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

Optimal. Leaf size=124 \[ \frac{3 x}{256 a^3 b^2 \left (a+b x^2\right )}+\frac{x}{128 a^2 b^2 \left (a+b x^2\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{7/2} b^{5/2}}+\frac{x}{160 a b^2 \left (a+b x^2\right )^3}-\frac{3 x}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^3}{10 b \left (a+b x^2\right )^5} \]

[Out]

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

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Rubi [A]  time = 0.0729037, antiderivative size = 124, 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} \[ \frac{3 x}{256 a^3 b^2 \left (a+b x^2\right )}+\frac{x}{128 a^2 b^2 \left (a+b x^2\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{7/2} b^{5/2}}+\frac{x}{160 a b^2 \left (a+b x^2\right )^3}-\frac{3 x}{80 b^2 \left (a+b x^2\right )^4}-\frac{x^3}{10 b \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0473166, size = 91, normalized size = 0.73 \[ \frac{128 a^2 b^2 x^5-70 a^3 b x^3-15 a^4 x+70 a b^3 x^7+15 b^4 x^9}{1280 a^3 b^2 \left (a+b x^2\right )^5}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 a^{7/2} b^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*a^4*x - 70*a^3*b*x^3 + 128*a^2*b^2*x^5 + 70*a*b^3*x^7 + 15*b^4*x^9)/(1280*a^3*b^2*(a + b*x^2)^5) + (3*Arc
Tan[(Sqrt[b]*x)/Sqrt[a]])/(256*a^(7/2)*b^(5/2))

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Maple [A]  time = 0.052, size = 78, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{5}} \left ({\frac{3\,{b}^{2}{x}^{9}}{256\,{a}^{3}}}+{\frac{7\,b{x}^{7}}{128\,{a}^{2}}}+{\frac{{x}^{5}}{10\,a}}-{\frac{7\,{x}^{3}}{128\,b}}-{\frac{3\,ax}{256\,{b}^{2}}} \right ) }+{\frac{3}{256\,{b}^{2}{a}^{3}}\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^4/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

(3/256*b^2/a^3*x^9+7/128*b/a^2*x^7+1/10/a*x^5-7/128/b*x^3-3/256/b^2*a*x)/(b*x^2+a)^5+3/256/b^2/a^3/(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^4/(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 = 1.50087, size = 838, normalized size = 6.76 \begin{align*} \left [\frac{30 \, a b^{5} x^{9} + 140 \, a^{2} b^{4} x^{7} + 256 \, a^{3} b^{3} x^{5} - 140 \, a^{4} b^{2} x^{3} - 30 \, a^{5} b x - 15 \,{\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^{4} b^{8} x^{10} + 5 \, a^{5} b^{7} x^{8} + 10 \, a^{6} b^{6} x^{6} + 10 \, a^{7} b^{5} x^{4} + 5 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}}, \frac{15 \, a b^{5} x^{9} + 70 \, a^{2} b^{4} x^{7} + 128 \, a^{3} b^{3} x^{5} - 70 \, a^{4} b^{2} x^{3} - 15 \, a^{5} b x + 15 \,{\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^{4} b^{8} x^{10} + 5 \, a^{5} b^{7} x^{8} + 10 \, a^{6} b^{6} x^{6} + 10 \, a^{7} b^{5} x^{4} + 5 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2560*(30*a*b^5*x^9 + 140*a^2*b^4*x^7 + 256*a^3*b^3*x^5 - 140*a^4*b^2*x^3 - 30*a^5*b*x - 15*(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^4*b^8*x^10 + 5*a^5*b^7*x^8 + 10*a^6*b^6*x^6 + 10*a^7*b^5*x^4 + 5*a^8*b^4*x^2 + a^9*b^3), 1/1280*
(15*a*b^5*x^9 + 70*a^2*b^4*x^7 + 128*a^3*b^3*x^5 - 70*a^4*b^2*x^3 - 15*a^5*b*x + 15*(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^4*b^8*x^10 + 5*a^5*b^
7*x^8 + 10*a^6*b^6*x^6 + 10*a^7*b^5*x^4 + 5*a^8*b^4*x^2 + a^9*b^3)]

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Sympy [A]  time = 1.11086, size = 196, normalized size = 1.58 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{7} b^{5}}} \log{\left (- a^{4} b^{2} \sqrt{- \frac{1}{a^{7} b^{5}}} + x \right )}}{512} + \frac{3 \sqrt{- \frac{1}{a^{7} b^{5}}} \log{\left (a^{4} b^{2} \sqrt{- \frac{1}{a^{7} b^{5}}} + x \right )}}{512} + \frac{- 15 a^{4} x - 70 a^{3} b x^{3} + 128 a^{2} b^{2} x^{5} + 70 a b^{3} x^{7} + 15 b^{4} x^{9}}{1280 a^{8} b^{2} + 6400 a^{7} b^{3} x^{2} + 12800 a^{6} b^{4} x^{4} + 12800 a^{5} b^{5} x^{6} + 6400 a^{4} b^{6} x^{8} + 1280 a^{3} b^{7} x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*sqrt(-1/(a**7*b**5))*log(-a**4*b**2*sqrt(-1/(a**7*b**5)) + x)/512 + 3*sqrt(-1/(a**7*b**5))*log(a**4*b**2*sq
rt(-1/(a**7*b**5)) + x)/512 + (-15*a**4*x - 70*a**3*b*x**3 + 128*a**2*b**2*x**5 + 70*a*b**3*x**7 + 15*b**4*x**
9)/(1280*a**8*b**2 + 6400*a**7*b**3*x**2 + 12800*a**6*b**4*x**4 + 12800*a**5*b**5*x**6 + 6400*a**4*b**6*x**8 +
1280*a**3*b**7*x**10)

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Giac [A]  time = 1.1785, size = 113, normalized size = 0.91 \begin{align*} \frac{3 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{256 \, \sqrt{a b} a^{3} b^{2}} + \frac{15 \, b^{4} x^{9} + 70 \, a b^{3} x^{7} + 128 \, a^{2} b^{2} x^{5} - 70 \, a^{3} b x^{3} - 15 \, a^{4} x}{1280 \,{\left (b x^{2} + a\right )}^{5} a^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/256*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3*b^2) + 1/1280*(15*b^4*x^9 + 70*a*b^3*x^7 + 128*a^2*b^2*x^5 - 70*a^3
*b*x^3 - 15*a^4*x)/((b*x^2 + a)^5*a^3*b^2)