3.66 \(\int \frac{\left (a+b x^2\right )^5}{x^{15}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac{\left (a+b x^2\right )^6}{14 a x^{14}} \]

[Out]

-(a + b*x^2)^6/(14*a*x^14) + (b*(a + b*x^2)^6)/(84*a^2*x^12)

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Rubi [A]  time = 0.0581051, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac{\left (a+b x^2\right )^6}{14 a x^{14}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^5/x^15,x]

[Out]

-(a + b*x^2)^6/(14*a*x^14) + (b*(a + b*x^2)^6)/(84*a^2*x^12)

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Rubi in Sympy [A]  time = 6.4845, size = 32, normalized size = 0.8 \[ - \frac{\left (a + b x^{2}\right )^{6}}{14 a x^{14}} + \frac{b \left (a + b x^{2}\right )^{6}}{84 a^{2} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**5/x**15,x)

[Out]

-(a + b*x**2)**6/(14*a*x**14) + b*(a + b*x**2)**6/(84*a**2*x**12)

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Mathematica [A]  time = 0.0110855, size = 67, normalized size = 1.68 \[ -\frac{a^5}{14 x^{14}}-\frac{5 a^4 b}{12 x^{12}}-\frac{a^3 b^2}{x^{10}}-\frac{5 a^2 b^3}{4 x^8}-\frac{5 a b^4}{6 x^6}-\frac{b^5}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^5/x^15,x]

[Out]

-a^5/(14*x^14) - (5*a^4*b)/(12*x^12) - (a^3*b^2)/x^10 - (5*a^2*b^3)/(4*x^8) - (5
*a*b^4)/(6*x^6) - b^5/(4*x^4)

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Maple [A]  time = 0.01, size = 58, normalized size = 1.5 \[ -{\frac{5\,{a}^{4}b}{12\,{x}^{12}}}-{\frac{5\,a{b}^{4}}{6\,{x}^{6}}}-{\frac{{b}^{5}}{4\,{x}^{4}}}-{\frac{5\,{a}^{2}{b}^{3}}{4\,{x}^{8}}}-{\frac{{a}^{5}}{14\,{x}^{14}}}-{\frac{{a}^{3}{b}^{2}}{{x}^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^5/x^15,x)

[Out]

-5/12*a^4*b/x^12-5/6*a*b^4/x^6-1/4*b^5/x^4-5/4*a^2*b^3/x^8-1/14*a^5/x^14-a^3*b^2
/x^10

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Maxima [A]  time = 1.33184, size = 80, normalized size = 2. \[ -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^15,x, algorithm="maxima")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*
x^2 + 6*a^5)/x^14

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Fricas [A]  time = 0.207042, size = 80, normalized size = 2. \[ -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^15,x, algorithm="fricas")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*
x^2 + 6*a^5)/x^14

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Sympy [A]  time = 2.48686, size = 63, normalized size = 1.58 \[ - \frac{6 a^{5} + 35 a^{4} b x^{2} + 84 a^{3} b^{2} x^{4} + 105 a^{2} b^{3} x^{6} + 70 a b^{4} x^{8} + 21 b^{5} x^{10}}{84 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**5/x**15,x)

[Out]

-(6*a**5 + 35*a**4*b*x**2 + 84*a**3*b**2*x**4 + 105*a**2*b**3*x**6 + 70*a*b**4*x
**8 + 21*b**5*x**10)/(84*x**14)

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GIAC/XCAS [A]  time = 0.206607, size = 80, normalized size = 2. \[ -\frac{21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5/x^15,x, algorithm="giac")

[Out]

-1/84*(21*b^5*x^10 + 70*a*b^4*x^8 + 105*a^2*b^3*x^6 + 84*a^3*b^2*x^4 + 35*a^4*b*
x^2 + 6*a^5)/x^14