3.2369 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0759707, antiderivative size = 54, 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 \[ -\frac{3 a^2}{2 b^3 \left (a+b \sqrt [3]{x}\right )^2}+\frac{6 a}{b^3 \left (a+b \sqrt [3]{x}\right )}+\frac{3 \log \left (a+b \sqrt [3]{x}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^(-3),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.5878, size = 49, normalized size = 0.91 \[ - \frac{3 a^{2}}{2 b^{3} \left (a + b \sqrt [3]{x}\right )^{2}} + \frac{6 a}{b^{3} \left (a + b \sqrt [3]{x}\right )} + \frac{3 \log{\left (a + b \sqrt [3]{x} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0368934, size = 45, normalized size = 0.83 \[ \frac{3 \left (\frac{a \left (3 a+4 b \sqrt [3]{x}\right )}{\left (a+b \sqrt [3]{x}\right )^2}+2 \log \left (a+b \sqrt [3]{x}\right )\right )}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.101, size = 330, normalized size = 6.1 \[ -{\frac{9\,{a}^{6}}{2\, \left ({b}^{3}x+{a}^{3} \right ) ^{2}{b}^{3}}}+9\,{\frac{{a}^{3}}{{b}^{3} \left ({b}^{3}x+{a}^{3} \right ) }}+{\frac{\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}}+2\,{\frac{\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-{\frac{{a}^{2}}{{b}^{3}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-{\frac{13\,{a}^{2}}{2\,b}{x}^{{\frac{2}{3}}} \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{-2}}+5\,{\frac{{a}^{3}\sqrt [3]{x}}{{b}^{2} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{2}}}-3\,{\frac{{a}^{4}}{{b}^{3} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{2}}}-{\frac{1}{2\,{b}^{3}}\ln \left ( b \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) \right ) }+{\frac{\sqrt{3}}{{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3\,a{b}^{2}} \left ( 2\,\sqrt [3]{x}{b}^{3}-a{b}^{2} \right ) } \right ) }+4\,{\frac{a}{{b}^{3} \left ( a+b\sqrt [3]{x} \right ) }}+2\,{\frac{ax}{ \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{2}}}-{\frac{1}{2\,{b}^{3}}\ln \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) }-{\frac{\sqrt{3}}{{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3\,ab} \left ( 2\,{b}^{2}\sqrt [3]{x}-ab \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^3,x)

[Out]

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

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Maxima [A]  time = 1.44246, size = 62, normalized size = 1.15 \[ \frac{3 \, \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{6 \, a}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} - \frac{3 \, a^{2}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2207, size = 93, normalized size = 1.72 \[ \frac{3 \,{\left (4 \, a b x^{\frac{1}{3}} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{\frac{2}{3}} + 2 \, a b x^{\frac{1}{3}} + a^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right )\right )}}{2 \,{\left (b^{5} x^{\frac{2}{3}} + 2 \, a b^{4} x^{\frac{1}{3}} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.43413, size = 228, normalized size = 4.22 \[ \begin{cases} \frac{6 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{3 a^{2}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{12 a b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} + \frac{6 b^{2} x^{\frac{2}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} - \frac{6 b^{2} x^{\frac{2}{3}}}{2 a^{2} b^{3} + 4 a b^{4} \sqrt [3]{x} + 2 b^{5} x^{\frac{2}{3}}} & \text{for}\: b \neq 0 \\\frac{x}{a^{3}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220071, size = 59, normalized size = 1.09 \[ \frac{3 \,{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \,{\left (4 \, a x^{\frac{1}{3}} + \frac{3 \, a^{2}}{b}\right )}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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