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

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

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Rubi [A]  time = 0.07, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {28, 199, 205} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [A]  time = 0.05, size = 89, normalized size = 0.79 \begin {gather*} \frac {\frac {\sqrt {a} x \left (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\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}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 386, normalized size = 3.42 \begin {gather*} \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 {gather*}

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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giac [A]  time = 0.17, size = 78, normalized size = 0.69 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.01, size = 96, normalized size = 0.85 \begin {gather*} \frac {x}{10 \left (b \,x^{2}+a \right )^{5} a}+\frac {9 x}{80 \left (b \,x^{2}+a \right )^{4} a^{2}}+\frac {21 x}{160 \left (b \,x^{2}+a \right )^{3} a^{3}}+\frac {21 x}{128 \left (b \,x^{2}+a \right )^{2} a^{4}}+\frac {63 x}{256 \left (b \,x^{2}+a \right ) a^{5}}+\frac {63 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}\, a^{5}} \end {gather*}

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(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.04, size = 124, normalized size = 1.10 \begin {gather*} \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 (a^{5} b^{5} x^{10} + 5 \, a^{6} b^{4} x^{8} + 10 \, a^{7} b^{3} x^{6} + 10 \, a^{8} b^{2} x^{4} + 5 \, a^{9} b x^{2} + a^{10}\right )}} + \frac {63 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{5}} \end {gather*}

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]

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)/(a^5*b^5*x^10 + 5*a^6*b^
4*x^8 + 10*a^7*b^3*x^6 + 10*a^8*b^2*x^4 + 5*a^9*b*x^2 + a^10) + 63/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5)

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mupad [B]  time = 4.71, size = 121, normalized size = 1.07 \begin {gather*} \frac {\frac {193\,x}{256\,a}+\frac {237\,b\,x^3}{128\,a^2}+\frac {21\,b^2\,x^5}{10\,a^3}+\frac {147\,b^3\,x^7}{128\,a^4}+\frac {63\,b^4\,x^9}{256\,a^5}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}}+\frac {63\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,a^{11/2}\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((193*x)/(256*a) + (237*b*x^3)/(128*a^2) + (21*b^2*x^5)/(10*a^3) + (147*b^3*x^7)/(128*a^4) + (63*b^4*x^9)/(256
*a^5))/(a^5 + b^5*x^10 + 5*a^4*b*x^2 + 5*a*b^4*x^8 + 10*a^3*b^2*x^4 + 10*a^2*b^3*x^6) + (63*atan((b^(1/2)*x)/a
^(1/2)))/(256*a^(11/2)*b^(1/2))

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sympy [A]  time = 0.68, size = 177, normalized size = 1.57 \begin {gather*} - \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 {gather*}

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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