3.2360 \(\int \frac{x}{\left (a+b \sqrt [3]{x}\right )^2} \, dx\)

Optimal. Leaf size=85 \[ \frac{3 a^5}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{15 a^4 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{12 a^3 \sqrt [3]{x}}{b^5}+\frac{9 a^2 x^{2/3}}{2 b^4}-\frac{2 a x}{b^3}+\frac{3 x^{4/3}}{4 b^2} \]

[Out]

(3*a^5)/(b^6*(a + b*x^(1/3))) - (12*a^3*x^(1/3))/b^5 + (9*a^2*x^(2/3))/(2*b^4) -
(2*a*x)/b^3 + (3*x^(4/3))/(4*b^2) + (15*a^4*Log[a + b*x^(1/3)])/b^6

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Rubi [A]  time = 0.130913, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 a^5}{b^6 \left (a+b \sqrt [3]{x}\right )}+\frac{15 a^4 \log \left (a+b \sqrt [3]{x}\right )}{b^6}-\frac{12 a^3 \sqrt [3]{x}}{b^5}+\frac{9 a^2 x^{2/3}}{2 b^4}-\frac{2 a x}{b^3}+\frac{3 x^{4/3}}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^(1/3))^2,x]

[Out]

(3*a^5)/(b^6*(a + b*x^(1/3))) - (12*a^3*x^(1/3))/b^5 + (9*a^2*x^(2/3))/(2*b^4) -
(2*a*x)/b^3 + (3*x^(4/3))/(4*b^2) + (15*a^4*Log[a + b*x^(1/3)])/b^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{5}}{b^{6} \left (a + b \sqrt [3]{x}\right )} + \frac{15 a^{4} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{6}} - \frac{12 a^{3} \sqrt [3]{x}}{b^{5}} + \frac{9 a^{2} \int ^{\sqrt [3]{x}} x\, dx}{b^{4}} - \frac{2 a x}{b^{3}} + \frac{3 x^{\frac{4}{3}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5/(b**6*(a + b*x**(1/3))) + 15*a**4*log(a + b*x**(1/3))/b**6 - 12*a**3*x**(
1/3)/b**5 + 9*a**2*Integral(x, (x, x**(1/3)))/b**4 - 2*a*x/b**3 + 3*x**(4/3)/(4*
b**2)

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Mathematica [A]  time = 0.0466647, size = 80, normalized size = 0.94 \[ \frac{\frac{12 a^5}{a+b \sqrt [3]{x}}+60 a^4 \log \left (a+b \sqrt [3]{x}\right )-48 a^3 b \sqrt [3]{x}+18 a^2 b^2 x^{2/3}-8 a b^3 x+3 b^4 x^{4/3}}{4 b^6} \]

Antiderivative was successfully verified.

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

[Out]

((12*a^5)/(a + b*x^(1/3)) - 48*a^3*b*x^(1/3) + 18*a^2*b^2*x^(2/3) - 8*a*b^3*x +
3*b^4*x^(4/3) + 60*a^4*Log[a + b*x^(1/3)])/(4*b^6)

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Maple [A]  time = 0.003, size = 72, normalized size = 0.9 \[ 3\,{\frac{{a}^{5}}{{b}^{6} \left ( a+b\sqrt [3]{x} \right ) }}-12\,{\frac{{a}^{3}\sqrt [3]{x}}{{b}^{5}}}+{\frac{9\,{a}^{2}}{2\,{b}^{4}}{x}^{{\frac{2}{3}}}}-2\,{\frac{ax}{{b}^{3}}}+{\frac{3}{4\,{b}^{2}}{x}^{{\frac{4}{3}}}}+15\,{\frac{{a}^{4}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/3))^2,x)

[Out]

3*a^5/b^6/(a+b*x^(1/3))-12*a^3*x^(1/3)/b^5+9/2*a^2*x^(2/3)/b^4-2*a*x/b^3+3/4*x^(
4/3)/b^2+15*a^4*ln(a+b*x^(1/3))/b^6

