3.2433 \(\int \frac{1}{\left (a+\frac{b}{\sqrt [3]{x}}\right )^3 x} \, dx\)

Optimal. Leaf size=54 \[ -\frac{3 b^2}{2 a^3 \left (a \sqrt [3]{x}+b\right )^2}+\frac{6 b}{a^3 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^3} \]

[Out]

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

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Rubi [A]  time = 0.0836381, antiderivative size = 54, 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 \[ -\frac{3 b^2}{2 a^3 \left (a \sqrt [3]{x}+b\right )^2}+\frac{6 b}{a^3 \left (a \sqrt [3]{x}+b\right )}+\frac{3 \log \left (a \sqrt [3]{x}+b\right )}{a^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^(1/3))^3*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.109, size = 49, normalized size = 0.91 \[ - \frac{3 b^{2}}{2 a^{3} \left (a \sqrt [3]{x} + b\right )^{2}} + \frac{6 b}{a^{3} \left (a \sqrt [3]{x} + b\right )} + \frac{3 \log{\left (a \sqrt [3]{x} + b \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0414173, size = 45, normalized size = 0.83 \[ \frac{3 \left (\frac{b \left (4 a \sqrt [3]{x}+3 b\right )}{\left (a \sqrt [3]{x}+b\right )^2}+2 \log \left (a \sqrt [3]{x}+b\right )\right )}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.112, size = 330, normalized size = 6.1 \[ -{\frac{9\,{b}^{6}}{2\, \left ({a}^{3}x+{b}^{3} \right ) ^{2}{a}^{3}}}+{\frac{\ln \left ({a}^{3}x+{b}^{3} \right ) }{{a}^{3}}}+9\,{\frac{{b}^{3}}{{a}^{3} \left ({a}^{3}x+{b}^{3} \right ) }}+2\,{\frac{\ln \left ( b+a\sqrt [3]{x} \right ) }{{a}^{3}}}-{\frac{{b}^{2}}{{a}^{3}} \left ( b+a\sqrt [3]{x} \right ) ^{-2}}-{\frac{13\,{b}^{2}}{2\,a}{x}^{{\frac{2}{3}}} \left ({a}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{-2}}+5\,{\frac{\sqrt [3]{x}{b}^{3}}{{a}^{2} \left ({a}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{2}}}-3\,{\frac{{b}^{4}}{{a}^{3} \left ({a}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{2}}}-{\frac{1}{2\,{a}^{3}}\ln \left ( a \left ({a}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{b}^{2} \right ) \right ) }+{\frac{\sqrt{3}}{{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3\,{a}^{2}b} \left ( 2\,\sqrt [3]{x}{a}^{3}-{a}^{2}b \right ) } \right ) }+4\,{\frac{b}{{a}^{3} \left ( b+a\sqrt [3]{x} \right ) }}+2\,{\frac{bx}{ \left ({a}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{b}^{2} \right ) ^{2}}}-{\frac{1}{2\,{a}^{3}}\ln \left ({a}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{b}^{2} \right ) }-{\frac{\sqrt{3}}{{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3\,ab} \left ( 2\,\sqrt [3]{x}{a}^{2}-ab \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^(1/3))^3/x,x)

[Out]

-9/2*b^6/(a^3*x+b^3)^2/a^3+1/a^3*ln(a^3*x+b^3)+9/a^3*b^3/(a^3*x+b^3)+2*ln(b+a*x^
(1/3))/a^3-b^2/a^3/(b+a*x^(1/3))^2-13/2/a*b^2/(a^2*x^(2/3)-a*b*x^(1/3)+b^2)^2*x^
(2/3)+5/a^2*b^3/(a^2*x^(2/3)-a*b*x^(1/3)+b^2)^2*x^(1/3)-3/a^3*b^4/(a^2*x^(2/3)-a
*b*x^(1/3)+b^2)^2-1/2/a^3*ln(a*(a^2*x^(2/3)-a*b*x^(1/3)+b^2))+1/a^3*3^(1/2)*arct
an(1/3*(2*x^(1/3)*a^3-a^2*b)*3^(1/2)/a^2/b)+4*b/a^3/(b+a*x^(1/3))+2*b/(a^2*x^(2/
3)-a*b*x^(1/3)+b^2)^2*x-1/2/a^3*ln(a^2*x^(2/3)-a*b*x^(1/3)+b^2)-1/a^3*3^(1/2)*ar
ctan(1/3*(2*x^(1/3)*a^2-a*b)*3^(1/2)/a/b)

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Maxima [A]  time = 1.42611, size = 77, normalized size = 1.43 \[ -\frac{3 \,{\left (3 \, a + \frac{2 \, b}{x^{\frac{1}{3}}}\right )}}{2 \,{\left (a^{4} + \frac{2 \, a^{3} b}{x^{\frac{1}{3}}} + \frac{a^{2} b^{2}}{x^{\frac{2}{3}}}\right )}} + \frac{3 \, \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{a^{3}} + \frac{\log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.229324, size = 93, normalized size = 1.72 \[ \frac{3 \,{\left (4 \, a b x^{\frac{1}{3}} + 3 \, b^{2} + 2 \,{\left (a^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + b^{2}\right )} \log \left (a x^{\frac{1}{3}} + b\right )\right )}}{2 \,{\left (a^{5} x^{\frac{2}{3}} + 2 \, a^{4} b x^{\frac{1}{3}} + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.24801, size = 240, normalized size = 4.44 \[ \begin{cases} \frac{6 a^{2} x^{\frac{4}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{12 a b x \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{12 a b x}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{6 b^{2} x^{\frac{2}{3}} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} + \frac{9 b^{2} x^{\frac{2}{3}}}{2 a^{5} x^{\frac{4}{3}} + 4 a^{4} b x + 2 a^{3} b^{2} x^{\frac{2}{3}}} & \text{for}\: a \neq 0 \\\frac{x}{b^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((6*a**2*x**(4/3)*log(x**(1/3) + b/a)/(2*a**5*x**(4/3) + 4*a**4*b*x + 2
*a**3*b**2*x**(2/3)) + 12*a*b*x*log(x**(1/3) + b/a)/(2*a**5*x**(4/3) + 4*a**4*b*
x + 2*a**3*b**2*x**(2/3)) + 12*a*b*x/(2*a**5*x**(4/3) + 4*a**4*b*x + 2*a**3*b**2
*x**(2/3)) + 6*b**2*x**(2/3)*log(x**(1/3) + b/a)/(2*a**5*x**(4/3) + 4*a**4*b*x +
2*a**3*b**2*x**(2/3)) + 9*b**2*x**(2/3)/(2*a**5*x**(4/3) + 4*a**4*b*x + 2*a**3*
b**2*x**(2/3)), Ne(a, 0)), (x/b**3, True))

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GIAC/XCAS [A]  time = 0.215782, size = 59, normalized size = 1.09 \[ \frac{3 \,{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{3}} + \frac{3 \,{\left (4 \, b x^{\frac{1}{3}} + \frac{3 \, b^{2}}{a}\right )}}{2 \,{\left (a x^{\frac{1}{3}} + b\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*ln(abs(a*x^(1/3) + b))/a^3 + 3/2*(4*b*x^(1/3) + 3*b^2/a)/((a*x^(1/3) + b)^2*a^
2)