3.202 \(\int \frac{x^5}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac{a}{8 b^3 \left (a+b x^2\right )^8}-\frac{1}{14 b^3 \left (a+b x^2\right )^7} \]

[Out]

-a^2/(18*b^3*(a + b*x^2)^9) + a/(8*b^3*(a + b*x^2)^8) - 1/(14*b^3*(a + b*x^2)^7)

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Rubi [A]  time = 0.100755, antiderivative size = 53, 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 \[ -\frac{a^2}{18 b^3 \left (a+b x^2\right )^9}+\frac{a}{8 b^3 \left (a+b x^2\right )^8}-\frac{1}{14 b^3 \left (a+b x^2\right )^7} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^2)^10,x]

[Out]

-a^2/(18*b^3*(a + b*x^2)^9) + a/(8*b^3*(a + b*x^2)^8) - 1/(14*b^3*(a + b*x^2)^7)

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Rubi in Sympy [A]  time = 14.7641, size = 46, normalized size = 0.87 \[ - \frac{a^{2}}{18 b^{3} \left (a + b x^{2}\right )^{9}} + \frac{a}{8 b^{3} \left (a + b x^{2}\right )^{8}} - \frac{1}{14 b^{3} \left (a + b x^{2}\right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)**10,x)

[Out]

-a**2/(18*b**3*(a + b*x**2)**9) + a/(8*b**3*(a + b*x**2)**8) - 1/(14*b**3*(a + b
*x**2)**7)

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Mathematica [A]  time = 0.0240419, size = 35, normalized size = 0.66 \[ -\frac{a^2+9 a b x^2+36 b^2 x^4}{504 b^3 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^2)^10,x]

[Out]

-(a^2 + 9*a*b*x^2 + 36*b^2*x^4)/(504*b^3*(a + b*x^2)^9)

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Maple [A]  time = 0.012, size = 48, normalized size = 0.9 \[ -{\frac{{a}^{2}}{18\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{a}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{1}{14\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^10,x)

[Out]

-1/18*a^2/b^3/(b*x^2+a)^9+1/8*a/b^3/(b*x^2+a)^8-1/14/b^3/(b*x^2+a)^7

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Maxima [A]  time = 1.38605, size = 167, normalized size = 3.15 \[ -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

-1/504*(36*b^2*x^4 + 9*a*b*x^2 + a^2)/(b^12*x^18 + 9*a*b^11*x^16 + 36*a^2*b^10*x
^14 + 84*a^3*b^9*x^12 + 126*a^4*b^8*x^10 + 126*a^5*b^7*x^8 + 84*a^6*b^6*x^6 + 36
*a^7*b^5*x^4 + 9*a^8*b^4*x^2 + a^9*b^3)

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Fricas [A]  time = 0.204608, size = 167, normalized size = 3.15 \[ -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b^{12} x^{18} + 9 \, a b^{11} x^{16} + 36 \, a^{2} b^{10} x^{14} + 84 \, a^{3} b^{9} x^{12} + 126 \, a^{4} b^{8} x^{10} + 126 \, a^{5} b^{7} x^{8} + 84 \, a^{6} b^{6} x^{6} + 36 \, a^{7} b^{5} x^{4} + 9 \, a^{8} b^{4} x^{2} + a^{9} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

-1/504*(36*b^2*x^4 + 9*a*b*x^2 + a^2)/(b^12*x^18 + 9*a*b^11*x^16 + 36*a^2*b^10*x
^14 + 84*a^3*b^9*x^12 + 126*a^4*b^8*x^10 + 126*a^5*b^7*x^8 + 84*a^6*b^6*x^6 + 36
*a^7*b^5*x^4 + 9*a^8*b^4*x^2 + a^9*b^3)

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Sympy [A]  time = 33.5177, size = 131, normalized size = 2.47 \[ - \frac{a^{2} + 9 a b x^{2} + 36 b^{2} x^{4}}{504 a^{9} b^{3} + 4536 a^{8} b^{4} x^{2} + 18144 a^{7} b^{5} x^{4} + 42336 a^{6} b^{6} x^{6} + 63504 a^{5} b^{7} x^{8} + 63504 a^{4} b^{8} x^{10} + 42336 a^{3} b^{9} x^{12} + 18144 a^{2} b^{10} x^{14} + 4536 a b^{11} x^{16} + 504 b^{12} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)**10,x)

[Out]

-(a**2 + 9*a*b*x**2 + 36*b**2*x**4)/(504*a**9*b**3 + 4536*a**8*b**4*x**2 + 18144
*a**7*b**5*x**4 + 42336*a**6*b**6*x**6 + 63504*a**5*b**7*x**8 + 63504*a**4*b**8*
x**10 + 42336*a**3*b**9*x**12 + 18144*a**2*b**10*x**14 + 4536*a*b**11*x**16 + 50
4*b**12*x**18)

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GIAC/XCAS [A]  time = 0.225358, size = 45, normalized size = 0.85 \[ -\frac{36 \, b^{2} x^{4} + 9 \, a b x^{2} + a^{2}}{504 \,{\left (b x^{2} + a\right )}^{9} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

-1/504*(36*b^2*x^4 + 9*a*b*x^2 + a^2)/((b*x^2 + a)^9*b^3)