3.2370 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^3 x} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{3}{a^2 \left (a+b \sqrt [3]{x}\right )}+\frac{3}{2 a \left (a+b \sqrt [3]{x}\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0791987, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{a^3}+\frac{\log (x)}{a^3}+\frac{3}{a^2 \left (a+b \sqrt [3]{x}\right )}+\frac{3}{2 a \left (a+b \sqrt [3]{x}\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*x^(1/3))^3*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0858, size = 54, normalized size = 0.96 \[ \frac{3}{2 a \left (a + b \sqrt [3]{x}\right )^{2}} + \frac{3}{a^{2} \left (a + b \sqrt [3]{x}\right )} + \frac{3 \log{\left (\sqrt [3]{x} \right )}}{a^{3}} - \frac{3 \log{\left (a + b \sqrt [3]{x} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0668966, size = 51, normalized size = 0.91 \[ \frac{3 \left (\frac{a \left (3 a+2 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}-2 \log \left (a+b \sqrt [3]{x}\right )+\frac{2 \log (x)}{3}\right )}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 49, normalized size = 0.9 \[{\frac{3}{2\,a} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}+3\,{\frac{1}{{a}^{2} \left ( a+b\sqrt [3]{x} \right ) }}-3\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{3}}}+{\frac{\ln \left ( x \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^3/x,x)

[Out]

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

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Maxima [A]  time = 1.44009, size = 77, normalized size = 1.38 \[ \frac{3 \,{\left (2 \, b x^{\frac{1}{3}} + 3 \, a\right )}}{2 \,{\left (a^{2} b^{2} x^{\frac{2}{3}} + 2 \, a^{3} b x^{\frac{1}{3}} + a^{4}\right )}} - \frac{3 \, \log \left (b x^{\frac{1}{3}} + a\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22849, size = 124, normalized size = 2.21 \[ \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + 3 \, a^{2} - 2 \,{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 2 \,{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )} \log \left (x^{\frac{1}{3}}\right )\right )}}{2 \,{\left (a^{3} b^{2} x^{\frac{2}{3}} + 2 \, a^{4} b x^{\frac{1}{3}} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.45339, size = 386, normalized size = 6.89 \[ \begin{cases} \frac{\tilde{\infty }}{x} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{b^{3} x} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a^{3}} & \text{for}\: b = 0 \\\frac{2 a^{2} x^{\frac{2}{3}} \log{\left (x \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{6 a^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{9 a^{2} x^{\frac{2}{3}}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{4 a b x \log{\left (x \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{12 a b x \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{6 a b x}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} + \frac{2 b^{2} x^{\frac{4}{3}} \log{\left (x \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} - \frac{6 b^{2} x^{\frac{4}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{5} x^{\frac{2}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{4}{3}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((zoo/x, Eq(a, 0) & Eq(b, 0)), (-1/(b**3*x), Eq(a, 0)), (log(x)/a**3, E
q(b, 0)), (2*a**2*x**(2/3)*log(x)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x*
*(4/3)) - 6*a**2*x**(2/3)*log(a/b + x**(1/3))/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*
a**3*b**2*x**(4/3)) + 9*a**2*x**(2/3)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**
2*x**(4/3)) + 4*a*b*x*log(x)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3
)) - 12*a*b*x*log(a/b + x**(1/3))/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x*
*(4/3)) + 6*a*b*x/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) + 2*b**2
*x**(4/3)*log(x)/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3)) - 6*b**2*
x**(4/3)*log(a/b + x**(1/3))/(2*a**5*x**(2/3) + 4*a**4*b*x + 2*a**3*b**2*x**(4/3
)), True))

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GIAC/XCAS [A]  time = 0.224045, size = 66, normalized size = 1.18 \[ -\frac{3 \,{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{3}} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} + 3 \, a^{2}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*ln(abs(b*x^(1/3) + a))/a^3 + ln(abs(x))/a^3 + 3/2*(2*a*b*x^(1/3) + 3*a^2)/((b
*x^(1/3) + a)^2*a^3)