Optimal. Leaf size=88 \[ \frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3} \]
[Out]
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Rubi [A] time = 0.15116, antiderivative size = 88, 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 \[ \frac{a^5}{b^6 \left (a+b \sqrt{x}\right )^2}-\frac{10 a^4}{b^6 \left (a+b \sqrt{x}\right )}-\frac{20 a^3 \log \left (a+b \sqrt{x}\right )}{b^6}+\frac{12 a^2 \sqrt{x}}{b^5}-\frac{3 a x}{b^4}+\frac{2 x^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a + b*Sqrt[x])^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{5}}{b^{6} \left (a + b \sqrt{x}\right )^{2}} - \frac{10 a^{4}}{b^{6} \left (a + b \sqrt{x}\right )} - \frac{20 a^{3} \log{\left (a + b \sqrt{x} \right )}}{b^{6}} + \frac{12 a^{2} \sqrt{x}}{b^{5}} - \frac{6 a \int ^{\sqrt{x}} x\, dx}{b^{4}} + \frac{2 x^{\frac{3}{2}}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a+b*x**(1/2))**3,x)
[Out]
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Mathematica [A] time = 0.0558735, size = 83, normalized size = 0.94 \[ \frac{\frac{3 a^5}{\left (a+b \sqrt{x}\right )^2}-\frac{30 a^4}{a+b \sqrt{x}}-60 a^3 \log \left (a+b \sqrt{x}\right )+36 a^2 b \sqrt{x}-9 a b^2 x+2 b^3 x^{3/2}}{3 b^6} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a + b*Sqrt[x])^3,x]
[Out]
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Maple [A] time = 0.011, size = 77, normalized size = 0.9 \[ -3\,{\frac{ax}{{b}^{4}}}+{\frac{2}{3\,{b}^{3}}{x}^{{\frac{3}{2}}}}-20\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{6}}}+12\,{\frac{{a}^{2}\sqrt{x}}{{b}^{5}}}+{\frac{{a}^{5}}{{b}^{6}} \left ( a+b\sqrt{x} \right ) ^{-2}}-10\,{\frac{{a}^{4}}{{b}^{6} \left ( a+b\sqrt{x} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a+b*x^(1/2))^3,x)
[Out]
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Maxima [A] time = 1.43531, size = 127, normalized size = 1.44 \[ -\frac{20 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{6}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{6}} - \frac{5 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{6}} + \frac{20 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{6}} - \frac{10 \, a^{4}}{{\left (b \sqrt{x} + a\right )} b^{6}} + \frac{a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*sqrt(x) + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233619, size = 147, normalized size = 1.67 \[ -\frac{5 \, a b^{4} x^{2} - 63 \, a^{3} b^{2} x + 27 \, a^{5} + 60 \,{\left (a^{3} b^{2} x + 2 \, a^{4} b \sqrt{x} + a^{5}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (b^{5} x^{2} + 10 \, a^{2} b^{3} x + 3 \, a^{4} b\right )} \sqrt{x}}{3 \,{\left (b^{8} x + 2 \, a b^{7} \sqrt{x} + a^{2} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*sqrt(x) + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.1606, size = 371, normalized size = 4.22 \[ \begin{cases} - \frac{60 a^{5} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{95 a^{5}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{120 a^{4} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{130 a^{4} b \sqrt{x}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{60 a^{3} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{5 a^{3} b^{2} x}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{20 a^{2} b^{3} x^{\frac{3}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} - \frac{5 a b^{4} x^{2}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} + \frac{2 b^{5} x^{\frac{5}{2}}}{3 a^{2} b^{6} + 6 a b^{7} \sqrt{x} + 3 b^{8} x} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{3}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a+b*x**(1/2))**3,x)
[Out]
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GIAC/XCAS [A] time = 0.291323, size = 107, normalized size = 1.22 \[ -\frac{20 \, a^{3}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{6}} - \frac{10 \, a^{4} b \sqrt{x} + 9 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{\frac{3}{2}} - 9 \, a b^{5} x + 36 \, a^{2} b^{4} \sqrt{x}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b*sqrt(x) + a)^3,x, algorithm="giac")
[Out]