Optimal. Leaf size=114 \[ \frac{a^7}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{14 a^6}{b^8 \left (a+b \sqrt{x}\right )}-\frac{42 a^5 \log \left (a+b \sqrt{x}\right )}{b^8}+\frac{30 a^4 \sqrt{x}}{b^7}-\frac{10 a^3 x}{b^6}+\frac{4 a^2 x^{3/2}}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{2 x^{5/2}}{5 b^3} \]
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Rubi [A] time = 0.21138, antiderivative size = 114, 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 \[ \frac{a^7}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{14 a^6}{b^8 \left (a+b \sqrt{x}\right )}-\frac{42 a^5 \log \left (a+b \sqrt{x}\right )}{b^8}+\frac{30 a^4 \sqrt{x}}{b^7}-\frac{10 a^3 x}{b^6}+\frac{4 a^2 x^{3/2}}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{2 x^{5/2}}{5 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a + b*Sqrt[x])^3,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{7}}{b^{8} \left (a + b \sqrt{x}\right )^{2}} - \frac{14 a^{6}}{b^{8} \left (a + b \sqrt{x}\right )} - \frac{42 a^{5} \log{\left (a + b \sqrt{x} \right )}}{b^{8}} + \frac{30 a^{4} \sqrt{x}}{b^{7}} - \frac{20 a^{3} \int ^{\sqrt{x}} x\, dx}{b^{6}} + \frac{4 a^{2} x^{\frac{3}{2}}}{b^{5}} - \frac{3 a x^{2}}{2 b^{4}} + \frac{2 x^{\frac{5}{2}}}{5 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(a+b*x**(1/2))**3,x)
[Out]
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Mathematica [A] time = 0.0769831, size = 107, normalized size = 0.94 \[ \frac{\frac{10 a^7}{\left (a+b \sqrt{x}\right )^2}-\frac{140 a^6}{a+b \sqrt{x}}-420 a^5 \log \left (a+b \sqrt{x}\right )+300 a^4 b \sqrt{x}-100 a^3 b^2 x+40 a^2 b^3 x^{3/2}-15 a b^4 x^2+4 b^5 x^{5/2}}{10 b^8} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a + b*Sqrt[x])^3,x]
[Out]
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Maple [A] time = 0.013, size = 99, normalized size = 0.9 \[ -10\,{\frac{{a}^{3}x}{{b}^{6}}}+4\,{\frac{{a}^{2}{x}^{3/2}}{{b}^{5}}}-{\frac{3\,a{x}^{2}}{2\,{b}^{4}}}+{\frac{2}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}-42\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+30\,{\frac{{a}^{4}\sqrt{x}}{{b}^{7}}}+{\frac{{a}^{7}}{{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-2}}-14\,{\frac{{a}^{6}}{{b}^{8} \left ( a+b\sqrt{x} \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(a+b*x^(1/2))^3,x)
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Maxima [A] time = 1.43945, size = 173, normalized size = 1.52 \[ -\frac{42 \, a^{5} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )} a^{4}}{b^{8}} - \frac{14 \, a^{6}}{{\left (b \sqrt{x} + a\right )} b^{8}} + \frac{a^{7}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b*sqrt(x) + a)^3,x, algorithm="maxima")
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Fricas [A] time = 0.237723, size = 178, normalized size = 1.56 \[ -\frac{7 \, a b^{6} x^{3} + 35 \, a^{3} b^{4} x^{2} - 500 \, a^{5} b^{2} x + 130 \, a^{7} + 420 \,{\left (a^{5} b^{2} x + 2 \, a^{6} b \sqrt{x} + a^{7}\right )} \log \left (b \sqrt{x} + a\right ) - 2 \,{\left (2 \, b^{7} x^{3} + 7 \, a^{2} b^{5} x^{2} + 70 \, a^{4} b^{3} x + 80 \, a^{6} b\right )} \sqrt{x}}{10 \,{\left (b^{10} x + 2 \, a b^{9} \sqrt{x} + a^{2} b^{8}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b*sqrt(x) + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.28267, size = 450, normalized size = 3.95 \[ \begin{cases} - \frac{420 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{665 a^{7}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{840 a^{6} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{910 a^{6} b \sqrt{x}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{420 a^{5} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{35 a^{5} b^{2} x}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{140 a^{4} b^{3} x^{\frac{3}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{35 a^{3} b^{4} x^{2}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{14 a^{2} b^{5} x^{\frac{5}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{7 a b^{6} x^{3}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{4 b^{7} x^{\frac{7}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{3}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(a+b*x**(1/2))**3,x)
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GIAC/XCAS [A] time = 0.275364, size = 136, normalized size = 1.19 \[ -\frac{42 \, a^{5}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} - \frac{14 \, a^{6} b \sqrt{x} + 13 \, a^{7}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} + \frac{4 \, b^{12} x^{\frac{5}{2}} - 15 \, a b^{11} x^{2} + 40 \, a^{2} b^{10} x^{\frac{3}{2}} - 100 \, a^{3} b^{9} x + 300 \, a^{4} b^{8} \sqrt{x}}{10 \, b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(b*sqrt(x) + a)^3,x, algorithm="giac")
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