Optimal. Leaf size=88 \[ \frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac{a x \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \]
[Out]
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Rubi [A] time = 0.0908995, antiderivative size = 88, 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 \[ \frac{3 b x^{4/3} \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{4 \left (a+b \sqrt [3]{x}\right )}+\frac{a x \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}}}{a+b \sqrt [3]{x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]
[Out]
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Rubi in Sympy [A] time = 9.32815, size = 70, normalized size = 0.8 \[ \frac{a x \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}}{4 \left (a + b \sqrt [3]{x}\right )} + \frac{3 x \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}}{4} \]
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))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0162209, size = 43, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (4 a x+3 b x^{4/3}\right )}{4 \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3)],x]
[Out]
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Maple [A] time = 0.004, size = 43, normalized size = 0.5 \[{\frac{1}{4}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 3\,b{x}^{4/3}+4\,ax \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]
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))^(1/2),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(sqrt(b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276833, size = 14, normalized size = 0.16 \[ \frac{3}{4} \, b x^{\frac{4}{3}} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}}\, dx \]
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))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270966, size = 35, normalized size = 0.4 \[ \frac{3}{4} \, b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b^2*x^(2/3) + 2*a*b*x^(1/3) + a^2),x, algorithm="giac")
[Out]