3.640 \(\int \frac{\left (a+b x^4\right )^3}{x^4} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a^3}{3 x^3}+3 a^2 b x+\frac{3}{5} a b^2 x^5+\frac{b^3 x^9}{9} \]

[Out]

-a^3/(3*x^3) + 3*a^2*b*x + (3*a*b^2*x^5)/5 + (b^3*x^9)/9

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Rubi [A]  time = 0.0363632, antiderivative size = 39, 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 \[ -\frac{a^3}{3 x^3}+3 a^2 b x+\frac{3}{5} a b^2 x^5+\frac{b^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^3/x^4,x]

[Out]

-a^3/(3*x^3) + 3*a^2*b*x + (3*a*b^2*x^5)/5 + (b^3*x^9)/9

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Rubi in Sympy [A]  time = 6.51733, size = 36, normalized size = 0.92 \[ - \frac{a^{3}}{3 x^{3}} + 3 a^{2} b x + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**3/x**4,x)

[Out]

-a**3/(3*x**3) + 3*a**2*b*x + 3*a*b**2*x**5/5 + b**3*x**9/9

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Mathematica [A]  time = 0.00672156, size = 39, normalized size = 1. \[ -\frac{a^3}{3 x^3}+3 a^2 b x+\frac{3}{5} a b^2 x^5+\frac{b^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^3/x^4,x]

[Out]

-a^3/(3*x^3) + 3*a^2*b*x + (3*a*b^2*x^5)/5 + (b^3*x^9)/9

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Maple [A]  time = 0.006, size = 34, normalized size = 0.9 \[ -{\frac{{a}^{3}}{3\,{x}^{3}}}+3\,{a}^{2}bx+{\frac{3\,a{b}^{2}{x}^{5}}{5}}+{\frac{{b}^{3}{x}^{9}}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^3/x^4,x)

[Out]

-1/3*a^3/x^3+3*a^2*b*x+3/5*a*b^2*x^5+1/9*b^3*x^9

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Maxima [A]  time = 1.4398, size = 45, normalized size = 1.15 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{5} \, a b^{2} x^{5} + 3 \, a^{2} b x - \frac{a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3/x^4,x, algorithm="maxima")

[Out]

1/9*b^3*x^9 + 3/5*a*b^2*x^5 + 3*a^2*b*x - 1/3*a^3/x^3

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Fricas [A]  time = 0.215726, size = 50, normalized size = 1.28 \[ \frac{5 \, b^{3} x^{12} + 27 \, a b^{2} x^{8} + 135 \, a^{2} b x^{4} - 15 \, a^{3}}{45 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3/x^4,x, algorithm="fricas")

[Out]

1/45*(5*b^3*x^12 + 27*a*b^2*x^8 + 135*a^2*b*x^4 - 15*a^3)/x^3

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Sympy [A]  time = 1.08477, size = 36, normalized size = 0.92 \[ - \frac{a^{3}}{3 x^{3}} + 3 a^{2} b x + \frac{3 a b^{2} x^{5}}{5} + \frac{b^{3} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**3/x**4,x)

[Out]

-a**3/(3*x**3) + 3*a**2*b*x + 3*a*b**2*x**5/5 + b**3*x**9/9

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GIAC/XCAS [A]  time = 0.216426, size = 45, normalized size = 1.15 \[ \frac{1}{9} \, b^{3} x^{9} + \frac{3}{5} \, a b^{2} x^{5} + 3 \, a^{2} b x - \frac{a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^3/x^4,x, algorithm="giac")

[Out]

1/9*b^3*x^9 + 3/5*a*b^2*x^5 + 3*a^2*b*x - 1/3*a^3/x^3