Optimal. Leaf size=80 \[ \frac{12 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}-\frac{3 b^3}{a^4 \left (a+b \sqrt [3]{x}\right )}-\frac{9 b^2}{a^4 \sqrt [3]{x}}+\frac{3 b}{a^3 x^{2/3}}-\frac{1}{a^2 x} \]
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Rubi [A] time = 0.120971, antiderivative size = 80, 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 \[ \frac{12 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^5}-\frac{4 b^3 \log (x)}{a^5}-\frac{3 b^3}{a^4 \left (a+b \sqrt [3]{x}\right )}-\frac{9 b^2}{a^4 \sqrt [3]{x}}+\frac{3 b}{a^3 x^{2/3}}-\frac{1}{a^2 x} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^(1/3))^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 17.4359, size = 82, normalized size = 1.02 \[ - \frac{1}{a^{2} x} + \frac{3 b}{a^{3} x^{\frac{2}{3}}} - \frac{3 b^{3}}{a^{4} \left (a + b \sqrt [3]{x}\right )} - \frac{9 b^{2}}{a^{4} \sqrt [3]{x}} - \frac{12 b^{3} \log{\left (\sqrt [3]{x} \right )}}{a^{5}} + \frac{12 b^{3} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**(1/3))**2/x**2,x)
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Mathematica [A] time = 0.154204, size = 77, normalized size = 0.96 \[ \frac{-\frac{a^4-2 a^3 b \sqrt [3]{x}+6 a^2 b^2 x^{2/3}+12 a b^3 x}{a x+b x^{4/3}}+12 b^3 \log \left (a+b \sqrt [3]{x}\right )-4 b^3 \log (x)}{a^5} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^(1/3))^2*x^2),x]
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Maple [A] time = 0.017, size = 73, normalized size = 0.9 \[ -3\,{\frac{{b}^{3}}{{a}^{4} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{x{a}^{2}}}+3\,{\frac{b}{{a}^{3}{x}^{2/3}}}-9\,{\frac{{b}^{2}}{{a}^{4}\sqrt [3]{x}}}+12\,{\frac{{b}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{5}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^(1/3))^2/x^2,x)
[Out]
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Maxima [A] time = 1.43324, size = 99, normalized size = 1.24 \[ -\frac{12 \, b^{3} x + 6 \, a b^{2} x^{\frac{2}{3}} - 2 \, a^{2} b x^{\frac{1}{3}} + a^{3}}{a^{4} b x^{\frac{4}{3}} + a^{5} x} + \frac{12 \, b^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{5}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(1/3) + a)^2*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.226877, size = 126, normalized size = 1.58 \[ -\frac{12 \, a b^{3} x + 6 \, a^{2} b^{2} x^{\frac{2}{3}} - 2 \, a^{3} b x^{\frac{1}{3}} + a^{4} - 12 \,{\left (b^{4} x^{\frac{4}{3}} + a b^{3} x\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 12 \,{\left (b^{4} x^{\frac{4}{3}} + a b^{3} x\right )} \log \left (x^{\frac{1}{3}}\right )}{a^{5} b x^{\frac{4}{3}} + a^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(1/3) + a)^2*x^2),x, algorithm="fricas")
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Sympy [A] time = 17.2018, size = 272, normalized size = 3.4 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{a^{2} x} & \text{for}\: b = 0 \\- \frac{3}{5 b^{2} x^{\frac{5}{3}}} & \text{for}\: a = 0 \\- \frac{a^{4} x^{\frac{2}{3}}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} + \frac{2 a^{3} b x}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} - \frac{6 a^{2} b^{2} x^{\frac{4}{3}}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} - \frac{4 a b^{3} x^{\frac{5}{3}} \log{\left (x \right )}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} + \frac{12 a b^{3} x^{\frac{5}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} - \frac{12 a b^{3} x^{\frac{5}{3}}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} - \frac{4 b^{4} x^{2} \log{\left (x \right )}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} + \frac{12 b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{a^{6} x^{\frac{5}{3}} + a^{5} b x^{2}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*x**(1/3))**2/x**2,x)
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GIAC/XCAS [A] time = 0.220681, size = 104, normalized size = 1.3 \[ \frac{12 \, b^{3}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{5}} - \frac{4 \, b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{5}} - \frac{12 \, a b^{3} x + 6 \, a^{2} b^{2} x^{\frac{2}{3}} - 2 \, a^{3} b x^{\frac{1}{3}} + a^{4}}{{\left (b x^{\frac{1}{3}} + a\right )} a^{5} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(1/3) + a)^2*x^2),x, algorithm="giac")
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