3.2415 \(\int \frac{x}{a+\frac{b}{\sqrt [3]{x}}} \, dx\)

Optimal. Leaf size=94 \[ \frac{3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac{3 b^5 \sqrt [3]{x}}{a^6}+\frac{3 b^4 x^{2/3}}{2 a^5}-\frac{b^3 x}{a^4}+\frac{3 b^2 x^{4/3}}{4 a^3}-\frac{3 b x^{5/3}}{5 a^2}+\frac{x^2}{2 a} \]

[Out]

(-3*b^5*x^(1/3))/a^6 + (3*b^4*x^(2/3))/(2*a^5) - (b^3*x)/a^4 + (3*b^2*x^(4/3))/(
4*a^3) - (3*b*x^(5/3))/(5*a^2) + x^2/(2*a) + (3*b^6*Log[b + a*x^(1/3)])/a^7

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Rubi [A]  time = 0.135585, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 b^6 \log \left (a \sqrt [3]{x}+b\right )}{a^7}-\frac{3 b^5 \sqrt [3]{x}}{a^6}+\frac{3 b^4 x^{2/3}}{2 a^5}-\frac{b^3 x}{a^4}+\frac{3 b^2 x^{4/3}}{4 a^3}-\frac{3 b x^{5/3}}{5 a^2}+\frac{x^2}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^(1/3)),x]

[Out]

(-3*b^5*x^(1/3))/a^6 + (3*b^4*x^(2/3))/(2*a^5) - (b^3*x)/a^4 + (3*b^2*x^(4/3))/(
4*a^3) - (3*b*x^(5/3))/(5*a^2) + x^2/(2*a) + (3*b^6*Log[b + a*x^(1/3)])/a^7

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 3 b^{5} \int ^{\sqrt [3]{x}} \frac{1}{a^{6}}\, dx + \frac{x^{2}}{2 a} - \frac{3 b x^{\frac{5}{3}}}{5 a^{2}} + \frac{3 b^{2} x^{\frac{4}{3}}}{4 a^{3}} - \frac{b^{3} x}{a^{4}} + \frac{3 b^{4} \int ^{\sqrt [3]{x}} x\, dx}{a^{5}} + \frac{3 b^{6} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b**5*Integral(a**(-6), (x, x**(1/3))) + x**2/(2*a) - 3*b*x**(5/3)/(5*a**2) +
3*b**2*x**(4/3)/(4*a**3) - b**3*x/a**4 + 3*b**4*Integral(x, (x, x**(1/3)))/a**5
+ 3*b**6*log(a*x**(1/3) + b)/a**7

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Mathematica [A]  time = 0.0253235, size = 88, normalized size = 0.94 \[ \frac{10 a^6 x^2-12 a^5 b x^{5/3}+15 a^4 b^2 x^{4/3}-20 a^3 b^3 x+30 a^2 b^4 x^{2/3}+60 b^6 \log \left (a \sqrt [3]{x}+b\right )-60 a b^5 \sqrt [3]{x}}{20 a^7} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^(1/3)),x]

[Out]

(-60*a*b^5*x^(1/3) + 30*a^2*b^4*x^(2/3) - 20*a^3*b^3*x + 15*a^4*b^2*x^(4/3) - 12
*a^5*b*x^(5/3) + 10*a^6*x^2 + 60*b^6*Log[b + a*x^(1/3)])/(20*a^7)

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Maple [A]  time = 0.005, size = 77, normalized size = 0.8 \[ -3\,{\frac{{b}^{5}\sqrt [3]{x}}{{a}^{6}}}+{\frac{3\,{b}^{4}}{2\,{a}^{5}}{x}^{{\frac{2}{3}}}}-{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{3\,{b}^{2}}{4\,{a}^{3}}{x}^{{\frac{4}{3}}}}-{\frac{3\,b}{5\,{a}^{2}}{x}^{{\frac{5}{3}}}}+{\frac{{x}^{2}}{2\,a}}+3\,{\frac{{b}^{6}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^(1/3)),x)

[Out]

