Optimal. Leaf size=268 \[ -\frac{60 a^2}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{6 \sqrt [6]{x} \left (a+b \sqrt [6]{x}\right )}{b^5 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \]
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Rubi [A] time = 0.294771, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{60 a^2}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{30 a \left (a+b \sqrt [6]{x}\right ) \log \left (a+b \sqrt [6]{x}\right )}{b^6 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{6 \sqrt [6]{x} \left (a+b \sqrt [6]{x}\right )}{b^5 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{3 a^5}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}-\frac{10 a^4}{b^6 \left (a+b \sqrt [6]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}}+\frac{30 a^3}{b^6 \left (a+b \sqrt [6]{x}\right ) \sqrt{a^2+2 a b \sqrt [6]{x}+b^2 \sqrt [3]{x}}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))^(-5/2),x]
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Rubi in Sympy [A] time = 27.63, size = 250, normalized size = 0.93 \[ - \frac{30 a \left (a + b \sqrt [6]{x}\right ) \log{\left (a + b \sqrt [6]{x} \right )}}{b^{6} \sqrt{a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}}} - \frac{3 x^{\frac{5}{6}} \left (2 a + 2 b \sqrt [6]{x}\right )}{4 b \left (a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}\right )^{\frac{5}{2}}} - \frac{5 x^{\frac{2}{3}}}{2 b^{2} \left (a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}\right )^{\frac{3}{2}}} - \frac{5 \sqrt{x} \left (2 a + 2 b \sqrt [6]{x}\right )}{2 b^{3} \left (a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}\right )^{\frac{3}{2}}} - \frac{15 \sqrt [3]{x}}{b^{4} \sqrt{a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}}} + \frac{30 \sqrt{a^{2} + 2 a b \sqrt [6]{x} + b^{2} \sqrt [3]{x}}}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a**2+2*a*b*x**(1/6)+b**2*x**(1/3))**(5/2),x)
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Mathematica [A] time = 0.0911913, size = 121, normalized size = 0.45 \[ \frac{-77 a^5-248 a^4 b \sqrt [6]{x}-252 a^3 b^2 \sqrt [3]{x}-48 a^2 b^3 \sqrt{x}+48 a b^4 x^{2/3}-60 a \left (a+b \sqrt [6]{x}\right )^4 \log \left (a+b \sqrt [6]{x}\right )+12 b^5 x^{5/6}}{2 b^6 \left (a+b \sqrt [6]{x}\right )^3 \sqrt{\left (a+b \sqrt [6]{x}\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^(1/6) + b^2*x^(1/3))^(-5/2),x]
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Maple [A] time = 0.022, size = 174, normalized size = 0.7 \[{\frac{1}{2\,{b}^{6}}\sqrt{{a}^{2}+2\,ab\sqrt [6]{x}+{b}^{2}\sqrt [3]{x}} \left ( 12\,{x}^{5/6}{b}^{5}-60\,{x}^{2/3}\ln \left ( a+b\sqrt [6]{x} \right ) a{b}^{4}+48\,{x}^{2/3}a{b}^{4}-240\,\sqrt{x}\ln \left ( a+b\sqrt [6]{x} \right ){a}^{2}{b}^{3}-48\,\sqrt{x}{a}^{2}{b}^{3}-360\,\sqrt [3]{x}\ln \left ( a+b\sqrt [6]{x} \right ){a}^{3}{b}^{2}-252\,\sqrt [3]{x}{a}^{3}{b}^{2}-240\,\sqrt [6]{x}\ln \left ( a+b\sqrt [6]{x} \right ){a}^{4}b-248\,\sqrt [6]{x}{a}^{4}b-60\,\ln \left ( a+b\sqrt [6]{x} \right ){a}^{5}-77\,{a}^{5} \right ) \left ( a+b\sqrt [6]{x} \right ) ^{-5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a^2+2*a*b*x^(1/6)+b^2*x^(1/3))^(5/2),x)
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Maxima [A] time = 0.794399, size = 161, normalized size = 0.6 \[ \frac{12 \, b^{5} x^{\frac{5}{6}} + 48 \, a b^{4} x^{\frac{2}{3}} - 48 \, a^{2} b^{3} \sqrt{x} - 252 \, a^{3} b^{2} x^{\frac{1}{3}} - 248 \, a^{4} b x^{\frac{1}{6}} - 77 \, a^{5}}{2 \,{\left (b^{10} x^{\frac{2}{3}} + 4 \, a b^{9} \sqrt{x} + 6 \, a^{2} b^{8} x^{\frac{1}{3}} + 4 \, a^{3} b^{7} x^{\frac{1}{6}} + a^{4} b^{6}\right )}} - \frac{30 \, a \log \left (b x^{\frac{1}{6}} + a\right )}{b^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(1/3) + 2*a*b*x^(1/6) + a^2)^(-5/2),x, algorithm="maxima")
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Fricas [A] time = 0.276934, size = 212, normalized size = 0.79 \[ \frac{12 \, b^{5} x^{\frac{5}{6}} + 48 \, a b^{4} x^{\frac{2}{3}} - 48 \, a^{2} b^{3} \sqrt{x} - 252 \, a^{3} b^{2} x^{\frac{1}{3}} - 248 \, a^{4} b x^{\frac{1}{6}} - 77 \, a^{5} - 60 \,{\left (a b^{4} x^{\frac{2}{3}} + 4 \, a^{2} b^{3} \sqrt{x} + 6 \, a^{3} b^{2} x^{\frac{1}{3}} + 4 \, a^{4} b x^{\frac{1}{6}} + a^{5}\right )} \log \left (b x^{\frac{1}{6}} + a\right )}{2 \,{\left (b^{10} x^{\frac{2}{3}} + 4 \, a b^{9} \sqrt{x} + 6 \, a^{2} b^{8} x^{\frac{1}{3}} + 4 \, a^{3} b^{7} x^{\frac{1}{6}} + a^{4} b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(1/3) + 2*a*b*x^(1/6) + a^2)^(-5/2),x, algorithm="fricas")
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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+2*a*b*x**(1/6)+b**2*x**(1/3))**(5/2),x)
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(1/3) + 2*a*b*x^(1/6) + a^2)^(-5/2),x, algorithm="giac")
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