Optimal. Leaf size=135 \[ -\frac{3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
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Rubi [A] time = 0.157291, antiderivative size = 135, 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 \[ -\frac{3 a^2}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}+\frac{2 a}{b^3 \left (a+b \sqrt [3]{x}\right )^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}-\frac{3}{2 b^3 \left (a+b \sqrt [3]{x}\right ) \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-5/2),x]
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Rubi in Sympy [A] time = 8.86532, size = 75, normalized size = 0.56 \[ \frac{3 x \left (2 a + 2 b \sqrt [3]{x}\right )}{8 a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}} + \frac{x}{4 a^{2} \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(5/2),x)
[Out]
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Mathematica [A] time = 0.0307225, size = 58, normalized size = 0.43 \[ \frac{-a^2-4 a b \sqrt [3]{x}-6 b^2 x^{2/3}}{4 b^3 \left (a+b \sqrt [3]{x}\right )^3 \sqrt{\left (a+b \sqrt [3]{x}\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(-5/2),x]
[Out]
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Maple [A] time = 0.01, size = 54, normalized size = 0.4 \[ -{\frac{1}{4\,{b}^{3}}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 6\,{b}^{2}{x}^{2/3}+4\,ab\sqrt [3]{x}+{a}^{2} \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(5/2),x)
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Maxima [A] time = 0.750146, size = 85, normalized size = 0.63 \[ -\frac{3 \, a^{2} b^{2}}{4 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{4}} + \frac{2 \, a b}{{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{3}} - \frac{3}{2 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{\frac{1}{3}} + \frac{a}{b}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(-5/2),x, algorithm="maxima")
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Fricas [A] time = 0.274416, size = 90, normalized size = 0.67 \[ -\frac{6 \, b^{2} x^{\frac{2}{3}} + 4 \, a b x^{\frac{1}{3}} + a^{2}}{4 \,{\left (4 \, a b^{6} x + 6 \, a^{2} b^{5} x^{\frac{2}{3}} + a^{4} b^{3} +{\left (b^{7} x + 4 \, a^{3} b^{4}\right )} x^{\frac{1}{3}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(-5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(5/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2)^(-5/2),x, algorithm="giac")
[Out]