3.6.15 \(\int \frac {x^9}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [515]

Optimal. Leaf size=19 \[ \frac {x^{10}}{10 a \left (a+b x^2\right )^5} \]

[Out]

1/10*x^10/a/(b*x^2+a)^5

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270} \begin {gather*} \frac {x^{10}}{10 a \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

x^10/(10*a*(a + b*x^2)^5)

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^9}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=b^6 \int \frac {x^9}{\left (a b+b^2 x^2\right )^6} \, dx\\ &=\frac {x^{10}}{10 a \left (a+b x^2\right )^5}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(19)=38\).
time = 0.01, size = 57, normalized size = 3.00 \begin {gather*} -\frac {a^4+5 a^3 b x^2+10 a^2 b^2 x^4+10 a b^3 x^6+5 b^4 x^8}{10 b^5 \left (a+b x^2\right )^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(17)=34\).
time = 0.04, size = 81, normalized size = 4.26

method	result	size
norman	\(\frac {-\frac {x^{8}}{2 b}-\frac {a \,x^{6}}{b^{2}}-\frac {a^{2} x^{4}}{b^{3}}-\frac {a^{3} x^{2}}{2 b^{4}}-\frac {a^{4}}{10 b^{5}}}{\left (b \,x^{2}+a \right )^{5}}\)	\(59\)
gosper	\(-\frac {5 b^{4} x^{8}+10 a \,b^{3} x^{6}+10 a^{2} b^{2} x^{4}+5 a^{3} b \,x^{2}+a^{4}}{10 \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2} b^{5}}\)	\(76\)
risch	\(\frac {-\frac {x^{8}}{2 b}-\frac {a \,x^{6}}{b^{2}}-\frac {a^{2} x^{4}}{b^{3}}-\frac {a^{3} x^{2}}{2 b^{4}}-\frac {a^{4}}{10 b^{5}}}{\left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}\)	\(79\)
default	\(\frac {a^{3}}{2 b^{5} \left (b \,x^{2}+a \right )^{4}}-\frac {1}{2 b^{5} \left (b \,x^{2}+a \right )}-\frac {a^{2}}{b^{5} \left (b \,x^{2}+a \right )^{3}}-\frac {a^{4}}{10 b^{5} \left (b \,x^{2}+a \right )^{5}}+\frac {a}{b^{5} \left (b \,x^{2}+a \right )^{2}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (17) = 34\).
time = 0.28, size = 102, normalized size = 5.37 \begin {gather*} -\frac {5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/10*(5*b^4*x^8 + 10*a*b^3*x^6 + 10*a^2*b^2*x^4 + 5*a^3*b*x^2 + a^4)/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^
6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (17) = 34\).
time = 0.35, size = 102, normalized size = 5.37 \begin {gather*} -\frac {5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \, {\left (b^{10} x^{10} + 5 \, a b^{9} x^{8} + 10 \, a^{2} b^{8} x^{6} + 10 \, a^{3} b^{7} x^{4} + 5 \, a^{4} b^{6} x^{2} + a^{5} b^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

-1/10*(5*b^4*x^8 + 10*a*b^3*x^6 + 10*a^2*b^2*x^4 + 5*a^3*b*x^2 + a^4)/(b^10*x^10 + 5*a*b^9*x^8 + 10*a^2*b^8*x^
6 + 10*a^3*b^7*x^4 + 5*a^4*b^6*x^2 + a^5*b^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (14) = 28\).
time = 0.32, size = 107, normalized size = 5.63 \begin {gather*} \frac {- a^{4} - 5 a^{3} b x^{2} - 10 a^{2} b^{2} x^{4} - 10 a b^{3} x^{6} - 5 b^{4} x^{8}}{10 a^{5} b^{5} + 50 a^{4} b^{6} x^{2} + 100 a^{3} b^{7} x^{4} + 100 a^{2} b^{8} x^{6} + 50 a b^{9} x^{8} + 10 b^{10} x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

(-a**4 - 5*a**3*b*x**2 - 10*a**2*b**2*x**4 - 10*a*b**3*x**6 - 5*b**4*x**8)/(10*a**5*b**5 + 50*a**4*b**6*x**2 +
100*a**3*b**7*x**4 + 100*a**2*b**8*x**6 + 50*a*b**9*x**8 + 10*b**10*x**10)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (17) = 34\).
time = 3.87, size = 55, normalized size = 2.89 \begin {gather*} -\frac {5 \, b^{4} x^{8} + 10 \, a b^{3} x^{6} + 10 \, a^{2} b^{2} x^{4} + 5 \, a^{3} b x^{2} + a^{4}}{10 \, {\left (b x^{2} + a\right )}^{5} b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.45, size = 104, normalized size = 5.47 \begin {gather*} -\frac {a^4+5\,a^3\,b\,x^2+10\,a^2\,b^2\,x^4+10\,a\,b^3\,x^6+5\,b^4\,x^8}{10\,a^5\,b^5+50\,a^4\,b^6\,x^2+100\,a^3\,b^7\,x^4+100\,a^2\,b^8\,x^6+50\,a\,b^9\,x^8+10\,b^{10}\,x^{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

-(a^4 + 5*b^4*x^8 + 5*a^3*b*x^2 + 10*a*b^3*x^6 + 10*a^2*b^2*x^4)/(10*a^5*b^5 + 10*b^10*x^10 + 50*a*b^9*x^8 + 5
0*a^4*b^6*x^2 + 100*a^3*b^7*x^4 + 100*a^2*b^8*x^6)

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