3.2435 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x^3} \, dx\)

Optimal. Leaf size=103 \[ \frac{30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}-\frac{12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{1}{b^3 x} \]

[Out]

(-3*a^3)/(2*b^4*(b + a*x^(1/3))^2) - (12*a^3)/(b^5*(b + a*x^(1/3))) - 1/(b^3*x)
+ (9*a)/(2*b^4*x^(2/3)) - (18*a^2)/(b^5*x^(1/3)) + (30*a^3*Log[b + a*x^(1/3)])/b
^6 - (10*a^3*Log[x])/b^6

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Rubi [A]  time = 0.168791, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{30 a^3 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{10 a^3 \log (x)}{b^6}-\frac{12 a^3}{b^5 \left (a \sqrt [3]{x}+b\right )}-\frac{3 a^3}{2 b^4 \left (a \sqrt [3]{x}+b\right )^2}-\frac{18 a^2}{b^5 \sqrt [3]{x}}+\frac{9 a}{2 b^4 x^{2/3}}-\frac{1}{b^3 x} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^3)/(2*b^4*(b + a*x^(1/3))^2) - (12*a^3)/(b^5*(b + a*x^(1/3))) - 1/(b^3*x)
+ (9*a)/(2*b^4*x^(2/3)) - (18*a^2)/(b^5*x^(1/3)) + (30*a^3*Log[b + a*x^(1/3)])/b
^6 - (10*a^3*Log[x])/b^6

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Rubi in Sympy [A]  time = 22.2117, size = 104, normalized size = 1.01 \[ - \frac{3 a^{3}}{2 b^{4} \left (a \sqrt [3]{x} + b\right )^{2}} - \frac{12 a^{3}}{b^{5} \left (a \sqrt [3]{x} + b\right )} - \frac{30 a^{3} \log{\left (\sqrt [3]{x} \right )}}{b^{6}} + \frac{30 a^{3} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{6}} - \frac{18 a^{2}}{b^{5} \sqrt [3]{x}} + \frac{9 a}{2 b^{4} x^{\frac{2}{3}}} - \frac{1}{b^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*a**3/(2*b**4*(a*x**(1/3) + b)**2) - 12*a**3/(b**5*(a*x**(1/3) + b)) - 30*a**3
*log(x**(1/3))/b**6 + 30*a**3*log(a*x**(1/3) + b)/b**6 - 18*a**2/(b**5*x**(1/3))
+ 9*a/(2*b**4*x**(2/3)) - 1/(b**3*x)

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Mathematica [A]  time = 0.208748, size = 93, normalized size = 0.9 \[ -\frac{-60 a^3 \log \left (a \sqrt [3]{x}+b\right )+20 a^3 \log (x)+\frac{b \left (60 a^4 x^{4/3}+90 a^3 b x+20 a^2 b^2 x^{2/3}-5 a b^3 \sqrt [3]{x}+2 b^4\right )}{x \left (a \sqrt [3]{x}+b\right )^2}}{2 b^6} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(2*b^4 - 5*a*b^3*x^(1/3) + 20*a^2*b^2*x^(2/3) + 90*a^3*b*x + 60*a^4*x^(4/3)
))/((b + a*x^(1/3))^2*x) - 60*a^3*Log[b + a*x^(1/3)] + 20*a^3*Log[x])/(2*b^6)

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Maple [A]  time = 0.018, size = 90, normalized size = 0.9 \[ -{\frac{3\,{a}^{3}}{2\,{b}^{4}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-12\,{\frac{{a}^{3}}{{b}^{5} \left ( b+a\sqrt [3]{x} \right ) }}-{\frac{1}{{b}^{3}x}}+{\frac{9\,a}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}-18\,{\frac{{a}^{2}}{{b}^{5}\sqrt [3]{x}}}+30\,{\frac{{a}^{3}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}-10\,{\frac{{a}^{3}\ln \left ( x \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^3/x^3,x)

[Out]

-3/2*a^3/b^4/(b+a*x^(1/3))^2-12*a^3/b^5/(b+a*x^(1/3))-1/b^3/x+9/2*a/b^4/x^(2/3)-
18*a^2/b^5/x^(1/3)+30*a^3*ln(b+a*x^(1/3))/b^6-10*a^3*ln(x)/b^6

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Maxima [A]  time = 1.44807, size = 128, normalized size = 1.24 \[ \frac{30 \, a^{3} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a}{2 \, b^{6}} - \frac{30 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{2}}{b^{6}} + \frac{15 \, a^{4}}{{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} b^{6}} - \frac{3 \, a^{5}}{2 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236942, size = 189, normalized size = 1.83 \[ -\frac{90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} + 2 \, b^{5} - 60 \,{\left (a^{5} x^{\frac{5}{3}} + 2 \, a^{4} b x^{\frac{4}{3}} + a^{3} b^{2} x\right )} \log \left (a x^{\frac{1}{3}} + b\right ) + 60 \,{\left (a^{5} x^{\frac{5}{3}} + 2 \, a^{4} b x^{\frac{4}{3}} + a^{3} b^{2} x\right )} \log \left (x^{\frac{1}{3}}\right ) + 5 \,{\left (12 \, a^{4} b x - a b^{4}\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{2} b^{6} x^{\frac{5}{3}} + 2 \, a b^{7} x^{\frac{4}{3}} + b^{8} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(90*a^3*b^2*x + 20*a^2*b^3*x^(2/3) + 2*b^5 - 60*(a^5*x^(5/3) + 2*a^4*b*x^(4
/3) + a^3*b^2*x)*log(a*x^(1/3) + b) + 60*(a^5*x^(5/3) + 2*a^4*b*x^(4/3) + a^3*b^
2*x)*log(x^(1/3)) + 5*(12*a^4*b*x - a*b^4)*x^(1/3))/(a^2*b^6*x^(5/3) + 2*a*b^7*x
^(4/3) + b^8*x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.217029, size = 122, normalized size = 1.18 \[ \frac{30 \, a^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} - \frac{10 \, a^{3}{\rm ln}\left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{\frac{4}{3}} + 90 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 5 \, a b^{4} x^{\frac{1}{3}} + 2 \, b^{5}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} b^{6} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

30*a^3*ln(abs(a*x^(1/3) + b))/b^6 - 10*a^3*ln(abs(x))/b^6 - 1/2*(60*a^4*b*x^(4/3
) + 90*a^3*b^2*x + 20*a^2*b^3*x^(2/3) - 5*a*b^4*x^(1/3) + 2*b^5)/((a*x^(1/3) + b
)^2*b^6*x)