3.282 \(\int \frac{\left (a+b x^3\right )^5}{x^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0567029, 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 \[ -\frac{a^5}{5 x^5}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 x+\frac{5}{2} a^2 b^3 x^4+\frac{5}{7} a b^4 x^7+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^6,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.9506, size = 63, normalized size = 0.97 \[ - \frac{a^{5}}{5 x^{5}} - \frac{5 a^{4} b}{2 x^{2}} + 10 a^{3} b^{2} x + \frac{5 a^{2} b^{3} x^{4}}{2} + \frac{5 a b^{4} x^{7}}{7} + \frac{b^{5} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5/x**6,x)

[Out]

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

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Mathematica [A]  time = 0.00733881, size = 65, normalized size = 1. \[ -\frac{a^5}{5 x^5}-\frac{5 a^4 b}{2 x^2}+10 a^3 b^2 x+\frac{5}{2} a^2 b^3 x^4+\frac{5}{7} a b^4 x^7+\frac{b^5 x^{10}}{10} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^6,x]

[Out]

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

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[ -{\frac{{a}^{5}}{5\,{x}^{5}}}-{\frac{5\,{a}^{4}b}{2\,{x}^{2}}}+10\,{a}^{3}{b}^{2}x+{\frac{5\,{a}^{2}{b}^{3}{x}^{4}}{2}}+{\frac{5\,a{b}^{4}{x}^{7}}{7}}+{\frac{{b}^{5}{x}^{10}}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^6,x)

[Out]

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

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Maxima [A]  time = 1.43288, size = 78, normalized size = 1.2 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 10 \, a^{3} b^{2} x - \frac{25 \, a^{4} b x^{3} + 2 \, a^{5}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^6,x, algorithm="maxima")

[Out]

1/10*b^5*x^10 + 5/7*a*b^4*x^7 + 5/2*a^2*b^3*x^4 + 10*a^3*b^2*x - 1/10*(25*a^4*b*
x^3 + 2*a^5)/x^5

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Fricas [A]  time = 0.203039, size = 80, normalized size = 1.23 \[ \frac{7 \, b^{5} x^{15} + 50 \, a b^{4} x^{12} + 175 \, a^{2} b^{3} x^{9} + 700 \, a^{3} b^{2} x^{6} - 175 \, a^{4} b x^{3} - 14 \, a^{5}}{70 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^6,x, algorithm="fricas")

[Out]

1/70*(7*b^5*x^15 + 50*a*b^4*x^12 + 175*a^2*b^3*x^9 + 700*a^3*b^2*x^6 - 175*a^4*b
*x^3 - 14*a^5)/x^5

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Sympy [A]  time = 1.36714, size = 63, normalized size = 0.97 \[ 10 a^{3} b^{2} x + \frac{5 a^{2} b^{3} x^{4}}{2} + \frac{5 a b^{4} x^{7}}{7} + \frac{b^{5} x^{10}}{10} - \frac{2 a^{5} + 25 a^{4} b x^{3}}{10 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**6,x)

[Out]

10*a**3*b**2*x + 5*a**2*b**3*x**4/2 + 5*a*b**4*x**7/7 + b**5*x**10/10 - (2*a**5
+ 25*a**4*b*x**3)/(10*x**5)

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GIAC/XCAS [A]  time = 0.220677, size = 78, normalized size = 1.2 \[ \frac{1}{10} \, b^{5} x^{10} + \frac{5}{7} \, a b^{4} x^{7} + \frac{5}{2} \, a^{2} b^{3} x^{4} + 10 \, a^{3} b^{2} x - \frac{25 \, a^{4} b x^{3} + 2 \, a^{5}}{10 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^5/x^6,x, algorithm="giac")

[Out]

1/10*b^5*x^10 + 5/7*a*b^4*x^7 + 5/2*a^2*b^3*x^4 + 10*a^3*b^2*x - 1/10*(25*a^4*b*
x^3 + 2*a^5)/x^5