3.2432 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}+\frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{18 b^2 \sqrt [3]{x}}{a^5}+\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3} \]

[Out]

(3*b^3)/(2*a^4*(a + b/x^(1/3))^2) + (12*b^3)/(a^5*(a + b/x^(1/3))) + (18*b^2*x^(
1/3))/a^5 - (9*b*x^(2/3))/(2*a^4) + x/a^3 - (30*b^3*Log[a + b/x^(1/3)])/a^6 - (1
0*b^3*Log[x])/a^6

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Rubi [A]  time = 0.159936, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{30 b^3 \log \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}+\frac{12 b^3}{a^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}+\frac{18 b^2 \sqrt [3]{x}}{a^5}+\frac{3 b^3}{2 a^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )^2}-\frac{9 b x^{2/3}}{2 a^4}+\frac{x}{a^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b/x^(1/3))^(-3),x]

[Out]

(3*b^3)/(2*a^4*(a + b/x^(1/3))^2) + (12*b^3)/(a^5*(a + b/x^(1/3))) + (18*b^2*x^(
1/3))/a^5 - (9*b*x^(2/3))/(2*a^4) + x/a^3 - (30*b^3*Log[a + b/x^(1/3)])/a^6 - (1
0*b^3*Log[x])/a^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 x}{2 a \left (a + \frac{b}{\sqrt [3]{x}}\right )^{2}} + \frac{30 b^{2} \int ^{\sqrt [3]{x}} \frac{1}{a^{3}}\, dx}{a^{2}} - \frac{15 x}{2 a^{2} \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{10 x}{a^{3}} - \frac{30 b \int ^{\sqrt [3]{x}} x\, dx}{a^{4}} - \frac{30 b^{3} \log{\left (a \sqrt [3]{x} + b \right )}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*x/(2*a*(a + b/x**(1/3))**2) + 30*b**2*Integral(a**(-3), (x, x**(1/3)))/a**2 -
15*x/(2*a**2*(a + b/x**(1/3))) + 10*x/a**3 - 30*b*Integral(x, (x, x**(1/3)))/a*
*4 - 30*b**3*log(a*x**(1/3) + b)/a**6

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Mathematica [A]  time = 0.0533127, size = 83, normalized size = 0.83 \[ \frac{2 a^3 x-9 a^2 b x^{2/3}+\frac{3 b^5}{\left (a \sqrt [3]{x}+b\right )^2}-\frac{30 b^4}{a \sqrt [3]{x}+b}-60 b^3 \log \left (a \sqrt [3]{x}+b\right )+36 a b^2 \sqrt [3]{x}}{2 a^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 77, normalized size = 0.8 \[{\frac{x}{{a}^{3}}}-{\frac{9\,b}{2\,{a}^{4}}{x}^{{\frac{2}{3}}}}+18\,{\frac{{b}^{2}\sqrt [3]{x}}{{a}^{5}}}-15\,{\frac{{b}^{4}}{ \left ( b+a\sqrt [3]{x} \right ){a}^{6}}}+{\frac{3\,{b}^{5}}{2\,{a}^{6}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-30\,{\frac{{b}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^3,x)

[Out]

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

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Maxima [A]  time = 1.42037, size = 136, normalized size = 1.36 \[ \frac{2 \, a^{4} - \frac{5 \, a^{3} b}{x^{\frac{1}{3}}} + \frac{20 \, a^{2} b^{2}}{x^{\frac{2}{3}}} + \frac{90 \, a b^{3}}{x} + \frac{60 \, b^{4}}{x^{\frac{4}{3}}}}{2 \,{\left (\frac{a^{7}}{x} + \frac{2 \, a^{6} b}{x^{\frac{4}{3}}} + \frac{a^{5} b^{2}}{x^{\frac{5}{3}}}\right )}} - \frac{30 \, b^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(2*a^4 - 5*a^3*b/x^(1/3) + 20*a^2*b^2/x^(2/3) + 90*a*b^3/x + 60*b^4/x^(4/3))
/(a^7/x + 2*a^6*b/x^(4/3) + a^5*b^2/x^(5/3)) - 30*b^3*log(a + b/x^(1/3))/a^6 - 1
0*b^3*log(x)/a^6

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Fricas [A]  time = 0.225781, size = 154, normalized size = 1.54 \[ \frac{20 \, a^{3} b^{2} x - 27 \, b^{5} - 60 \,{\left (a^{2} b^{3} x^{\frac{2}{3}} + 2 \, a b^{4} x^{\frac{1}{3}} + b^{5}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) +{\left (2 \, a^{5} x + 63 \, a^{2} b^{3}\right )} x^{\frac{2}{3}} -{\left (5 \, a^{4} b x - 6 \, a b^{4}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{8} x^{\frac{2}{3}} + 2 \, a^{7} b x^{\frac{1}{3}} + a^{6} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.90255, size = 364, normalized size = 3.64 \[ \begin{cases} \frac{2 a^{5} x^{\frac{5}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{5 a^{4} b x^{\frac{4}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac{20 a^{3} b^{2} x}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{60 a^{2} b^{3} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} + \frac{60 a^{2} b^{3} x^{\frac{2}{3}}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{120 a b^{4} \sqrt [3]{x} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{60 b^{5} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} - \frac{30 b^{5}}{2 a^{8} x^{\frac{2}{3}} + 4 a^{7} b \sqrt [3]{x} + 2 a^{6} b^{2}} & \text{for}\: a \neq 0 \\\frac{x^{2}}{2 b^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a**5*x**(5/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) -
5*a**4*b*x**(4/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) + 20*a**3
*b**2*x/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 60*a**2*b**3*x**(2
/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) + 60
*a**2*b**3*x**(2/3)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2) - 120*a*
b**4*x**(1/3)*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*
b**2) - 60*b**5*log(x**(1/3) + b/a)/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**
6*b**2) - 30*b**5/(2*a**8*x**(2/3) + 4*a**7*b*x**(1/3) + 2*a**6*b**2), Ne(a, 0))
, (x**2/(2*b**3), True))

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GIAC/XCAS [A]  time = 0.21633, size = 107, normalized size = 1.07 \[ -\frac{30 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{6}} - \frac{3 \,{\left (10 \, a b^{4} x^{\frac{1}{3}} + 9 \, b^{5}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{6}} + \frac{2 \, a^{6} x - 9 \, a^{5} b x^{\frac{2}{3}} + 36 \, a^{4} b^{2} x^{\frac{1}{3}}}{2 \, a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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