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

Optimal. Leaf size=63 \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{11}}{11} \]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (
b^5*x^11)/11

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Rubi [A]  time = 0.0579662, antiderivative size = 63, 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 \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (
b^5*x^11)/11

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{5}}{4 x^{4}} - \frac{5 a^{4} b}{x} + 10 a^{3} b^{2} \int x\, dx + 2 a^{2} b^{3} x^{5} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**5/(4*x**4) - 5*a**4*b/x + 10*a**3*b**2*Integral(x, x) + 2*a**2*b**3*x**5 + 5
*a*b**4*x**8/8 + b**5*x**11/11

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Mathematica [A]  time = 0.0112509, size = 63, normalized size = 1. \[ -\frac{a^5}{4 x^4}-\frac{5 a^4 b}{x}+5 a^3 b^2 x^2+2 a^2 b^3 x^5+\frac{5}{8} a b^4 x^8+\frac{b^5 x^{11}}{11} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^5/x^5,x]

[Out]

-a^5/(4*x^4) - (5*a^4*b)/x + 5*a^3*b^2*x^2 + 2*a^2*b^3*x^5 + (5*a*b^4*x^8)/8 + (
b^5*x^11)/11

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Maple [A]  time = 0.008, size = 58, normalized size = 0.9 \[ -{\frac{{a}^{5}}{4\,{x}^{4}}}-5\,{\frac{{a}^{4}b}{x}}+5\,{a}^{3}{b}^{2}{x}^{2}+2\,{a}^{2}{b}^{3}{x}^{5}+{\frac{5\,a{b}^{4}{x}^{8}}{8}}+{\frac{{b}^{5}{x}^{11}}{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5/x^5,x)

[Out]

-1/4*a^5/x^4-5*a^4*b/x+5*a^3*b^2*x^2+2*a^2*b^3*x^5+5/8*a*b^4*x^8+1/11*b^5*x^11

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Maxima [A]  time = 1.42435, size = 78, normalized size = 1.24 \[ \frac{1}{11} \, b^{5} x^{11} + \frac{5}{8} \, a b^{4} x^{8} + 2 \, a^{2} b^{3} x^{5} + 5 \, a^{3} b^{2} x^{2} - \frac{20 \, a^{4} b x^{3} + a^{5}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^5*x^11 + 5/8*a*b^4*x^8 + 2*a^2*b^3*x^5 + 5*a^3*b^2*x^2 - 1/4*(20*a^4*b*x^
3 + a^5)/x^4

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Fricas [A]  time = 0.205073, size = 80, normalized size = 1.27 \[ \frac{8 \, b^{5} x^{15} + 55 \, a b^{4} x^{12} + 176 \, a^{2} b^{3} x^{9} + 440 \, a^{3} b^{2} x^{6} - 440 \, a^{4} b x^{3} - 22 \, a^{5}}{88 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/88*(8*b^5*x^15 + 55*a*b^4*x^12 + 176*a^2*b^3*x^9 + 440*a^3*b^2*x^6 - 440*a^4*b
*x^3 - 22*a^5)/x^4

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Sympy [A]  time = 1.39178, size = 61, normalized size = 0.97 \[ 5 a^{3} b^{2} x^{2} + 2 a^{2} b^{3} x^{5} + \frac{5 a b^{4} x^{8}}{8} + \frac{b^{5} x^{11}}{11} - \frac{a^{5} + 20 a^{4} b x^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5/x**5,x)

[Out]

5*a**3*b**2*x**2 + 2*a**2*b**3*x**5 + 5*a*b**4*x**8/8 + b**5*x**11/11 - (a**5 +
20*a**4*b*x**3)/(4*x**4)

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GIAC/XCAS [A]  time = 0.21861, size = 78, normalized size = 1.24 \[ \frac{1}{11} \, b^{5} x^{11} + \frac{5}{8} \, a b^{4} x^{8} + 2 \, a^{2} b^{3} x^{5} + 5 \, a^{3} b^{2} x^{2} - \frac{20 \, a^{4} b x^{3} + a^{5}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/11*b^5*x^11 + 5/8*a*b^4*x^8 + 2*a^2*b^3*x^5 + 5*a^3*b^2*x^2 - 1/4*(20*a^4*b*x^
3 + a^5)/x^4