Optimal. Leaf size=103 \[ \frac{30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac{18 b^2}{a^5 \sqrt [3]{x}}-\frac{3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}+\frac{9 b}{2 a^4 x^{2/3}}-\frac{1}{a^3 x} \]
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Rubi [A] time = 0.153794, antiderivative size = 103, 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 \[ \frac{30 b^3 \log \left (a+b \sqrt [3]{x}\right )}{a^6}-\frac{10 b^3 \log (x)}{a^6}-\frac{12 b^3}{a^5 \left (a+b \sqrt [3]{x}\right )}-\frac{18 b^2}{a^5 \sqrt [3]{x}}-\frac{3 b^3}{2 a^4 \left (a+b \sqrt [3]{x}\right )^2}+\frac{9 b}{2 a^4 x^{2/3}}-\frac{1}{a^3 x} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^(1/3))^3*x^2),x]
[Out]
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Rubi in Sympy [A] time = 21.3877, size = 104, normalized size = 1.01 \[ - \frac{1}{a^{3} x} - \frac{3 b^{3}}{2 a^{4} \left (a + b \sqrt [3]{x}\right )^{2}} + \frac{9 b}{2 a^{4} x^{\frac{2}{3}}} - \frac{12 b^{3}}{a^{5} \left (a + b \sqrt [3]{x}\right )} - \frac{18 b^{2}}{a^{5} \sqrt [3]{x}} - \frac{30 b^{3} \log{\left (\sqrt [3]{x} \right )}}{a^{6}} + \frac{30 b^{3} \log{\left (a + b \sqrt [3]{x} \right )}}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b*x**(1/3))**3/x**2,x)
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Mathematica [A] time = 0.215498, size = 93, normalized size = 0.9 \[ -\frac{\frac{a \left (2 a^4-5 a^3 b \sqrt [3]{x}+20 a^2 b^2 x^{2/3}+90 a b^3 x+60 b^4 x^{4/3}\right )}{x \left (a+b \sqrt [3]{x}\right )^2}-60 b^3 \log \left (a+b \sqrt [3]{x}\right )+20 b^3 \log (x)}{2 a^6} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x^(1/3))^3*x^2),x]
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Maple [A] time = 0.017, size = 90, normalized size = 0.9 \[ -{\frac{3\,{b}^{3}}{2\,{a}^{4}} \left ( a+b\sqrt [3]{x} \right ) ^{-2}}-12\,{\frac{{b}^{3}}{{a}^{5} \left ( a+b\sqrt [3]{x} \right ) }}-{\frac{1}{{a}^{3}x}}+{\frac{9\,b}{2\,{a}^{4}}{x}^{-{\frac{2}{3}}}}-18\,{\frac{{b}^{2}}{{a}^{5}\sqrt [3]{x}}}+30\,{\frac{{b}^{3}\ln \left ( a+b\sqrt [3]{x} \right ) }{{a}^{6}}}-10\,{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b*x^(1/3))^3/x^2,x)
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Maxima [A] time = 1.44166, size = 131, normalized size = 1.27 \[ -\frac{60 \, b^{4} x^{\frac{4}{3}} + 90 \, a b^{3} x + 20 \, a^{2} b^{2} x^{\frac{2}{3}} - 5 \, a^{3} b x^{\frac{1}{3}} + 2 \, a^{4}}{2 \,{\left (a^{5} b^{2} x^{\frac{5}{3}} + 2 \, a^{6} b x^{\frac{4}{3}} + a^{7} x\right )}} + \frac{30 \, b^{3} \log \left (b x^{\frac{1}{3}} + a\right )}{a^{6}} - \frac{10 \, b^{3} \log \left (x\right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(1/3) + a)^3*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.237774, size = 189, normalized size = 1.83 \[ -\frac{90 \, a^{2} b^{3} x + 20 \, a^{3} b^{2} x^{\frac{2}{3}} + 2 \, a^{5} - 60 \,{\left (b^{5} x^{\frac{5}{3}} + 2 \, a b^{4} x^{\frac{4}{3}} + a^{2} b^{3} x\right )} \log \left (b x^{\frac{1}{3}} + a\right ) + 60 \,{\left (b^{5} x^{\frac{5}{3}} + 2 \, a b^{4} x^{\frac{4}{3}} + a^{2} b^{3} x\right )} \log \left (x^{\frac{1}{3}}\right ) + 5 \,{\left (12 \, a b^{4} x - a^{4} b\right )} x^{\frac{1}{3}}}{2 \,{\left (a^{6} b^{2} x^{\frac{5}{3}} + 2 \, a^{7} b x^{\frac{4}{3}} + a^{8} x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(1/3) + a)^3*x^2),x, algorithm="fricas")
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Sympy [A] time = 28.3452, size = 561, normalized size = 5.45 \[ \begin{cases} \frac{\tilde{\infty }}{x^{2}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{2 b^{3} x^{2}} & \text{for}\: a = 0 \\- \frac{1}{a^{3} x} & \text{for}\: b = 0 \\- \frac{2 a^{5} x^{\frac{2}{3}}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{5 a^{4} b x}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{20 a^{3} b^{2} x^{\frac{4}{3}}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{20 a^{2} b^{3} x^{\frac{5}{3}} \log{\left (x \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{60 a^{2} b^{3} x^{\frac{5}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{90 a^{2} b^{3} x^{\frac{5}{3}}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{40 a b^{4} x^{2} \log{\left (x \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{120 a b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{60 a b^{4} x^{2}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} - \frac{20 b^{5} x^{\frac{7}{3}} \log{\left (x \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} + \frac{60 b^{5} x^{\frac{7}{3}} \log{\left (\frac{a}{b} + \sqrt [3]{x} \right )}}{2 a^{8} x^{\frac{5}{3}} + 4 a^{7} b x^{2} + 2 a^{6} b^{2} x^{\frac{7}{3}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b*x**(1/3))**3/x**2,x)
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GIAC/XCAS [A] time = 0.226235, size = 122, normalized size = 1.18 \[ \frac{30 \, b^{3}{\rm ln}\left ({\left | b x^{\frac{1}{3}} + a \right |}\right )}{a^{6}} - \frac{10 \, b^{3}{\rm ln}\left ({\left | x \right |}\right )}{a^{6}} - \frac{60 \, a b^{4} x^{\frac{4}{3}} + 90 \, a^{2} b^{3} x + 20 \, a^{3} b^{2} x^{\frac{2}{3}} - 5 \, a^{4} b x^{\frac{1}{3}} + 2 \, a^{5}}{2 \,{\left (b x^{\frac{1}{3}} + a\right )}^{2} a^{6} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^(1/3) + a)^3*x^2),x, algorithm="giac")
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