3.2311 \(\int \frac{\left (a+b \sqrt [3]{x}\right )^5}{x^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{a^5}{x}-\frac{15 a^4 b}{2 x^{2/3}}-\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]

[Out]

-(a^5/x) - (15*a^4*b)/(2*x^(2/3)) - (30*a^3*b^2)/x^(1/3) + 15*a*b^4*x^(1/3) + (3
*b^5*x^(2/3))/2 + 10*a^2*b^3*Log[x]

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Rubi [A]  time = 0.0839101, antiderivative size = 68, 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 \[ -\frac{a^5}{x}-\frac{15 a^4 b}{2 x^{2/3}}-\frac{30 a^3 b^2}{\sqrt [3]{x}}+10 a^2 b^3 \log (x)+15 a b^4 \sqrt [3]{x}+\frac{3}{2} b^5 x^{2/3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^5/x^2,x]

[Out]

-(a^5/x) - (15*a^4*b)/(2*x^(2/3)) - (30*a^3*b^2)/x^(1/3) + 15*a*b^4*x^(1/3) + (3
*b^5*x^(2/3))/2 + 10*a^2*b^3*Log[x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*x**(1/3))**5/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0229316, size = 69, normalized size = 1.01 \[ \frac{-2 a^5-15 a^4 b \sqrt [3]{x}-60 a^3 b^2 x^{2/3}+20 a^2 b^3 x \log (x)+30 a b^4 x^{4/3}+3 b^5 x^{5/3}}{2 x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^5/x^2,x]

[Out]

(-2*a^5 - 15*a^4*b*x^(1/3) - 60*a^3*b^2*x^(2/3) + 30*a*b^4*x^(4/3) + 3*b^5*x^(5/
3) + 20*a^2*b^3*x*Log[x])/(2*x)

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Maple [A]  time = 0.01, size = 57, normalized size = 0.8 \[ -{\frac{{a}^{5}}{x}}-{\frac{15\,{a}^{4}b}{2}{x}^{-{\frac{2}{3}}}}-30\,{\frac{{a}^{3}{b}^{2}}{\sqrt [3]{x}}}+15\,a{b}^{4}\sqrt [3]{x}+{\frac{3\,{b}^{5}}{2}{x}^{{\frac{2}{3}}}}+10\,{a}^{2}{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^(1/3))^5/x^2,x)

[Out]

-a^5/x-15/2*a^4*b/x^(2/3)-30*a^3*b^2/x^(1/3)+15*a*b^4*x^(1/3)+3/2*b^5*x^(2/3)+10
*a^2*b^3*ln(x)

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Maxima [A]  time = 1.43707, size = 80, normalized size = 1.18 \[ 10 \, a^{2} b^{3} \log \left (x\right ) + \frac{3}{2} \, b^{5} x^{\frac{2}{3}} + 15 \, a b^{4} x^{\frac{1}{3}} - \frac{60 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5/x^2,x, algorithm="maxima")

[Out]

10*a^2*b^3*log(x) + 3/2*b^5*x^(2/3) + 15*a*b^4*x^(1/3) - 1/2*(60*a^3*b^2*x^(2/3)
+ 15*a^4*b*x^(1/3) + 2*a^5)/x

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Fricas [A]  time = 0.218582, size = 84, normalized size = 1.24 \[ \frac{60 \, a^{2} b^{3} x \log \left (x^{\frac{1}{3}}\right ) - 2 \, a^{5} + 3 \,{\left (b^{5} x - 20 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} + 15 \,{\left (2 \, a b^{4} x - a^{4} b\right )} x^{\frac{1}{3}}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5/x^2,x, algorithm="fricas")

[Out]

1/2*(60*a^2*b^3*x*log(x^(1/3)) - 2*a^5 + 3*(b^5*x - 20*a^3*b^2)*x^(2/3) + 15*(2*
a*b^4*x - a^4*b)*x^(1/3))/x

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Sympy [A]  time = 2.37137, size = 66, normalized size = 0.97 \[ - \frac{a^{5}}{x} - \frac{15 a^{4} b}{2 x^{\frac{2}{3}}} - \frac{30 a^{3} b^{2}}{\sqrt [3]{x}} + 10 a^{2} b^{3} \log{\left (x \right )} + 15 a b^{4} \sqrt [3]{x} + \frac{3 b^{5} x^{\frac{2}{3}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**(1/3))**5/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.242628, size = 81, normalized size = 1.19 \[ 10 \, a^{2} b^{3}{\rm ln}\left ({\left | x \right |}\right ) + \frac{3}{2} \, b^{5} x^{\frac{2}{3}} + 15 \, a b^{4} x^{\frac{1}{3}} - \frac{60 \, a^{3} b^{2} x^{\frac{2}{3}} + 15 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^5/x^2,x, algorithm="giac")

[Out]

10*a^2*b^3*ln(abs(x)) + 3/2*b^5*x^(2/3) + 15*a*b^4*x^(1/3) - 1/2*(60*a^3*b^2*x^(
2/3) + 15*a^4*b*x^(1/3) + 2*a^5)/x