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

Optimal. Leaf size=125 \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}}+\frac{7 x}{256 a^4 b \left (a+b x^2\right )}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{x}{80 a b \left (a+b x^2\right )^4}-\frac{x}{10 b \left (a+b x^2\right )^5} \]

[Out]

-x/(10*b*(a + b*x^2)^5) + x/(80*a*b*(a + b*x^2)^4) + (7*x)/(480*a^2*b*(a + b*x^2)^3) + (7*x)/(384*a^3*b*(a + b
*x^2)^2) + (7*x)/(256*a^4*b*(a + b*x^2)) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))

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Rubi [A]  time = 0.0746596, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 288, 199, 205} \[ \frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}}+\frac{7 x}{256 a^4 b \left (a+b x^2\right )}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{x}{80 a b \left (a+b x^2\right )^4}-\frac{x}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x/(10*b*(a + b*x^2)^5) + x/(80*a*b*(a + b*x^2)^4) + (7*x)/(480*a^2*b*(a + b*x^2)^3) + (7*x)/(384*a^3*b*(a + b
*x^2)^2) + (7*x)/(256*a^4*b*(a + b*x^2)) + (7*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))

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 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{x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac{x^2}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=-\frac{x}{10 b \left (a+b x^2\right )^5}+\frac{1}{10} b^4 \int \frac{1}{\left (a b+b^2 x^2\right )^5} \, dx\\ &=-\frac{x}{10 b \left (a+b x^2\right )^5}+\frac{x}{80 a b \left (a+b x^2\right )^4}+\frac{\left (7 b^3\right ) \int \frac{1}{\left (a b+b^2 x^2\right )^4} \, dx}{80 a}\\ &=-\frac{x}{10 b \left (a+b x^2\right )^5}+\frac{x}{80 a b \left (a+b x^2\right )^4}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{\left (7 b^2\right ) \int \frac{1}{\left (a b+b^2 x^2\right )^3} \, dx}{96 a^2}\\ &=-\frac{x}{10 b \left (a+b x^2\right )^5}+\frac{x}{80 a b \left (a+b x^2\right )^4}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{(7 b) \int \frac{1}{\left (a b+b^2 x^2\right )^2} \, dx}{128 a^3}\\ &=-\frac{x}{10 b \left (a+b x^2\right )^5}+\frac{x}{80 a b \left (a+b x^2\right )^4}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{7 x}{256 a^4 b \left (a+b x^2\right )}+\frac{7 \int \frac{1}{a b+b^2 x^2} \, dx}{256 a^4}\\ &=-\frac{x}{10 b \left (a+b x^2\right )^5}+\frac{x}{80 a b \left (a+b x^2\right )^4}+\frac{7 x}{480 a^2 b \left (a+b x^2\right )^3}+\frac{7 x}{384 a^3 b \left (a+b x^2\right )^2}+\frac{7 x}{256 a^4 b \left (a+b x^2\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0513873, size = 91, normalized size = 0.73 \[ \frac{896 a^2 b^2 x^5+790 a^3 b x^3-105 a^4 x+490 a b^3 x^7+105 b^4 x^9}{3840 a^4 b \left (a+b x^2\right )^5}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{9/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-105*a^4*x + 790*a^3*b*x^3 + 896*a^2*b^2*x^5 + 490*a*b^3*x^7 + 105*b^4*x^9)/(3840*a^4*b*(a + b*x^2)^5) + (7*A
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(9/2)*b^(3/2))

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Maple [A]  time = 0.051, size = 80, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{7\,{b}^{3}{x}^{9}}{256\,{a}^{4}}}+{\frac{49\,{b}^{2}{x}^{7}}{384\,{a}^{3}}}+{\frac{7\,b{x}^{5}}{30\,{a}^{2}}}+{\frac{79\,{x}^{3}}{384\,a}}-{\frac{7\,x}{256\,b}} \right ) }+{\frac{7}{256\,{a}^{4}b}\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^2/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

