3.2368 \(\int \frac{x}{\left (a+b \sqrt [3]{x}\right )^3} \, dx\)

Optimal. Leaf size=90 \[ \frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3} \]

[Out]

(3*a^5)/(2*b^6*(a + b*x^(1/3))^2) - (15*a^4)/(b^6*(a + b*x^(1/3))) + (18*a^2*x^(
1/3))/b^5 - (9*a*x^(2/3))/(2*b^4) + x/b^3 - (30*a^3*Log[a + b*x^(1/3)])/b^6

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Rubi [A]  time = 0.147201, antiderivative size = 90, 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 \[ \frac{3 a^5}{2 b^6 \left (a+b \sqrt [3]{x}\right )^2}-\frac{15 a^4}{b^6 \left (a+b \sqrt [3]{x}\right )}-\frac{30 a^3 \log \left (a+b \sqrt [3]{x}\right )}{b^6}+\frac{18 a^2 \sqrt [3]{x}}{b^5}-\frac{9 a x^{2/3}}{2 b^4}+\frac{x}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^(1/3))^3,x]

[Out]

(3*a^5)/(2*b^6*(a + b*x^(1/3))^2) - (15*a^4)/(b^6*(a + b*x^(1/3))) + (18*a^2*x^(
1/3))/b^5 - (9*a*x^(2/3))/(2*b^4) + x/b^3 - (30*a^3*Log[a + b*x^(1/3)])/b^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{5}}{2 b^{6} \left (a + b \sqrt [3]{x}\right )^{2}} - \frac{15 a^{4}}{b^{6} \left (a + b \sqrt [3]{x}\right )} - \frac{30 a^{3} \log{\left (a + b \sqrt [3]{x} \right )}}{b^{6}} + \frac{18 a^{2} \sqrt [3]{x}}{b^{5}} - \frac{9 a \int ^{\sqrt [3]{x}} x\, dx}{b^{4}} + \frac{x}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a**5/(2*b**6*(a + b*x**(1/3))**2) - 15*a**4/(b**6*(a + b*x**(1/3))) - 30*a**3*
log(a + b*x**(1/3))/b**6 + 18*a**2*x**(1/3)/b**5 - 9*a*Integral(x, (x, x**(1/3))
)/b**4 + x/b**3

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Mathematica [A]  time = 0.0606054, size = 83, normalized size = 0.92 \[ \frac{\frac{3 a^5}{\left (a+b \sqrt [3]{x}\right )^2}-\frac{30 a^4}{a+b \sqrt [3]{x}}-60 a^3 \log \left (a+b \sqrt [3]{x}\right )+36 a^2 b \sqrt [3]{x}-9 a b^2 x^{2/3}+2 b^3 x}{2 b^6} \]

Antiderivative was successfully verified.

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

[Out]

((3*a^5)/(a + b*x^(1/3))^2 - (30*a^4)/(a + b*x^(1/3)) + 36*a^2*b*x^(1/3) - 9*a*b
^2*x^(2/3) + 2*b^3*x - 60*a^3*Log[a + b*x^(1/3)])/(2*b^6)

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Maple [A]  time = 0.004, size = 77, normalized size = 0.9 \[{\frac{3\,{a}^{5}}{2\,{b}^{6}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-15\,{\frac{{a}^{4}}{{b}^{6} \left ( a+b\sqrt [3]{x} \right ) }}+18\,{\frac{{a}^{2}\sqrt [3]{x}}{{b}^{5}}}-{\frac{9\,a}{2\,{b}^{4}}{x}^{{\frac{2}{3}}}}+{\frac{x}{{b}^{3}}}-30\,{\frac{{a}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b*x^(1/3))^3,x)

[Out]

3/2*a^5/b^6/(a+b*x^(1/3))^2-15*a^4/b^6/(a+b*x^(1/3))+18*a^2*x^(1/3)/b^5-9/2*a*x^
(2/3)/b^4+x/b^3-30*a^3*ln(a+b*x^(1/3))/b^6

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Maxima [A]  time = 1.43712, size = 127, normalized size = 1.41 \[ -\frac{30 \, a^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{b^{6}} + \frac{{\left (b x^{\frac{1}{3}} + a\right )}^{3}}{b^{6}} - \frac{15 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a}{2 \, b^{6}} + \frac{30 \,{\left (b x^{\frac{1}{3}} + a\right )} a^{2}}{b^{6}} - \frac{15 \, a^{4}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{6}} + \frac{3 \, a^{5}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.226502, size = 154, normalized size = 1.71 \[ \frac{20 \, a^{2} b^{3} x - 27 \, a^{5} - 60 \,{\left (a^{3} b^{2} x^{\frac{2}{3}} + 2 \, a^{4} b x^{\frac{1}{3}} + a^{5}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) +{\left (2 \, b^{5} x + 63 \, a^{3} b^{2}\right )} x^{\frac{2}{3}} -{\left (5 \, a b^{4} x - 6 \, a^{4} b\right )} x^{\frac{1}{3}}}{2 \,{\left (b^{8} x^{\frac{2}{3}} + 2 \, a b^{7} x^{\frac{1}{3}} + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(20*a^2*b^3*x - 27*a^5 - 60*(a^3*b^2*x^(2/3) + 2*a^4*b*x^(1/3) + a^5)*log(b*
x^(1/3) + a) + (2*b^5*x + 63*a^3*b^2)*x^(2/3) - (5*a*b^4*x - 6*a^4*b)*x^(1/3))/(
b^8*x^(2/3) + 2*a*b^7*x^(1/3) + a^2*b^6)