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Maxima [A]  time = 1.44379, size = 128, normalized size = 1.51 \[ \frac{15 \, a^{4} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{6}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}^{4}}{4 \, b^{6}} - \frac{5 \,{\left (b x^{\frac{1}{3}} + a\right )}^{3} a}{b^{6}} + \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{2}}{b^{6}} - \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{3}}{b^{6}} + \frac{3 \, a^{5}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*a^4*log(b*x^(1/3) + a)/b^6 + 3/4*(b*x^(1/3) + a)^4/b^6 - 5*(b*x^(1/3) + a)^3*
a/b^6 + 15*(b*x^(1/3) + a)^2*a^2/b^6 - 30*(b*x^(1/3) + a)*a^3/b^6 + 3*a^5/((b*x^
(1/3) + a)*b^6)

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Fricas [A]  time = 0.218904, size = 124, normalized size = 1.46 \[ \frac{10 \, a^{2} b^{3} x + 12 \, a^{5} + 60 \,{\left (a^{4} b x^{\frac{1}{3}} + a^{5}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 3 \,{\left (b^{5} x - 10 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} -{\left (5 \, a b^{4} x + 48 \, a^{4} b\right )} x^{\frac{1}{3}}}{4 \,{\left (b^{7} x^{\frac{1}{3}} + a b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(10*a^2*b^3*x + 12*a^5 + 60*(a^4*b*x^(1/3) + a^5)*log(b*x^(1/3) + a) + 3*(b^
5*x - 10*a^3*b^2)*x^(2/3) - (5*a*b^4*x + 48*a^4*b)*x^(1/3))/(b^7*x^(1/3) + a*b^6
)

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Sympy [A]  time = 24.9744, size = 243, normalized size = 2.86 \[ \frac{60 a^{5} x^{\frac{80}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} + \frac{60 a^{4} b x^{27} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} - \frac{60 a^{4} b x^{27}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} - \frac{30 a^{3} b^{2} x^{\frac{82}{3}}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} + \frac{10 a^{2} b^{3} x^{\frac{83}{3}}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} - \frac{5 a b^{4} x^{28}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} + \frac{3 b^{5} x^{\frac{85}{3}}}{4 a b^{6} x^{\frac{80}{3}} + 4 b^{7} x^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

60*a**5*x**(80/3)*log(1 + b*x**(1/3)/a)/(4*a*b**6*x**(80/3) + 4*b**7*x**27) + 60
*a**4*b*x**27*log(1 + b*x**(1/3)/a)/(4*a*b**6*x**(80/3) + 4*b**7*x**27) - 60*a**
4*b*x**27/(4*a*b**6*x**(80/3) + 4*b**7*x**27) - 30*a**3*b**2*x**(82/3)/(4*a*b**6
*x**(80/3) + 4*b**7*x**27) + 10*a**2*b**3*x**(83/3)/(4*a*b**6*x**(80/3) + 4*b**7
*x**27) - 5*a*b**4*x**28/(4*a*b**6*x**(80/3) + 4*b**7*x**27) + 3*b**5*x**(85/3)/
(4*a*b**6*x**(80/3) + 4*b**7*x**27)

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GIAC/XCAS [A]  time = 0.21626, size = 105, normalized size = 1.24 \[ \frac{15 \, a^{4}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{6}} + \frac{3 \, a^{5}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} + \frac{3 \, b^{6} x^{\frac{4}{3}} - 8 \, a b^{5} x + 18 \, a^{2} b^{4} x^{\frac{2}{3}} - 48 \, a^{3} b^{3} x^{\frac{1}{3}}}{4 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

15*a^4*ln(abs(b*x^(1/3) + a))/b^6 + 3*a^5/((b*x^(1/3) + a)*b^6) + 1/4*(3*b^6*x^(
4/3) - 8*a*b^5*x + 18*a^2*b^4*x^(2/3) - 48*a^3*b^3*x^(1/3))/b^8