3.41 \(\int \frac{\left (a+b x^2\right )^3}{x^{15}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^3}{14 x^{14}}-\frac{a^2 b}{4 x^{12}}-\frac{3 a b^2}{10 x^{10}}-\frac{b^3}{8 x^8} \]

[Out]

-a^3/(14*x^14) - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)

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Rubi [A]  time = 0.0555315, antiderivative size = 43, 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 \[ -\frac{a^3}{14 x^{14}}-\frac{a^2 b}{4 x^{12}}-\frac{3 a b^2}{10 x^{10}}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^15,x]

[Out]

-a^3/(14*x^14) - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)

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Rubi in Sympy [A]  time = 9.21817, size = 39, normalized size = 0.91 \[ - \frac{a^{3}}{14 x^{14}} - \frac{a^{2} b}{4 x^{12}} - \frac{3 a b^{2}}{10 x^{10}} - \frac{b^{3}}{8 x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**15,x)

[Out]

-a**3/(14*x**14) - a**2*b/(4*x**12) - 3*a*b**2/(10*x**10) - b**3/(8*x**8)

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Mathematica [A]  time = 0.0120506, size = 43, normalized size = 1. \[ -\frac{a^3}{14 x^{14}}-\frac{a^2 b}{4 x^{12}}-\frac{3 a b^2}{10 x^{10}}-\frac{b^3}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^15,x]

[Out]

-a^3/(14*x^14) - (a^2*b)/(4*x^12) - (3*a*b^2)/(10*x^10) - b^3/(8*x^8)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \[ -{\frac{{a}^{3}}{14\,{x}^{14}}}-{\frac{{a}^{2}b}{4\,{x}^{12}}}-{\frac{3\,a{b}^{2}}{10\,{x}^{10}}}-{\frac{{b}^{3}}{8\,{x}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^15,x)

[Out]

-1/14*a^3/x^14-1/4*a^2*b/x^12-3/10*a*b^2/x^10-1/8*b^3/x^8

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Maxima [A]  time = 1.34464, size = 50, normalized size = 1.16 \[ -\frac{35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^15,x, algorithm="maxima")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14

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Fricas [A]  time = 0.193665, size = 50, normalized size = 1.16 \[ -\frac{35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^15,x, algorithm="fricas")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14

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Sympy [A]  time = 1.90044, size = 39, normalized size = 0.91 \[ - \frac{20 a^{3} + 70 a^{2} b x^{2} + 84 a b^{2} x^{4} + 35 b^{3} x^{6}}{280 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**15,x)

[Out]

-(20*a**3 + 70*a**2*b*x**2 + 84*a*b**2*x**4 + 35*b**3*x**6)/(280*x**14)

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GIAC/XCAS [A]  time = 0.21349, size = 50, normalized size = 1.16 \[ -\frac{35 \, b^{3} x^{6} + 84 \, a b^{2} x^{4} + 70 \, a^{2} b x^{2} + 20 \, a^{3}}{280 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^15,x, algorithm="giac")

[Out]

-1/280*(35*b^3*x^6 + 84*a*b^2*x^4 + 70*a^2*b*x^2 + 20*a^3)/x^14