3.2314 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^5}{x^5} \, dx\)

Optimal. Leaf size=75 \[ -\frac{a^5}{4 x^4}-\frac{15 a^4 b}{11 x^{11/3}}-\frac{3 a^3 b^2}{x^{10/3}}-\frac{10 a^2 b^3}{3 x^3}-\frac{15 a b^4}{8 x^{8/3}}-\frac{3 b^5}{7 x^{7/3}} \]

[Out]

-a^5/(4*x^4) - (15*a^4*b)/(11*x^(11/3)) - (3*a^3*b^2)/x^(10/3) - (10*a^2*b^3)/(3
*x^3) - (15*a*b^4)/(8*x^(8/3)) - (3*b^5)/(7*x^(7/3))

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Rubi [A]  time = 0.0844925, antiderivative size = 75, 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 \[ -\frac{a^5}{4 x^4}-\frac{15 a^4 b}{11 x^{11/3}}-\frac{3 a^3 b^2}{x^{10/3}}-\frac{10 a^2 b^3}{3 x^3}-\frac{15 a b^4}{8 x^{8/3}}-\frac{3 b^5}{7 x^{7/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5/x^5,x]

[Out]

-a^5/(4*x^4) - (15*a^4*b)/(11*x^(11/3)) - (3*a^3*b^2)/x^(10/3) - (10*a^2*b^3)/(3
*x^3) - (15*a*b^4)/(8*x^(8/3)) - (3*b^5)/(7*x^(7/3))

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Rubi in Sympy [A]  time = 14.3219, size = 75, normalized size = 1. \[ - \frac{a^{5}}{4 x^{4}} - \frac{15 a^{4} b}{11 x^{\frac{11}{3}}} - \frac{3 a^{3} b^{2}}{x^{\frac{10}{3}}} - \frac{10 a^{2} b^{3}}{3 x^{3}} - \frac{15 a b^{4}}{8 x^{\frac{8}{3}}} - \frac{3 b^{5}}{7 x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**5/(4*x**4) - 15*a**4*b/(11*x**(11/3)) - 3*a**3*b**2/x**(10/3) - 10*a**2*b**3
/(3*x**3) - 15*a*b**4/(8*x**(8/3)) - 3*b**5/(7*x**(7/3))

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Mathematica [A]  time = 0.0242013, size = 67, normalized size = 0.89 \[ -\frac{462 a^5+2520 a^4 b \sqrt [3]{x}+5544 a^3 b^2 x^{2/3}+6160 a^2 b^3 x+3465 a b^4 x^{4/3}+792 b^5 x^{5/3}}{1848 x^4} \]

Antiderivative was successfully verified.

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

[Out]

-(462*a^5 + 2520*a^4*b*x^(1/3) + 5544*a^3*b^2*x^(2/3) + 6160*a^2*b^3*x + 3465*a*
b^4*x^(4/3) + 792*b^5*x^(5/3))/(1848*x^4)

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Maple [A]  time = 0.009, size = 58, normalized size = 0.8 \[ -{\frac{{a}^{5}}{4\,{x}^{4}}}-{\frac{15\,{a}^{4}b}{11}{x}^{-{\frac{11}{3}}}}-3\,{\frac{{a}^{3}{b}^{2}}{{x}^{10/3}}}-{\frac{10\,{a}^{2}{b}^{3}}{3\,{x}^{3}}}-{\frac{15\,a{b}^{4}}{8}{x}^{-{\frac{8}{3}}}}-{\frac{3\,{b}^{5}}{7}{x}^{-{\frac{7}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5/x^5,x)

[Out]

-1/4*a^5/x^4-15/11*a^4*b/x^(11/3)-3*a^3*b^2/x^(10/3)-10/3*a^2*b^3/x^3-15/8*a*b^4
/x^(8/3)-3/7*b^5/x^(7/3)

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Maxima [A]  time = 1.44568, size = 77, normalized size = 1.03 \[ -\frac{792 \, b^{5} x^{\frac{5}{3}} + 3465 \, a b^{4} x^{\frac{4}{3}} + 6160 \, a^{2} b^{3} x + 5544 \, a^{3} b^{2} x^{\frac{2}{3}} + 2520 \, a^{4} b x^{\frac{1}{3}} + 462 \, a^{5}}{1848 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1848*(792*b^5*x^(5/3) + 3465*a*b^4*x^(4/3) + 6160*a^2*b^3*x + 5544*a^3*b^2*x^
(2/3) + 2520*a^4*b*x^(1/3) + 462*a^5)/x^4

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Fricas [A]  time = 0.222502, size = 78, normalized size = 1.04 \[ -\frac{6160 \, a^{2} b^{3} x + 462 \, a^{5} + 792 \,{\left (b^{5} x + 7 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 315 \,{\left (11 \, a b^{4} x + 8 \, a^{4} b\right )} x^{\frac{1}{3}}}{1848 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1848*(6160*a^2*b^3*x + 462*a^5 + 792*(b^5*x + 7*a^3*b^2)*x^(2/3) + 315*(11*a*
b^4*x + 8*a^4*b)*x^(1/3))/x^4

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Sympy [A]  time = 14.6929, size = 75, normalized size = 1. \[ - \frac{a^{5}}{4 x^{4}} - \frac{15 a^{4} b}{11 x^{\frac{11}{3}}} - \frac{3 a^{3} b^{2}}{x^{\frac{10}{3}}} - \frac{10 a^{2} b^{3}}{3 x^{3}} - \frac{15 a b^{4}}{8 x^{\frac{8}{3}}} - \frac{3 b^{5}}{7 x^{\frac{7}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**5/(4*x**4) - 15*a**4*b/(11*x**(11/3)) - 3*a**3*b**2/x**(10/3) - 10*a**2*b**3
/(3*x**3) - 15*a*b**4/(8*x**(8/3)) - 3*b**5/(7*x**(7/3))

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GIAC/XCAS [A]  time = 0.242489, size = 77, normalized size = 1.03 \[ -\frac{792 \, b^{5} x^{\frac{5}{3}} + 3465 \, a b^{4} x^{\frac{4}{3}} + 6160 \, a^{2} b^{3} x + 5544 \, a^{3} b^{2} x^{\frac{2}{3}} + 2520 \, a^{4} b x^{\frac{1}{3}} + 462 \, a^{5}}{1848 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/1848*(792*b^5*x^(5/3) + 3465*a*b^4*x^(4/3) + 6160*a^2*b^3*x + 5544*a^3*b^2*x^
(2/3) + 2520*a^4*b*x^(1/3) + 462*a^5)/x^4