3.2295 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^2}{x^4} \, dx\)

Optimal. Leaf size=34 \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{4 x^{8/3}}-\frac{3 b^2}{7 x^{7/3}} \]

[Out]

-a^2/(3*x^3) - (3*a*b)/(4*x^(8/3)) - (3*b^2)/(7*x^(7/3))

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Rubi [A]  time = 0.0436681, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{4 x^{8/3}}-\frac{3 b^2}{7 x^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^2/x^4,x]

[Out]

-a^2/(3*x^3) - (3*a*b)/(4*x^(8/3)) - (3*b^2)/(7*x^(7/3))

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Rubi in Sympy [A]  time = 6.90215, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{3 x^{3}} - \frac{3 a b}{4 x^{\frac{8}{3}}} - \frac{3 b^{2}}{7 x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**2/x**4,x)

[Out]

-a**2/(3*x**3) - 3*a*b/(4*x**(8/3)) - 3*b**2/(7*x**(7/3))

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Mathematica [A]  time = 0.0150763, size = 34, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{3 a b}{4 x^{8/3}}-\frac{3 b^2}{7 x^{7/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^2/x^4,x]

[Out]

-a^2/(3*x^3) - (3*a*b)/(4*x^(8/3)) - (3*b^2)/(7*x^(7/3))

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{{a}^{2}}{3\,{x}^{3}}}-{\frac{3\,ab}{4}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{b}^{2}}{7}{x}^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^2/x^4,x)

[Out]

-1/3*a^2/x^3-3/4*a*b/x^(8/3)-3/7*b^2/x^(7/3)

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Maxima [A]  time = 1.41781, size = 35, normalized size = 1.03 \[ -\frac{36 \, b^{2} x^{\frac{2}{3}} + 63 \, a b x^{\frac{1}{3}} + 28 \, a^{2}}{84 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^2/x^4,x, algorithm="maxima")

[Out]

-1/84*(36*b^2*x^(2/3) + 63*a*b*x^(1/3) + 28*a^2)/x^3

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Fricas [A]  time = 0.212264, size = 35, normalized size = 1.03 \[ -\frac{36 \, b^{2} x^{\frac{2}{3}} + 63 \, a b x^{\frac{1}{3}} + 28 \, a^{2}}{84 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^2/x^4,x, algorithm="fricas")

[Out]

-1/84*(36*b^2*x^(2/3) + 63*a*b*x^(1/3) + 28*a^2)/x^3

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Sympy [A]  time = 7.37126, size = 32, normalized size = 0.94 \[ - \frac{a^{2}}{3 x^{3}} - \frac{3 a b}{4 x^{\frac{8}{3}}} - \frac{3 b^{2}}{7 x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**2/x**4,x)

[Out]

-a**2/(3*x**3) - 3*a*b/(4*x**(8/3)) - 3*b**2/(7*x**(7/3))

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GIAC/XCAS [A]  time = 0.22022, size = 35, normalized size = 1.03 \[ -\frac{36 \, b^{2} x^{\frac{2}{3}} + 63 \, a b x^{\frac{1}{3}} + 28 \, a^{2}}{84 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^2/x^4,x, algorithm="giac")

[Out]

-1/84*(36*b^2*x^(2/3) + 63*a*b*x^(1/3) + 28*a^2)/x^3