3.2310 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^5}{x} \, dx\)

Optimal. Leaf size=65 \[ a^5 \log (x)+15 a^4 b \sqrt [3]{x}+15 a^3 b^2 x^{2/3}+10 a^2 b^3 x+\frac{15}{4} a b^4 x^{4/3}+\frac{3}{5} b^5 x^{5/3} \]

[Out]

15*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 10*a^2*b^3*x + (15*a*b^4*x^(4/3))/4 + (3
*b^5*x^(5/3))/5 + a^5*Log[x]

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Rubi [A]  time = 0.0767114, antiderivative size = 65, 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 \[ a^5 \log (x)+15 a^4 b \sqrt [3]{x}+15 a^3 b^2 x^{2/3}+10 a^2 b^3 x+\frac{15}{4} a b^4 x^{4/3}+\frac{3}{5} b^5 x^{5/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5/x,x]

[Out]

15*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 10*a^2*b^3*x + (15*a*b^4*x^(4/3))/4 + (3
*b^5*x^(5/3))/5 + a^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 3 a^{5} \log{\left (\sqrt [3]{x} \right )} + 15 a^{4} b \sqrt [3]{x} + 30 a^{3} b^{2} \int ^{\sqrt [3]{x}} x\, dx + 10 a^{2} b^{3} x + \frac{15 a b^{4} x^{\frac{4}{3}}}{4} + \frac{3 b^{5} x^{\frac{5}{3}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5*log(x**(1/3)) + 15*a**4*b*x**(1/3) + 30*a**3*b**2*Integral(x, (x, x**(1/3
))) + 10*a**2*b**3*x + 15*a*b**4*x**(4/3)/4 + 3*b**5*x**(5/3)/5

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Mathematica [A]  time = 0.0202114, size = 65, normalized size = 1. \[ a^5 \log (x)+15 a^4 b \sqrt [3]{x}+15 a^3 b^2 x^{2/3}+10 a^2 b^3 x+\frac{15}{4} a b^4 x^{4/3}+\frac{3}{5} b^5 x^{5/3} \]

Antiderivative was successfully verified.

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

[Out]

15*a^4*b*x^(1/3) + 15*a^3*b^2*x^(2/3) + 10*a^2*b^3*x + (15*a*b^4*x^(4/3))/4 + (3
*b^5*x^(5/3))/5 + a^5*Log[x]

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Maple [A]  time = 0.004, size = 54, normalized size = 0.8 \[ 15\,{a}^{4}b\sqrt [3]{x}+15\,{a}^{3}{b}^{2}{x}^{2/3}+10\,{a}^{2}{b}^{3}x+{\frac{15\,a{b}^{4}}{4}{x}^{{\frac{4}{3}}}}+{\frac{3\,{b}^{5}}{5}{x}^{{\frac{5}{3}}}}+{a}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5/x,x)

[Out]

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

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Maxima [A]  time = 1.52388, size = 72, normalized size = 1.11 \[ \frac{3}{5} \, b^{5} x^{\frac{5}{3}} + \frac{15}{4} \, a b^{4} x^{\frac{4}{3}} + 10 \, a^{2} b^{3} x + a^{5} \log \left (x\right ) + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218309, size = 76, normalized size = 1.17 \[ 10 \, a^{2} b^{3} x + 3 \, a^{5} \log \left (x^{\frac{1}{3}}\right ) + \frac{3}{5} \,{\left (b^{5} x + 25 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + \frac{15}{4} \,{\left (a b^{4} x + 4 \, a^{4} b\right )} x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*a^2*b^3*x + 3*a^5*log(x^(1/3)) + 3/5*(b^5*x + 25*a^3*b^2)*x^(2/3) + 15/4*(a*b
^4*x + 4*a^4*b)*x^(1/3)

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Sympy [A]  time = 3.52726, size = 66, normalized size = 1.02 \[ a^{5} \log{\left (x \right )} + 15 a^{4} b \sqrt [3]{x} + 15 a^{3} b^{2} x^{\frac{2}{3}} + 10 a^{2} b^{3} x + \frac{15 a b^{4} x^{\frac{4}{3}}}{4} + \frac{3 b^{5} x^{\frac{5}{3}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.239783, size = 73, normalized size = 1.12 \[ \frac{3}{5} \, b^{5} x^{\frac{5}{3}} + \frac{15}{4} \, a b^{4} x^{\frac{4}{3}} + 10 \, a^{2} b^{3} x + a^{5}{\rm ln}\left ({\left | x \right |}\right ) + 15 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/5*b^5*x^(5/3) + 15/4*a*b^4*x^(4/3) + 10*a^2*b^3*x + a^5*ln(abs(x)) + 15*a^3*b^
2*x^(2/3) + 15*a^4*b*x^(1/3)