3.200 \(\int \frac{x^9}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac{a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac{3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac{a}{3 b^5 \left (a+b x^2\right )^6}-\frac{1}{10 b^5 \left (a+b x^2\right )^5} \]

[Out]

-a^4/(18*b^5*(a + b*x^2)^9) + a^3/(4*b^5*(a + b*x^2)^8) - (3*a^2)/(7*b^5*(a + b*
x^2)^7) + a/(3*b^5*(a + b*x^2)^6) - 1/(10*b^5*(a + b*x^2)^5)

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Rubi [A]  time = 0.163549, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^4}{18 b^5 \left (a+b x^2\right )^9}+\frac{a^3}{4 b^5 \left (a+b x^2\right )^8}-\frac{3 a^2}{7 b^5 \left (a+b x^2\right )^7}+\frac{a}{3 b^5 \left (a+b x^2\right )^6}-\frac{1}{10 b^5 \left (a+b x^2\right )^5} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + b*x^2)^10,x]

[Out]

-a^4/(18*b^5*(a + b*x^2)^9) + a^3/(4*b^5*(a + b*x^2)^8) - (3*a^2)/(7*b^5*(a + b*
x^2)^7) + a/(3*b^5*(a + b*x^2)^6) - 1/(10*b^5*(a + b*x^2)^5)

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Rubi in Sympy [A]  time = 23.7135, size = 82, normalized size = 0.9 \[ - \frac{a^{4}}{18 b^{5} \left (a + b x^{2}\right )^{9}} + \frac{a^{3}}{4 b^{5} \left (a + b x^{2}\right )^{8}} - \frac{3 a^{2}}{7 b^{5} \left (a + b x^{2}\right )^{7}} + \frac{a}{3 b^{5} \left (a + b x^{2}\right )^{6}} - \frac{1}{10 b^{5} \left (a + b x^{2}\right )^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(b*x**2+a)**10,x)

[Out]

-a**4/(18*b**5*(a + b*x**2)**9) + a**3/(4*b**5*(a + b*x**2)**8) - 3*a**2/(7*b**5
*(a + b*x**2)**7) + a/(3*b**5*(a + b*x**2)**6) - 1/(10*b**5*(a + b*x**2)**5)

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Mathematica [A]  time = 0.0267643, size = 57, normalized size = 0.63 \[ -\frac{a^4+9 a^3 b x^2+36 a^2 b^2 x^4+84 a b^3 x^6+126 b^4 x^8}{1260 b^5 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + b*x^2)^10,x]

[Out]

-(a^4 + 9*a^3*b*x^2 + 36*a^2*b^2*x^4 + 84*a*b^3*x^6 + 126*b^4*x^8)/(1260*b^5*(a
+ b*x^2)^9)

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Maple [A]  time = 0.011, size = 82, normalized size = 0.9 \[ -{\frac{{a}^{4}}{18\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{{a}^{3}}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{3\,{a}^{2}}{7\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{7}}}+{\frac{a}{3\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{1}{10\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(b*x^2+a)^10,x)

[Out]

-1/18*a^4/b^5/(b*x^2+a)^9+1/4*a^3/b^5/(b*x^2+a)^8-3/7*a^2/b^5/(b*x^2+a)^7+1/3*a/
b^5/(b*x^2+a)^6-1/10/b^5/(b*x^2+a)^5

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Maxima [A]  time = 1.36678, size = 197, normalized size = 2.16 \[ -\frac{126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \,{\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/(b^14*
x^18 + 9*a*b^13*x^16 + 36*a^2*b^12*x^14 + 84*a^3*b^11*x^12 + 126*a^4*b^10*x^10 +
126*a^5*b^9*x^8 + 84*a^6*b^8*x^6 + 36*a^7*b^7*x^4 + 9*a^8*b^6*x^2 + a^9*b^5)

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Fricas [A]  time = 0.206562, size = 197, normalized size = 2.16 \[ -\frac{126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \,{\left (b^{14} x^{18} + 9 \, a b^{13} x^{16} + 36 \, a^{2} b^{12} x^{14} + 84 \, a^{3} b^{11} x^{12} + 126 \, a^{4} b^{10} x^{10} + 126 \, a^{5} b^{9} x^{8} + 84 \, a^{6} b^{8} x^{6} + 36 \, a^{7} b^{7} x^{4} + 9 \, a^{8} b^{6} x^{2} + a^{9} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/(b^14*
x^18 + 9*a*b^13*x^16 + 36*a^2*b^12*x^14 + 84*a^3*b^11*x^12 + 126*a^4*b^10*x^10 +
126*a^5*b^9*x^8 + 84*a^6*b^8*x^6 + 36*a^7*b^7*x^4 + 9*a^8*b^6*x^2 + a^9*b^5)

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Sympy [A]  time = 33.929, size = 155, normalized size = 1.7 \[ - \frac{a^{4} + 9 a^{3} b x^{2} + 36 a^{2} b^{2} x^{4} + 84 a b^{3} x^{6} + 126 b^{4} x^{8}}{1260 a^{9} b^{5} + 11340 a^{8} b^{6} x^{2} + 45360 a^{7} b^{7} x^{4} + 105840 a^{6} b^{8} x^{6} + 158760 a^{5} b^{9} x^{8} + 158760 a^{4} b^{10} x^{10} + 105840 a^{3} b^{11} x^{12} + 45360 a^{2} b^{12} x^{14} + 11340 a b^{13} x^{16} + 1260 b^{14} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(b*x**2+a)**10,x)

[Out]

-(a**4 + 9*a**3*b*x**2 + 36*a**2*b**2*x**4 + 84*a*b**3*x**6 + 126*b**4*x**8)/(12
60*a**9*b**5 + 11340*a**8*b**6*x**2 + 45360*a**7*b**7*x**4 + 105840*a**6*b**8*x*
*6 + 158760*a**5*b**9*x**8 + 158760*a**4*b**10*x**10 + 105840*a**3*b**11*x**12 +
45360*a**2*b**12*x**14 + 11340*a*b**13*x**16 + 1260*b**14*x**18)

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GIAC/XCAS [A]  time = 0.211509, size = 74, normalized size = 0.81 \[ -\frac{126 \, b^{4} x^{8} + 84 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 9 \, a^{3} b x^{2} + a^{4}}{1260 \,{\left (b x^{2} + a\right )}^{9} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

-1/1260*(126*b^4*x^8 + 84*a*b^3*x^6 + 36*a^2*b^2*x^4 + 9*a^3*b*x^2 + a^4)/((b*x^
2 + a)^9*b^5)