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

Optimal. Leaf size=65 \[ -\frac{a^5}{2 x^2}+5 a^4 b x+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^2 b^3 x^7+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13} \]

[Out]

-a^5/(2*x^2) + 5*a^4*b*x + (5*a^3*b^2*x^4)/2 + (10*a^2*b^3*x^7)/7 + (a*b^4*x^10)
/2 + (b^5*x^13)/13

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Rubi [A]  time = 0.0562805, 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 \[ -\frac{a^5}{2 x^2}+5 a^4 b x+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^2 b^3 x^7+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^3,x]

[Out]

-a^5/(2*x^2) + 5*a^4*b*x + (5*a^3*b^2*x^4)/2 + (10*a^2*b^3*x^7)/7 + (a*b^4*x^10)
/2 + (b^5*x^13)/13

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Rubi in Sympy [A]  time = 10.864, size = 61, normalized size = 0.94 \[ - \frac{a^{5}}{2 x^{2}} + 5 a^{4} b x + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{10 a^{2} b^{3} x^{7}}{7} + \frac{a b^{4} x^{10}}{2} + \frac{b^{5} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**5/(2*x**2) + 5*a**4*b*x + 5*a**3*b**2*x**4/2 + 10*a**2*b**3*x**7/7 + a*b**4*
x**10/2 + b**5*x**13/13

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Mathematica [A]  time = 0.00778775, size = 65, normalized size = 1. \[ -\frac{a^5}{2 x^2}+5 a^4 b x+\frac{5}{2} a^3 b^2 x^4+\frac{10}{7} a^2 b^3 x^7+\frac{1}{2} a b^4 x^{10}+\frac{b^5 x^{13}}{13} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^3,x]

[Out]

-a^5/(2*x^2) + 5*a^4*b*x + (5*a^3*b^2*x^4)/2 + (10*a^2*b^3*x^7)/7 + (a*b^4*x^10)
/2 + (b^5*x^13)/13

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \[ -{\frac{{a}^{5}}{2\,{x}^{2}}}+5\,{a}^{4}bx+{\frac{5\,{a}^{3}{b}^{2}{x}^{4}}{2}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{7}}{7}}+{\frac{a{b}^{4}{x}^{10}}{2}}+{\frac{{b}^{5}{x}^{13}}{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^3,x)

[Out]

-1/2*a^5/x^2+5*a^4*b*x+5/2*a^3*b^2*x^4+10/7*a^2*b^3*x^7+1/2*a*b^4*x^10+1/13*b^5*
x^13

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Maxima [A]  time = 1.44252, size = 74, normalized size = 1.14 \[ \frac{1}{13} \, b^{5} x^{13} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{2} \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x - \frac{a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^5*x^13 + 1/2*a*b^4*x^10 + 10/7*a^2*b^3*x^7 + 5/2*a^3*b^2*x^4 + 5*a^4*b*x
- 1/2*a^5/x^2

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Fricas [A]  time = 0.203451, size = 80, normalized size = 1.23 \[ \frac{14 \, b^{5} x^{15} + 91 \, a b^{4} x^{12} + 260 \, a^{2} b^{3} x^{9} + 455 \, a^{3} b^{2} x^{6} + 910 \, a^{4} b x^{3} - 91 \, a^{5}}{182 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/182*(14*b^5*x^15 + 91*a*b^4*x^12 + 260*a^2*b^3*x^9 + 455*a^3*b^2*x^6 + 910*a^4
*b*x^3 - 91*a^5)/x^2

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Sympy [A]  time = 1.20857, size = 61, normalized size = 0.94 \[ - \frac{a^{5}}{2 x^{2}} + 5 a^{4} b x + \frac{5 a^{3} b^{2} x^{4}}{2} + \frac{10 a^{2} b^{3} x^{7}}{7} + \frac{a b^{4} x^{10}}{2} + \frac{b^{5} x^{13}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**3,x)

[Out]

-a**5/(2*x**2) + 5*a**4*b*x + 5*a**3*b**2*x**4/2 + 10*a**2*b**3*x**7/7 + a*b**4*
x**10/2 + b**5*x**13/13

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GIAC/XCAS [A]  time = 0.224384, size = 74, normalized size = 1.14 \[ \frac{1}{13} \, b^{5} x^{13} + \frac{1}{2} \, a b^{4} x^{10} + \frac{10}{7} \, a^{2} b^{3} x^{7} + \frac{5}{2} \, a^{3} b^{2} x^{4} + 5 \, a^{4} b x - \frac{a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/13*b^5*x^13 + 1/2*a*b^4*x^10 + 10/7*a^2*b^3*x^7 + 5/2*a^3*b^2*x^4 + 5*a^4*b*x
- 1/2*a^5/x^2