a^{6} b^{6}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

[-1/2560*(1930*a*b^5*x^9 + 4740*a^2*b^4*x^7 + 5376*a^3*b^3*x^5 + 2940*a^4*b^2*x^3 + 630*a^5*b*x + 315*(b^5*x^1
0 + 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)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x
- a)/(b*x^2 + a)))/(a*b^11*x^10 + 5*a^2*b^10*x^8 + 10*a^3*b^9*x^6 + 10*a^4*b^8*x^4 + 5*a^5*b^7*x^2 + a^6*b^6),
-1/1280*(965*a*b^5*x^9 + 2370*a^2*b^4*x^7 + 2688*a^3*b^3*x^5 + 1470*a^4*b^2*x^3 + 315*a^5*b*x - 315*(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)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a*b^11
*x^10 + 5*a^2*b^10*x^8 + 10*a^3*b^9*x^6 + 10*a^4*b^8*x^4 + 5*a^5*b^7*x^2 + a^6*b^6)]

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Sympy [A]  time = 1.27807, size = 180, normalized size = 1.49 \begin{align*} - \frac{63 \sqrt{- \frac{1}{a b^{11}}} \log{\left (- a b^{5} \sqrt{- \frac{1}{a b^{11}}} + x \right )}}{512} + \frac{63 \sqrt{- \frac{1}{a b^{11}}} \log{\left (a b^{5} \sqrt{- \frac{1}{a b^{11}}} + x \right )}}{512} - \frac{315 a^{4} x + 1470 a^{3} b x^{3} + 2688 a^{2} b^{2} x^{5} + 2370 a b^{3} x^{7} + 965 b^{4} x^{9}}{1280 a^{5} b^{5} + 6400 a^{4} b^{6} x^{2} + 12800 a^{3} b^{7} x^{4} + 12800 a^{2} b^{8} x^{6} + 6400 a b^{9} x^{8} + 1280 b^{10} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-63*sqrt(-1/(a*b**11))*log(-a*b**5*sqrt(-1/(a*b**11)) + x)/512 + 63*sqrt(-1/(a*b**11))*log(a*b**5*sqrt(-1/(a*b
**11)) + x)/512 - (315*a**4*x + 1470*a**3*b*x**3 + 2688*a**2*b**2*x**5 + 2370*a*b**3*x**7 + 965*b**4*x**9)/(12
80*a**5*b**5 + 6400*a**4*b**6*x**2 + 12800*a**3*b**7*x**4 + 12800*a**2*b**8*x**6 + 6400*a*b**9*x**8 + 1280*b**
10*x**10)

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Giac [A]  time = 1.1735, size = 105, normalized size = 0.87 \begin{align*} \frac{63 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} b^{5}} - \frac{965 \, b^{4} x^{9} + 2370 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 1470 \, a^{3} b x^{3} + 315 \, a^{4} x}{1280 \,{\left (b x^{2} + a\right )}^{5} b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

63/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/1280*(965*b^4*x^9 + 2370*a*b^3*x^7 + 2688*a^2*b^2*x^5 + 1470*
a^3*b*x^3 + 315*a^4*x)/((b*x^2 + a)^5*b^5)