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Sympy [A]  time = 25.5478, size = 622, normalized size = 6.91 \[ - \frac{60 a^{6} x^{30} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} - \frac{180 a^{5} b x^{\frac{91}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} + \frac{60 a^{5} b x^{\frac{91}{3}}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} - \frac{180 a^{4} b^{2} x^{\frac{92}{3}} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} + \frac{150 a^{4} b^{2} x^{\frac{92}{3}}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} - \frac{60 a^{3} b^{3} x^{31} \log{\left (1 + \frac{b \sqrt [3]{x}}{a} \right )}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} + \frac{110 a^{3} b^{3} x^{31}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} + \frac{15 a^{2} b^{4} x^{\frac{94}{3}}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} - \frac{3 a b^{5} x^{\frac{95}{3}}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} + \frac{2 b^{6} x^{32}}{2 a^{3} b^{6} x^{30} + 6 a^{2} b^{7} x^{\frac{91}{3}} + 6 a b^{8} x^{\frac{92}{3}} + 2 b^{9} x^{31}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-60*a**6*x**30*log(1 + b*x**(1/3)/a)/(2*a**3*b**6*x**30 + 6*a**2*b**7*x**(91/3)
+ 6*a*b**8*x**(92/3) + 2*b**9*x**31) - 180*a**5*b*x**(91/3)*log(1 + b*x**(1/3)/a
)/(2*a**3*b**6*x**30 + 6*a**2*b**7*x**(91/3) + 6*a*b**8*x**(92/3) + 2*b**9*x**31
) + 60*a**5*b*x**(91/3)/(2*a**3*b**6*x**30 + 6*a**2*b**7*x**(91/3) + 6*a*b**8*x*
*(92/3) + 2*b**9*x**31) - 180*a**4*b**2*x**(92/3)*log(1 + b*x**(1/3)/a)/(2*a**3*
b**6*x**30 + 6*a**2*b**7*x**(91/3) + 6*a*b**8*x**(92/3) + 2*b**9*x**31) + 150*a*
*4*b**2*x**(92/3)/(2*a**3*b**6*x**30 + 6*a**2*b**7*x**(91/3) + 6*a*b**8*x**(92/3
) + 2*b**9*x**31) - 60*a**3*b**3*x**31*log(1 + b*x**(1/3)/a)/(2*a**3*b**6*x**30
+ 6*a**2*b**7*x**(91/3) + 6*a*b**8*x**(92/3) + 2*b**9*x**31) + 110*a**3*b**3*x**
31/(2*a**3*b**6*x**30 + 6*a**2*b**7*x**(91/3) + 6*a*b**8*x**(92/3) + 2*b**9*x**3
1) + 15*a**2*b**4*x**(94/3)/(2*a**3*b**6*x**30 + 6*a**2*b**7*x**(91/3) + 6*a*b**
8*x**(92/3) + 2*b**9*x**31) - 3*a*b**5*x**(95/3)/(2*a**3*b**6*x**30 + 6*a**2*b**
7*x**(91/3) + 6*a*b**8*x**(92/3) + 2*b**9*x**31) + 2*b**6*x**32/(2*a**3*b**6*x**
30 + 6*a**2*b**7*x**(91/3) + 6*a*b**8*x**(92/3) + 2*b**9*x**31)

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GIAC/XCAS [A]  time = 0.2179, size = 107, normalized size = 1.19 \[ -\frac{30 \, a^{3}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{6}} - \frac{3 \,{\left (10 \, a^{4} b x^{\frac{1}{3}} + 9 \, a^{5}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x - 9 \, a b^{5} x^{\frac{2}{3}} + 36 \, a^{2} b^{4} x^{\frac{1}{3}}}{2 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30*a^3*ln(abs(b*x^(1/3) + a))/b^6 - 3/2*(10*a^4*b*x^(1/3) + 9*a^5)/((b*x^(1/3)
+ a)^2*b^6) + 1/2*(2*b^6*x - 9*a*b^5*x^(2/3) + 36*a^2*b^4*x^(1/3))/b^9