3.272 \(\int \frac{\left (a+b x^3\right )^5}{x^{25}} \, dx\)

Optimal. Leaf size=62 \[ -\frac{b^2 \left (a+b x^3\right )^6}{504 a^3 x^{18}}+\frac{b \left (a+b x^3\right )^6}{84 a^2 x^{21}}-\frac{\left (a+b x^3\right )^6}{24 a x^{24}} \]

[Out]

-(a + b*x^3)^6/(24*a*x^24) + (b*(a + b*x^3)^6)/(84*a^2*x^21) - (b^2*(a + b*x^3)^
6)/(504*a^3*x^18)

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Rubi [A]  time = 0.077408, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b^2 \left (a+b x^3\right )^6}{504 a^3 x^{18}}+\frac{b \left (a+b x^3\right )^6}{84 a^2 x^{21}}-\frac{\left (a+b x^3\right )^6}{24 a x^{24}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^25,x]

[Out]

-(a + b*x^3)^6/(24*a*x^24) + (b*(a + b*x^3)^6)/(84*a^2*x^21) - (b^2*(a + b*x^3)^
6)/(504*a^3*x^18)

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Rubi in Sympy [A]  time = 14.0305, size = 68, normalized size = 1.1 \[ - \frac{a^{5}}{24 x^{24}} - \frac{5 a^{4} b}{21 x^{21}} - \frac{5 a^{3} b^{2}}{9 x^{18}} - \frac{2 a^{2} b^{3}}{3 x^{15}} - \frac{5 a b^{4}}{12 x^{12}} - \frac{b^{5}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5/x**25,x)

[Out]

-a**5/(24*x**24) - 5*a**4*b/(21*x**21) - 5*a**3*b**2/(9*x**18) - 2*a**2*b**3/(3*
x**15) - 5*a*b**4/(12*x**12) - b**5/(9*x**9)

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Mathematica [A]  time = 0.0128134, size = 69, normalized size = 1.11 \[ -\frac{a^5}{24 x^{24}}-\frac{5 a^4 b}{21 x^{21}}-\frac{5 a^3 b^2}{9 x^{18}}-\frac{2 a^2 b^3}{3 x^{15}}-\frac{5 a b^4}{12 x^{12}}-\frac{b^5}{9 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^25,x]

[Out]

-a^5/(24*x^24) - (5*a^4*b)/(21*x^21) - (5*a^3*b^2)/(9*x^18) - (2*a^2*b^3)/(3*x^1
5) - (5*a*b^4)/(12*x^12) - b^5/(9*x^9)

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Maple [A]  time = 0.009, size = 58, normalized size = 0.9 \[ -{\frac{5\,{a}^{3}{b}^{2}}{9\,{x}^{18}}}-{\frac{5\,a{b}^{4}}{12\,{x}^{12}}}-{\frac{2\,{a}^{2}{b}^{3}}{3\,{x}^{15}}}-{\frac{{a}^{5}}{24\,{x}^{24}}}-{\frac{{b}^{5}}{9\,{x}^{9}}}-{\frac{5\,{a}^{4}b}{21\,{x}^{21}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^25,x)

[Out]

-5/9*a^3*b^2/x^18-5/12*a*b^4/x^12-2/3*a^2*b^3/x^15-1/24*a^5/x^24-1/9*b^5/x^9-5/2
1*a^4*b/x^21

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Maxima [A]  time = 1.44933, size = 80, normalized size = 1.29 \[ -\frac{56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^25,x, algorithm="maxima")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a
^4*b*x^3 + 21*a^5)/x^24

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Fricas [A]  time = 0.207343, size = 80, normalized size = 1.29 \[ -\frac{56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^25,x, algorithm="fricas")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a
^4*b*x^3 + 21*a^5)/x^24

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Sympy [A]  time = 3.23828, size = 63, normalized size = 1.02 \[ - \frac{21 a^{5} + 120 a^{4} b x^{3} + 280 a^{3} b^{2} x^{6} + 336 a^{2} b^{3} x^{9} + 210 a b^{4} x^{12} + 56 b^{5} x^{15}}{504 x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**25,x)

[Out]

-(21*a**5 + 120*a**4*b*x**3 + 280*a**3*b**2*x**6 + 336*a**2*b**3*x**9 + 210*a*b*
*4*x**12 + 56*b**5*x**15)/(504*x**24)

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GIAC/XCAS [A]  time = 0.222404, size = 80, normalized size = 1.29 \[ -\frac{56 \, b^{5} x^{15} + 210 \, a b^{4} x^{12} + 336 \, a^{2} b^{3} x^{9} + 280 \, a^{3} b^{2} x^{6} + 120 \, a^{4} b x^{3} + 21 \, a^{5}}{504 \, x^{24}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^25,x, algorithm="giac")

[Out]

-1/504*(56*b^5*x^15 + 210*a*b^4*x^12 + 336*a^2*b^3*x^9 + 280*a^3*b^2*x^6 + 120*a
^4*b*x^3 + 21*a^5)/x^24