3.2419 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right ) x^3} \, dx\)

Optimal. Leaf size=93 \[ \frac{3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{a^5 \log (x)}{b^6}-\frac{3 a^4}{b^5 \sqrt [3]{x}}+\frac{3 a^3}{2 b^4 x^{2/3}}-\frac{a^2}{b^3 x}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{3}{5 b x^{5/3}} \]

[Out]

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

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Rubi [A]  time = 0.126447, antiderivative size = 93, 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 \[ \frac{3 a^5 \log \left (a \sqrt [3]{x}+b\right )}{b^6}-\frac{a^5 \log (x)}{b^6}-\frac{3 a^4}{b^5 \sqrt [3]{x}}+\frac{3 a^3}{2 b^4 x^{2/3}}-\frac{a^2}{b^3 x}+\frac{3 a}{4 b^2 x^{4/3}}-\frac{3}{5 b x^{5/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))*x^3),x]

[Out]

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

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Rubi in Sympy [A]  time = 18.9429, size = 94, normalized size = 1.01 \[ - \frac{3 a^{5} \log{\left (\sqrt [3]{x} \right )}}{b^{6}} + \frac{3 a^{5} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{6}} - \frac{3 a^{4}}{b^{5} \sqrt [3]{x}} + \frac{3 a^{3}}{2 b^{4} x^{\frac{2}{3}}} - \frac{a^{2}}{b^{3} x} + \frac{3 a}{4 b^{2} x^{\frac{4}{3}}} - \frac{3}{5 b x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.103311, size = 84, normalized size = 0.9 \[ \frac{60 a^5 \log \left (a \sqrt [3]{x}+b\right )-20 a^5 \log (x)+\frac{b \left (-60 a^4 x^{4/3}+30 a^3 b x-20 a^2 b^2 x^{2/3}+15 a b^3 \sqrt [3]{x}-12 b^4\right )}{x^{5/3}}}{20 b^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 78, normalized size = 0.8 \[ -{\frac{3}{5\,b}{x}^{-{\frac{5}{3}}}}+{\frac{3\,a}{4\,{b}^{2}}{x}^{-{\frac{4}{3}}}}-{\frac{{a}^{2}}{{b}^{3}x}}+{\frac{3\,{a}^{3}}{2\,{b}^{4}}{x}^{-{\frac{2}{3}}}}-3\,{\frac{{a}^{4}}{{b}^{5}\sqrt [3]{x}}}+3\,{\frac{{a}^{5}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{6}}}-{\frac{{a}^{5}\ln \left ( x \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))/x^3,x)

[Out]

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

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Maxima [A]  time = 1.4422, size = 128, normalized size = 1.38 \[ \frac{3 \, a^{5} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{6}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{5}}{5 \, b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{4} a}{4 \, b^{6}} - \frac{10 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{3} a^{2}}{b^{6}} + \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2} a^{3}}{b^{6}} - \frac{15 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a^{4}}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237978, size = 109, normalized size = 1.17 \[ \frac{60 \, a^{5} x^{\frac{5}{3}} \log \left (a x^{\frac{1}{3}} + b\right ) - 60 \, a^{5} x^{\frac{5}{3}} \log \left (x^{\frac{1}{3}}\right ) + 30 \, a^{3} b^{2} x - 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 12 \, b^{5} - 15 \,{\left (4 \, a^{4} b x - a b^{4}\right )} x^{\frac{1}{3}}}{20 \, b^{6} x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 18.5777, size = 116, normalized size = 1.25 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 a x^{2}} & \text{for}\: b = 0 \\- \frac{3}{5 b x^{\frac{5}{3}}} & \text{for}\: a = 0 \\- \frac{a^{5} \log{\left (x \right )}}{b^{6}} + \frac{3 a^{5} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{6}} - \frac{3 a^{4}}{b^{5} \sqrt [3]{x}} + \frac{3 a^{3}}{2 b^{4} x^{\frac{2}{3}}} - \frac{a^{2}}{b^{3} x} + \frac{3 a}{4 b^{2} x^{\frac{4}{3}}} - \frac{3}{5 b x^{\frac{5}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x**(5/3), Eq(a, 0) & Eq(b, 0)), (-1/(2*a*x**2), Eq(b, 0)), (-3/(5
*b*x**(5/3)), Eq(a, 0)), (-a**5*log(x)/b**6 + 3*a**5*log(x**(1/3) + b/a)/b**6 -
3*a**4/(b**5*x**(1/3)) + 3*a**3/(2*b**4*x**(2/3)) - a**2/(b**3*x) + 3*a/(4*b**2*
x**(4/3)) - 3/(5*b*x**(5/3)), True))

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GIAC/XCAS [A]  time = 0.217522, size = 109, normalized size = 1.17 \[ \frac{3 \, a^{5}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{6}} - \frac{a^{5}{\rm ln}\left ({\left | x \right |}\right )}{b^{6}} - \frac{60 \, a^{4} b x^{\frac{4}{3}} - 30 \, a^{3} b^{2} x + 20 \, a^{2} b^{3} x^{\frac{2}{3}} - 15 \, a b^{4} x^{\frac{1}{3}} + 12 \, b^{5}}{20 \, b^{6} x^{\frac{5}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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