Optimal. Leaf size=410 \[ \frac{3 b^7 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac{7 b^6 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac{63 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{105 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac{105 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{45 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^7 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{15 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
[Out]
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Rubi [A] time = 0.521337, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{3 b^7 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{4 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^4}-\frac{7 b^6 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^3}+\frac{63 b^5 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )^2}-\frac{105 b^4 \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}} \left (a \sqrt [3]{x}+b\right )}-\frac{105 b^3 \left (a+\frac{b}{\sqrt [3]{x}}\right ) \log \left (a \sqrt [3]{x}+b\right )}{a^8 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{45 b^2 \sqrt [3]{x} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^7 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}-\frac{15 b x^{2/3} \left (a+\frac{b}{\sqrt [3]{x}}\right )}{2 a^6 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}}+\frac{x \left (a+\frac{b}{\sqrt [3]{x}}\right )}{a^5 \sqrt{a^2+\frac{2 a b}{\sqrt [3]{x}}+\frac{b^2}{x^{2/3}}}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(-5/2),x]
[Out]
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Rubi in Sympy [A] time = 68.3089, size = 396, normalized size = 0.97 \[ - \frac{3 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{8 a \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{5}{2}}} - \frac{7 x}{4 a^{2} \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}} - \frac{21 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{8 a^{3} \left (a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}\right )^{\frac{3}{2}}} - \frac{105 x}{4 a^{4} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{35 x \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{2 a^{5} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} - \frac{105 b x^{\frac{2}{3}} \left (2 a + \frac{2 b}{\sqrt [3]{x}}\right )}{4 a^{6} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}} + \frac{105 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (\frac{1}{\sqrt [3]{x}} \right )}}{a^{8} \left (a + \frac{b}{\sqrt [3]{x}}\right )} - \frac{105 b^{3} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}} \log{\left (a + \frac{b}{\sqrt [3]{x}} \right )}}{a^{8} \left (a + \frac{b}{\sqrt [3]{x}}\right )} + \frac{105 b^{2} \sqrt [3]{x} \sqrt{a^{2} + \frac{2 a b}{\sqrt [3]{x}} + \frac{b^{2}}{x^{\frac{2}{3}}}}}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(5/2),x)
[Out]
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Mathematica [A] time = 0.128563, size = 152, normalized size = 0.37 \[ \frac{\left (a \sqrt [3]{x}+b\right ) \left (4 a^7 x^{7/3}-14 a^6 b x^2+84 a^5 b^2 x^{5/3}+556 a^4 b^3 x^{4/3}+544 a^3 b^4 x-444 a^2 b^5 x^{2/3}-856 a b^6 \sqrt [3]{x}-420 b^3 \left (a \sqrt [3]{x}+b\right )^4 \log \left (a \sqrt [3]{x}+b\right )-319 b^7\right )}{4 a^8 x^{5/3} \left (\frac{\left (a \sqrt [3]{x}+b\right )^2}{x^{2/3}}\right )^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + b^2/x^(2/3) + (2*a*b)/x^(1/3))^(-5/2),x]
[Out]
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Maple [A] time = 0.018, size = 199, normalized size = 0.5 \[{\frac{1}{4\,{a}^{8}} \left ( 4\,{x}^{7/3}{a}^{7}+84\,{a}^{5}{b}^{2}{x}^{5/3}-420\,{x}^{4/3}\ln \left ( b+a\sqrt [3]{x} \right ){a}^{4}{b}^{3}+556\,{x}^{4/3}{a}^{4}{b}^{3}-2520\,{x}^{2/3}\ln \left ( b+a\sqrt [3]{x} \right ){a}^{2}{b}^{5}-444\,{x}^{2/3}{a}^{2}{b}^{5}-1680\,\sqrt [3]{x}\ln \left ( b+a\sqrt [3]{x} \right ) a{b}^{6}-1680\,x\ln \left ( b+a\sqrt [3]{x} \right ){a}^{3}{b}^{4}-14\,{a}^{6}b{x}^{2}-856\,\sqrt [3]{x}a{b}^{6}-420\,\ln \left ( b+a\sqrt [3]{x} \right ){b}^{7}+544\,x{a}^{3}{b}^{4}-319\,{b}^{7} \right ) \left ( b+a\sqrt [3]{x} \right ) \left ({1 \left ({a}^{2}{x}^{{\frac{2}{3}}}+2\,ab\sqrt [3]{x}+{b}^{2} \right ){x}^{-{\frac{2}{3}}}} \right ) ^{-{\frac{5}{2}}}{x}^{-{\frac{5}{3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a^2+b^2/x^(2/3)+2*a*b/x^(1/3))^(5/2),x)
[Out]
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Maxima [A] time = 0.750663, size = 188, normalized size = 0.46 \[ \frac{4 \, a^{7} x^{\frac{7}{3}} - 14 \, a^{6} b x^{2} + 84 \, a^{5} b^{2} x^{\frac{5}{3}} + 556 \, a^{4} b^{3} x^{\frac{4}{3}} + 544 \, a^{3} b^{4} x - 444 \, a^{2} b^{5} x^{\frac{2}{3}} - 856 \, a b^{6} x^{\frac{1}{3}} - 319 \, b^{7}}{4 \,{\left (a^{12} x^{\frac{4}{3}} + 4 \, a^{11} b x + 6 \, a^{10} b^{2} x^{\frac{2}{3}} + 4 \, a^{9} b^{3} x^{\frac{1}{3}} + a^{8} b^{4}\right )}} - \frac{105 \, b^{3} \log \left (a x^{\frac{1}{3}} + b\right )}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(-5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275565, size = 238, normalized size = 0.58 \[ -\frac{14 \, a^{6} b x^{2} - 544 \, a^{3} b^{4} x + 319 \, b^{7} + 420 \,{\left (4 \, a^{3} b^{4} x + 6 \, a^{2} b^{5} x^{\frac{2}{3}} + b^{7} +{\left (a^{4} b^{3} x + 4 \, a b^{6}\right )} x^{\frac{1}{3}}\right )} \log \left (a x^{\frac{1}{3}} + b\right ) - 12 \,{\left (7 \, a^{5} b^{2} x - 37 \, a^{2} b^{5}\right )} x^{\frac{2}{3}} - 4 \,{\left (a^{7} x^{2} + 139 \, a^{4} b^{3} x - 214 \, a b^{6}\right )} x^{\frac{1}{3}}}{4 \,{\left (4 \, a^{11} b x + 6 \, a^{10} b^{2} x^{\frac{2}{3}} + a^{8} b^{4} +{\left (a^{12} x + 4 \, a^{9} b^{3}\right )} x^{\frac{1}{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(-5/2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a**2+b**2/x**(2/3)+2*a*b/x**(1/3))**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.319103, size = 190, normalized size = 0.46 \[ -\frac{105 \, b^{3}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{a^{8}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} - \frac{420 \, a^{3} b^{4} x + 1134 \, a^{2} b^{5} x^{\frac{2}{3}} + 1036 \, a b^{6} x^{\frac{1}{3}} + 319 \, b^{7}}{4 \,{\left (a x^{\frac{1}{3}} + b\right )}^{4} a^{8}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} + \frac{2 \, a^{10} x - 15 \, a^{9} b x^{\frac{2}{3}} + 90 \, a^{8} b^{2} x^{\frac{1}{3}}}{2 \, a^{15}{\rm sign}\left (a x^{\frac{2}{3}} + b x^{\frac{1}{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a^2 + 2*a*b/x^(1/3) + b^2/x^(2/3))^(-5/2),x, algorithm="giac")
[Out]