3.517 \(\int \frac {x^5}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\)

Optimal. Leaf size=53 \[ -\frac {a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac {a}{4 b^3 \left (a+b x^2\right )^4}-\frac {1}{6 b^3 \left (a+b x^2\right )^3} \]

[Out]

-1/10*a^2/b^3/(b*x^2+a)^5+1/4*a/b^3/(b*x^2+a)^4-1/6/b^3/(b*x^2+a)^3

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Rubi [A]  time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ -\frac {a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac {a}{4 b^3 \left (a+b x^2\right )^4}-\frac {1}{6 b^3 \left (a+b x^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-a^2/(10*b^3*(a + b*x^2)^5) + a/(4*b^3*(a + b*x^2)^4) - 1/(6*b^3*(a + b*x^2)^3)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^5}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^5}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \frac {x^2}{\left (a b+b^2 x\right )^6} \, dx,x,x^2\right )\\ &=\frac {1}{2} b^6 \operatorname {Subst}\left (\int \left (\frac {a^2}{b^8 (a+b x)^6}-\frac {2 a}{b^8 (a+b x)^5}+\frac {1}{b^8 (a+b x)^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{10 b^3 \left (a+b x^2\right )^5}+\frac {a}{4 b^3 \left (a+b x^2\right )^4}-\frac {1}{6 b^3 \left (a+b x^2\right )^3}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 0.66 \[ -\frac {a^2+5 a b x^2+10 b^2 x^4}{60 b^3 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

-1/60*(a^2 + 5*a*b*x^2 + 10*b^2*x^4)/(b^3*(a + b*x^2)^5)

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fricas [A]  time = 1.01, size = 80, normalized size = 1.51 \[ -\frac {10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \, {\left (b^{8} x^{10} + 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} + 10 \, a^{3} b^{5} x^{4} + 5 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/60*(10*b^2*x^4 + 5*a*b*x^2 + a^2)/(b^8*x^10 + 5*a*b^7*x^8 + 10*a^2*b^6*x^6 + 10*a^3*b^5*x^4 + 5*a^4*b^4*x^2
+ a^5*b^3)

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giac [A]  time = 0.19, size = 33, normalized size = 0.62 \[ -\frac {10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \, {\left (b x^{2} + a\right )}^{5} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

-1/60*(10*b^2*x^4 + 5*a*b*x^2 + a^2)/((b*x^2 + a)^5*b^3)

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maple [A]  time = 0.01, size = 48, normalized size = 0.91 \[ -\frac {a^{2}}{10 \left (b \,x^{2}+a \right )^{5} b^{3}}+\frac {a}{4 \left (b \,x^{2}+a \right )^{4} b^{3}}-\frac {1}{6 \left (b \,x^{2}+a \right )^{3} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x)

[Out]

-1/10*a^2/b^3/(b*x^2+a)^5+1/4*a/b^3/(b*x^2+a)^4-1/6/b^3/(b*x^2+a)^3

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maxima [A]  time = 1.35, size = 80, normalized size = 1.51 \[ -\frac {10 \, b^{2} x^{4} + 5 \, a b x^{2} + a^{2}}{60 \, {\left (b^{8} x^{10} + 5 \, a b^{7} x^{8} + 10 \, a^{2} b^{6} x^{6} + 10 \, a^{3} b^{5} x^{4} + 5 \, a^{4} b^{4} x^{2} + a^{5} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/60*(10*b^2*x^4 + 5*a*b*x^2 + a^2)/(b^8*x^10 + 5*a*b^7*x^8 + 10*a^2*b^6*x^6 + 10*a^3*b^5*x^4 + 5*a^4*b^4*x^2
+ a^5*b^3)

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mupad [B]  time = 4.62, size = 81, normalized size = 1.53 \[ -\frac {\frac {a^2}{60\,b^3}+\frac {x^4}{6\,b}+\frac {a\,x^2}{12\,b^2}}{a^5+5\,a^4\,b\,x^2+10\,a^3\,b^2\,x^4+10\,a^2\,b^3\,x^6+5\,a\,b^4\,x^8+b^5\,x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

-(a^2/(60*b^3) + x^4/(6*b) + (a*x^2)/(12*b^2))/(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)

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sympy [A]  time = 0.62, size = 83, normalized size = 1.57 \[ \frac {- a^{2} - 5 a b x^{2} - 10 b^{2} x^{4}}{60 a^{5} b^{3} + 300 a^{4} b^{4} x^{2} + 600 a^{3} b^{5} x^{4} + 600 a^{2} b^{6} x^{6} + 300 a b^{7} x^{8} + 60 b^{8} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

(-a**2 - 5*a*b*x**2 - 10*b**2*x**4)/(60*a**5*b**3 + 300*a**4*b**4*x**2 + 600*a**3*b**5*x**4 + 600*a**2*b**6*x*
*6 + 300*a*b**7*x**8 + 60*b**8*x**10)

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