3.2312 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^5}{x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \]

[Out]

-(a + b*x^(1/3))^6/(2*a*x^2)

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Rubi [A]  time = 0.0156913, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{\left (a+b \sqrt [3]{x}\right )^6}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5/x^3,x]

[Out]

-(a + b*x^(1/3))^6/(2*a*x^2)

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Rubi in Sympy [A]  time = 2.79034, size = 17, normalized size = 0.81 \[ - \frac{\left (a + b \sqrt [3]{x}\right )^{6}}{2 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**5/x**3,x)

[Out]

-(a + b*x**(1/3))**6/(2*a*x**2)

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Mathematica [B]  time = 0.0203644, size = 65, normalized size = 3.1 \[ -\frac{a^5+6 a^4 b \sqrt [3]{x}+15 a^3 b^2 x^{2/3}+20 a^2 b^3 x+15 a b^4 x^{4/3}+6 b^5 x^{5/3}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^5/x^3,x]

[Out]

-(a^5 + 6*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 20*a^2*b^3*x + 15*a*b^4*x^(4/3) +
6*b^5*x^(5/3))/(2*x^2)

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Maple [B]  time = 0.01, size = 58, normalized size = 2.8 \[ -{\frac{{a}^{5}}{2\,{x}^{2}}}-{\frac{15\,{a}^{3}{b}^{2}}{2}{x}^{-{\frac{4}{3}}}}-10\,{\frac{{a}^{2}{b}^{3}}{x}}-{\frac{15\,a{b}^{4}}{2}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{a}^{4}b}{{x}^{5/3}}}-3\,{\frac{{b}^{5}}{\sqrt [3]{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5/x^3,x)

[Out]

-1/2*a^5/x^2-15/2*a^3*b^2/x^(4/3)-10*a^2*b^3/x-15/2*a*b^4/x^(2/3)-3*a^4*b/x^(5/3
)-3*b^5/x^(1/3)

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Maxima [A]  time = 1.44903, size = 74, normalized size = 3.52 \[ -\frac{6 \, b^{5} x^{\frac{5}{3}} + 15 \, a b^{4} x^{\frac{4}{3}} + 20 \, a^{2} b^{3} x + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 6 \, a^{4} b x^{\frac{1}{3}} + a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5/x^3,x, algorithm="maxima")

[Out]

-1/2*(6*b^5*x^(5/3) + 15*a*b^4*x^(4/3) + 20*a^2*b^3*x + 15*a^3*b^2*x^(2/3) + 6*a
^4*b*x^(1/3) + a^5)/x^2

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Fricas [A]  time = 0.215366, size = 77, normalized size = 3.67 \[ -\frac{20 \, a^{2} b^{3} x + a^{5} + 3 \,{\left (2 \, b^{5} x + 5 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 3 \,{\left (5 \, a b^{4} x + 2 \, a^{4} b\right )} x^{\frac{1}{3}}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.22533, size = 70, normalized size = 3.33 \[ - \frac{a^{5}}{2 x^{2}} - \frac{3 a^{4} b}{x^{\frac{5}{3}}} - \frac{15 a^{3} b^{2}}{2 x^{\frac{4}{3}}} - \frac{10 a^{2} b^{3}}{x} - \frac{15 a b^{4}}{2 x^{\frac{2}{3}}} - \frac{3 b^{5}}{\sqrt [3]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**5/x**3,x)

[Out]

-a**5/(2*x**2) - 3*a**4*b/x**(5/3) - 15*a**3*b**2/(2*x**(4/3)) - 10*a**2*b**3/x
- 15*a*b**4/(2*x**(2/3)) - 3*b**5/x**(1/3)

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GIAC/XCAS [A]  time = 0.258683, size = 74, normalized size = 3.52 \[ -\frac{6 \, b^{5} x^{\frac{5}{3}} + 15 \, a b^{4} x^{\frac{4}{3}} + 20 \, a^{2} b^{3} x + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 6 \, a^{4} b x^{\frac{1}{3}} + a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5/x^3,x, algorithm="giac")

[Out]

-1/2*(6*b^5*x^(5/3) + 15*a*b^4*x^(4/3) + 20*a^2*b^3*x + 15*a^3*b^2*x^(2/3) + 6*a
^4*b*x^(1/3) + a^5)/x^2