Optimal. Leaf size=137 \[ \frac{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^4}{5 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^3}{4 b^3} \]
[Out]
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Rubi [A] time = 0.122619, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{\sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^4}{5 b^3}+\frac{3 a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^3}{4 b^3} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 15.0208, size = 122, normalized size = 0.89 \[ \frac{a^{2} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{8 b^{3}} - \frac{a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{5 b^{3}} + \frac{x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(3/2),x)
[Out]
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Mathematica [A] time = 0.0332571, size = 65, normalized size = 0.47 \[ \frac{x \sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (20 a^3+45 a^2 b \sqrt [3]{x}+36 a b^2 x^{2/3}+10 b^3 x\right )}{20 \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 65, normalized size = 0.5 \[{\frac{1}{20}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 36\,a{x}^{5/3}{b}^{2}+45\,{x}^{4/3}{a}^{2}b+10\,{b}^{3}{x}^{2}+20\,{a}^{3}x \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a^2+2*a*b*x^(1/3)+b^2*x^(2/3))^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
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)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272695, size = 43, normalized size = 0.31 \[ \frac{1}{2} \, b^{3} x^{2} + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}} + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}} + a^{3} x \]
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)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{3}{2}}\, dx \]
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))**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.282533, size = 86, normalized size = 0.63 \[ \frac{1}{2} \, b^{3} x^{2}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{9}{5} \, a b^{2} x^{\frac{5}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{9}{4} \, a^{2} b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a^{3} x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]
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)^(3/2),x, algorithm="giac")
[Out]