3.531 \(\int \frac{1}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=113 \[ \frac{63 x}{256 a^5 \left (a+b x^2\right )}+\frac{21 x}{128 a^4 \left (a+b x^2\right )^2}+\frac{21 x}{160 a^3 \left (a+b x^2\right )^3}+\frac{9 x}{80 a^2 \left (a+b x^2\right )^4}+\frac{63 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{11/2} \sqrt{b}}+\frac{x}{10 a \left (a+b x^2\right )^5} \]

[Out]

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

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Rubi [A]  time = 0.0663419, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {28, 199, 205} \[ \frac{63 x}{256 a^5 \left (a+b x^2\right )}+\frac{21 x}{128 a^4 \left (a+b x^2\right )^2}+\frac{21 x}{160 a^3 \left (a+b x^2\right )^3}+\frac{9 x}{80 a^2 \left (a+b x^2\right )^4}+\frac{63 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{11/2} \sqrt{b}}+\frac{x}{10 a \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

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

Mathematica [A]  time = 0.0442066, size = 89, normalized size = 0.79 \[ \frac{\frac{\sqrt{a} x \left (2688 a^2 b^2 x^4+2370 a^3 b x^2+965 a^4+1470 a b^3 x^6+315 b^4 x^8\right )}{\left (a+b x^2\right )^5}+\frac{315 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}}{1280 a^{11/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 96, normalized size = 0.9 \begin{align*}{\frac{x}{10\,a \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{9\,x}{80\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{21\,x}{160\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{21\,x}{128\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{63\,x}{256\,{a}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{63}{256\,{a}^{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(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

1/10*x/a/(b*x^2+a)^5+9/80*x/a^2/(b*x^2+a)^4+21/160*x/a^3/(b*x^2+a)^3+21/128*x/a^4/(b*x^2+a)^2+63/256*x/a^5/(b*
x^2+a)+63/256/a^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(1/(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 = 1.51128, size = 859, normalized size = 7.6 \begin{align*} \left [\frac{630 \, a b^{5} x^{9} + 2940 \, a^{2} b^{4} x^{7} + 5376 \, a^{3} b^{3} x^{5} + 4740 \, a^{4} b^{2} x^{3} + 1930 \, 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^{6} b^{6} x^{10} + 5 \, a^{7} b^{5} x^{8} + 10 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{4} + 5 \, a^{10} b^{2} x^{2} + a^{11} b\right )}}, \frac{315 \, a b^{5} x^{9} + 1470 \, a^{2} b^{4} x^{7} + 2688 \, a^{3} b^{3} x^{5} + 2370 \, a^{4} b^{2} x^{3} + 965 \, 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^{6} b^{6} x^{10} + 5 \, a^{7} b^{5} x^{8} + 10 \, a^{8} b^{4} x^{6} + 10 \, a^{9} b^{3} x^{4} + 5 \, a^{10} b^{2} x^{2} + a^{11} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.12022, size = 177, normalized size = 1.57 \begin{align*} - \frac{63 \sqrt{- \frac{1}{a^{11} b}} \log{\left (- a^{6} \sqrt{- \frac{1}{a^{11} b}} + x \right )}}{512} + \frac{63 \sqrt{- \frac{1}{a^{11} b}} \log{\left (a^{6} \sqrt{- \frac{1}{a^{11} b}} + x \right )}}{512} + \frac{965 a^{4} x + 2370 a^{3} b x^{3} + 2688 a^{2} b^{2} x^{5} + 1470 a b^{3} x^{7} + 315 b^{4} x^{9}}{1280 a^{10} + 6400 a^{9} b x^{2} + 12800 a^{8} b^{2} x^{4} + 12800 a^{7} b^{3} x^{6} + 6400 a^{6} b^{4} x^{8} + 1280 a^{5} b^{5} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

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

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Giac [A]  time = 1.14911, size = 105, normalized size = 0.93 \begin{align*} \frac{63 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{5}} + \frac{315 \, b^{4} x^{9} + 1470 \, a b^{3} x^{7} + 2688 \, a^{2} b^{2} x^{5} + 2370 \, a^{3} b x^{3} + 965 \, a^{4} x}{1280 \,{\left (b x^{2} + a\right )}^{5} a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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)*a^5) + 1/1280*(315*b^4*x^9 + 1470*a*b^3*x^7 + 2688*a^2*b^2*x^5 + 2370*
a^3*b*x^3 + 965*a^4*x)/((b*x^2 + a)^5*a^5)