3.80 \(\int \frac{\left (a+b x^2\right )^5}{x^{12}} \, dx\)

Optimal. Leaf size=65 \[ -\frac{a^5}{11 x^{11}}-\frac{5 a^4 b}{9 x^9}-\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x} \]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5
*a*b^4)/(3*x^3) - b^5/x

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Rubi [A]  time = 0.0640907, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a^5}{11 x^{11}}-\frac{5 a^4 b}{9 x^9}-\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5
*a*b^4)/(3*x^3) - b^5/x

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Rubi in Sympy [A]  time = 12.106, size = 63, normalized size = 0.97 \[ - \frac{a^{5}}{11 x^{11}} - \frac{5 a^{4} b}{9 x^{9}} - \frac{10 a^{3} b^{2}}{7 x^{7}} - \frac{2 a^{2} b^{3}}{x^{5}} - \frac{5 a b^{4}}{3 x^{3}} - \frac{b^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5/x**12,x)

[Out]

-a**5/(11*x**11) - 5*a**4*b/(9*x**9) - 10*a**3*b**2/(7*x**7) - 2*a**2*b**3/x**5
- 5*a*b**4/(3*x**3) - b**5/x

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Mathematica [A]  time = 0.00754584, size = 65, normalized size = 1. \[ -\frac{a^5}{11 x^{11}}-\frac{5 a^4 b}{9 x^9}-\frac{10 a^3 b^2}{7 x^7}-\frac{2 a^2 b^3}{x^5}-\frac{5 a b^4}{3 x^3}-\frac{b^5}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5
*a*b^4)/(3*x^3) - b^5/x

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Maple [A]  time = 0.008, size = 58, normalized size = 0.9 \[ -{\frac{{a}^{5}}{11\,{x}^{11}}}-{\frac{5\,{a}^{4}b}{9\,{x}^{9}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,{x}^{7}}}-2\,{\frac{{a}^{2}{b}^{3}}{{x}^{5}}}-{\frac{5\,a{b}^{4}}{3\,{x}^{3}}}-{\frac{{b}^{5}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5/x^12,x)

[Out]

-1/11*a^5/x^11-5/9*a^4*b/x^9-10/7*a^3*b^2/x^7-2*a^2*b^3/x^5-5/3*a*b^4/x^3-b^5/x

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Maxima [A]  time = 1.34133, size = 80, normalized size = 1.23 \[ -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^12,x, algorithm="maxima")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385
*a^4*b*x^2 + 63*a^5)/x^11

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Fricas [A]  time = 0.195432, size = 80, normalized size = 1.23 \[ -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^12,x, algorithm="fricas")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385
*a^4*b*x^2 + 63*a^5)/x^11

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Sympy [A]  time = 2.23197, size = 63, normalized size = 0.97 \[ - \frac{63 a^{5} + 385 a^{4} b x^{2} + 990 a^{3} b^{2} x^{4} + 1386 a^{2} b^{3} x^{6} + 1155 a b^{4} x^{8} + 693 b^{5} x^{10}}{693 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5/x**12,x)

[Out]

-(63*a**5 + 385*a**4*b*x**2 + 990*a**3*b**2*x**4 + 1386*a**2*b**3*x**6 + 1155*a*
b**4*x**8 + 693*b**5*x**10)/(693*x**11)

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GIAC/XCAS [A]  time = 0.206826, size = 80, normalized size = 1.23 \[ -\frac{693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^12,x, algorithm="giac")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385
*a^4*b*x^2 + 63*a^5)/x^11