(7/256*b^3/a^4*x^9+49/384*b^2/a^3*x^7+7/30*b/a^2*x^5+79/384/a*x^3-7/256*x/b)/(b*x^2+a)^5+7/256/a^4/b/(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^2/(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 = 1.47419, size = 855, normalized size = 6.84 \begin{align*} \left [\frac{210 \, a b^{5} x^{9} + 980 \, a^{2} b^{4} x^{7} + 1792 \, a^{3} b^{3} x^{5} + 1580 \, a^{4} b^{2} x^{3} - 210 \, a^{5} b x - 105 \,{\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 )}{7680 \,{\left (a^{5} b^{7} x^{10} + 5 \, a^{6} b^{6} x^{8} + 10 \, a^{7} b^{5} x^{6} + 10 \, a^{8} b^{4} x^{4} + 5 \, a^{9} b^{3} x^{2} + a^{10} b^{2}\right )}}, \frac{105 \, a b^{5} x^{9} + 490 \, a^{2} b^{4} x^{7} + 896 \, a^{3} b^{3} x^{5} + 790 \, a^{4} b^{2} x^{3} - 105 \, a^{5} b x + 105 \,{\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 )}{3840 \,{\left (a^{5} b^{7} x^{10} + 5 \, a^{6} b^{6} x^{8} + 10 \, a^{7} b^{5} x^{6} + 10 \, a^{8} b^{4} x^{4} + 5 \, a^{9} b^{3} x^{2} + a^{10} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/7680*(210*a*b^5*x^9 + 980*a^2*b^4*x^7 + 1792*a^3*b^3*x^5 + 1580*a^4*b^2*x^3 - 210*a^5*b*x - 105*(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^5*b^7*x^10 + 5*a^6*b^6*x^8 + 10*a^7*b^5*x^6 + 10*a^8*b^4*x^4 + 5*a^9*b^3*x^2 + a^10*b^2), 1
/3840*(105*a*b^5*x^9 + 490*a^2*b^4*x^7 + 896*a^3*b^3*x^5 + 790*a^4*b^2*x^3 - 105*a^5*b*x + 105*(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^5*b^7*x^10
+ 5*a^6*b^6*x^8 + 10*a^7*b^5*x^6 + 10*a^8*b^4*x^4 + 5*a^9*b^3*x^2 + a^10*b^2)]

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Sympy [A]  time = 1.10457, size = 190, normalized size = 1.52 \begin{align*} - \frac{7 \sqrt{- \frac{1}{a^{9} b^{3}}} \log{\left (- a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{512} + \frac{7 \sqrt{- \frac{1}{a^{9} b^{3}}} \log{\left (a^{5} b \sqrt{- \frac{1}{a^{9} b^{3}}} + x \right )}}{512} + \frac{- 105 a^{4} x + 790 a^{3} b x^{3} + 896 a^{2} b^{2} x^{5} + 490 a b^{3} x^{7} + 105 b^{4} x^{9}}{3840 a^{9} b + 19200 a^{8} b^{2} x^{2} + 38400 a^{7} b^{3} x^{4} + 38400 a^{6} b^{4} x^{6} + 19200 a^{5} b^{5} x^{8} + 3840 a^{4} b^{6} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*sqrt(-1/(a**9*b**3))*log(-a**5*b*sqrt(-1/(a**9*b**3)) + x)/512 + 7*sqrt(-1/(a**9*b**3))*log(a**5*b*sqrt(-1/
(a**9*b**3)) + x)/512 + (-105*a**4*x + 790*a**3*b*x**3 + 896*a**2*b**2*x**5 + 490*a*b**3*x**7 + 105*b**4*x**9)
/(3840*a**9*b + 19200*a**8*b**2*x**2 + 38400*a**7*b**3*x**4 + 38400*a**6*b**4*x**6 + 19200*a**5*b**5*x**8 + 38
40*a**4*b**6*x**10)

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Giac [A]  time = 1.16046, size = 113, normalized size = 0.9 \begin{align*} \frac{7 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{4} b} + \frac{105 \, b^{4} x^{9} + 490 \, a b^{3} x^{7} + 896 \, a^{2} b^{2} x^{5} + 790 \, a^{3} b x^{3} - 105 \, a^{4} x}{3840 \,{\left (b x^{2} + a\right )}^{5} a^{4} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^4*b) + 1/3840*(105*b^4*x^9 + 490*a*b^3*x^7 + 896*a^2*b^2*x^5 + 790*a^
3*b*x^3 - 105*a^4*x)/((b*x^2 + a)^5*a^4*b)