3.2361 \(\int \frac{1}{\left (a+b \sqrt [3]{x}\right )^2} \, dx\)

Optimal. Leaf size=46 \[ -\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}+\frac{3 \sqrt [3]{x}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0622783, antiderivative size = 46, 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 \[ -\frac{3 a^2}{b^3 \left (a+b \sqrt [3]{x}\right )}-\frac{6 a \log \left (a+b \sqrt [3]{x}\right )}{b^3}+\frac{3 \sqrt [3]{x}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^(1/3))^(-2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 a^{2}}{b^{3} \left (a + b \sqrt [3]{x}\right )} - \frac{6 a \log{\left (a + b \sqrt [3]{x} \right )}}{b^{3}} + 3 \int ^{\sqrt [3]{x}} \frac{1}{b^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b*x**(1/3))**2,x)

[Out]

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

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Mathematica [A]  time = 0.031182, size = 42, normalized size = 0.91 \[ \frac{3 \left (-\frac{a^2}{a+b \sqrt [3]{x}}-2 a \log \left (a+b \sqrt [3]{x}\right )+b \sqrt [3]{x}\right )}{b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^(1/3))^(-2),x]

[Out]

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

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Maple [B]  time = 0.071, size = 257, normalized size = 5.6 \[ -3\,{\frac{{a}^{4}}{ \left ({b}^{3}x+{a}^{3} \right ){b}^{3}}}+3\,{\frac{\sqrt [3]{x}}{{b}^{2}}}-4\,{\frac{a\ln \left ( a+b\sqrt [3]{x} \right ) }{{b}^{3}}}-2\,{\frac{{a}^{2}}{{b}^{3} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{{a}^{2}}{{b}^{2}}\sqrt [3]{x} \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) ^{-1}}+2\,{\frac{{a}^{3}}{{b}^{3} \left ({b}^{2}{x}^{2/3}-ab\sqrt [3]{x}+{a}^{2} \right ) }}+{\frac{5\,a}{3\,{b}^{3}}\ln \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) }+{\frac{2\,a\sqrt{3}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3\,ab} \left ( 2\,{b}^{2}\sqrt [3]{x}-ab \right ) } \right ) }+{\frac{a}{3\,{b}^{3}}\ln \left ( b \left ({b}^{2}{x}^{{\frac{2}{3}}}-ab\sqrt [3]{x}+{a}^{2} \right ) \right ) }-{\frac{2\,a\sqrt{3}}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3\,a{b}^{2}} \left ( 2\,\sqrt [3]{x}{b}^{3}-a{b}^{2} \right ) } \right ) }-2\,{\frac{a\ln \left ({b}^{3}x+{a}^{3} \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b*x^(1/3))^2,x)

[Out]

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

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Maxima [A]  time = 1.43303, size = 59, normalized size = 1.28 \[ -\frac{6 \, a \log \left (b x^{\frac{1}{3}} + a\right )}{b^{3}} + \frac{3 \,{\left (b x^{\frac{1}{3}} + a\right )}}{b^{3}} - \frac{3 \, a^{2}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^(-2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.21563, size = 76, normalized size = 1.65 \[ \frac{3 \,{\left (b^{2} x^{\frac{2}{3}} + a b x^{\frac{1}{3}} - a^{2} - 2 \,{\left (a b x^{\frac{1}{3}} + a^{2}\right )} \log \left (b x^{\frac{1}{3}} + a\right )\right )}}{b^{4} x^{\frac{1}{3}} + a b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^(-2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.68711, size = 109, normalized size = 2.37 \[ \begin{cases} - \frac{6 a^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a b^{3} + b^{4} \sqrt [3]{x}} - \frac{6 a^{2}}{a b^{3} + b^{4} \sqrt [3]{x}} - \frac{6 a b \sqrt [3]{x} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a b^{3} + b^{4} \sqrt [3]{x}} + \frac{3 b^{2} x^{\frac{2}{3}}}{a b^{3} + b^{4} \sqrt [3]{x}} & \text{for}\: b \neq 0 \\\frac{x}{a^{2}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b*x**(1/3))**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217199, size = 55, normalized size = 1.2 \[ -\frac{6 \, a{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{b^{3}} + \frac{3 \, x^{\frac{1}{3}}}{b^{2}} - \frac{3 \, a^{2}}{{\left (b x^{\frac{1}{3}} + a\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(1/3) + a)^(-2),x, algorithm="giac")

[Out]

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