-3*b^5*x^(1/3)/a^6+3/2*b^4*x^(2/3)/a^5-b^3*x/a^4+3/4*b^2*x^(4/3)/a^3-3/5*b*x^(5/
3)/a^2+1/2*x^2/a+3*b^6*ln(b+a*x^(1/3))/a^7

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Maxima [A]  time = 1.43698, size = 119, normalized size = 1.27 \[ \frac{3 \, b^{6} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{7}} + \frac{b^{6} \log \left (x\right )}{a^{7}} + \frac{{\left (10 \, a^{5} - \frac{12 \, a^{4} b}{x^{\frac{1}{3}}} + \frac{15 \, a^{3} b^{2}}{x^{\frac{2}{3}}} - \frac{20 \, a^{2} b^{3}}{x} + \frac{30 \, a b^{4}}{x^{\frac{4}{3}}} - \frac{60 \, b^{5}}{x^{\frac{5}{3}}}\right )} x^{2}}{20 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b^6*log(a + b/x^(1/3))/a^7 + b^6*log(x)/a^7 + 1/20*(10*a^5 - 12*a^4*b/x^(1/3)
+ 15*a^3*b^2/x^(2/3) - 20*a^2*b^3/x + 30*a*b^4/x^(4/3) - 60*b^5/x^(5/3))*x^2/a^6

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Fricas [A]  time = 0.226243, size = 104, normalized size = 1.11 \[ \frac{10 \, a^{6} x^{2} - 20 \, a^{3} b^{3} x + 60 \, b^{6} \log \left (a x^{\frac{1}{3}} + b\right ) - 6 \,{\left (2 \, a^{5} b x - 5 \, a^{2} b^{4}\right )} x^{\frac{2}{3}} + 15 \,{\left (a^{4} b^{2} x - 4 \, a b^{5}\right )} x^{\frac{1}{3}}}{20 \, a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(10*a^6*x^2 - 20*a^3*b^3*x + 60*b^6*log(a*x^(1/3) + b) - 6*(2*a^5*b*x - 5*a
^2*b^4)*x^(2/3) + 15*(a^4*b^2*x - 4*a*b^5)*x^(1/3))/a^7

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Sympy [A]  time = 35.3014, size = 92, normalized size = 0.98 \[ \frac{x^{2}}{2 a} - \frac{3 b x^{\frac{5}{3}}}{5 a^{2}} + \frac{3 b^{2} x^{\frac{4}{3}}}{4 a^{3}} - \frac{b^{3} x}{a^{4}} + \frac{3 b^{4} x^{\frac{2}{3}}}{2 a^{5}} - \frac{3 b^{5} \sqrt [3]{x}}{a^{6}} + \frac{3 b^{6} \log{\left (\frac{a \sqrt [3]{x}}{b} + 1 \right )}}{a^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**(1/3)),x)

[Out]

x**2/(2*a) - 3*b*x**(5/3)/(5*a**2) + 3*b**2*x**(4/3)/(4*a**3) - b**3*x/a**4 + 3*
b**4*x**(2/3)/(2*a**5) - 3*b**5*x**(1/3)/a**6 + 3*b**6*log(a*x**(1/3)/b + 1)/a**
7

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GIAC/XCAS [A]  time = 0.216151, size = 105, normalized size = 1.12 \[ \frac{3 \, b^{6}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{7}} + \frac{10 \, a^{5} x^{2} - 12 \, a^{4} b x^{\frac{5}{3}} + 15 \, a^{3} b^{2} x^{\frac{4}{3}} - 20 \, a^{2} b^{3} x + 30 \, a b^{4} x^{\frac{2}{3}} - 60 \, b^{5} x^{\frac{1}{3}}}{20 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*b^6*ln(abs(a*x^(1/3) + b))/a^7 + 1/20*(10*a^5*x^2 - 12*a^4*b*x^(5/3) + 15*a^3*
b^2*x^(4/3) - 20*a^2*b^3*x + 30*a*b^4*x^(2/3) - 60*b^5*x^(1/3))/a^6