Optimal. Leaf size=137 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]
[Out]
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Rubi [A] time = 0.156943, antiderivative size = 137, 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 \[ \frac{3 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^7}{8 b^3}-\frac{6 a \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^6}{7 b^3}+\frac{a^2 \sqrt{a^2+2 a b \sqrt [3]{x}+b^2 x^{2/3}} \left (a+b \sqrt [3]{x}\right )^5}{2 b^3} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 14.9486, size = 126, normalized size = 0.92 \[ \frac{a^{2} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{16 b^{3}} - \frac{3 a \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{7}{2}}}{28 b^{3}} + \frac{3 x^{\frac{2}{3}} \left (2 a + 2 b \sqrt [3]{x}\right ) \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}}{16 b} \]
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))**(5/2),x)
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Mathematica [A] time = 0.0423923, size = 91, normalized size = 0.66 \[ \frac{x \sqrt{\left (a+b \sqrt [3]{x}\right )^2} \left (56 a^5+210 a^4 b \sqrt [3]{x}+336 a^3 b^2 x^{2/3}+280 a^2 b^3 x+120 a b^4 x^{4/3}+21 b^5 x^{5/3}\right )}{56 \left (a+b \sqrt [3]{x}\right )} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x^(1/3) + b^2*x^(2/3))^(5/2),x]
[Out]
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Maple [A] time = 0.004, size = 87, normalized size = 0.6 \[{\frac{1}{56}\sqrt{{a}^{2}+2\,ab\sqrt [3]{x}+{b}^{2}{x}^{{\frac{2}{3}}}} \left ( 21\,{b}^{5}{x}^{8/3}+120\,a{b}^{4}{x}^{7/3}+336\,{a}^{3}{b}^{2}{x}^{5/3}+210\,{a}^{4}b{x}^{4/3}+280\,{a}^{2}{b}^{3}{x}^{2}+56\,{a}^{5}x \right ) \left ( a+b\sqrt [3]{x} \right ) ^{-1}} \]
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))^(5/2),x)
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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)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274681, size = 82, normalized size = 0.6 \[ 5 \, a^{2} b^{3} x^{2} + a^{5} x + \frac{3}{8} \,{\left (b^{5} x^{2} + 16 \, a^{3} b^{2} x\right )} x^{\frac{2}{3}} + \frac{15}{28} \,{\left (4 \, a b^{4} x^{2} + 7 \, a^{4} b x\right )} x^{\frac{1}{3}} \]
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. \[ \int \left (a^{2} + 2 a b \sqrt [3]{x} + b^{2} x^{\frac{2}{3}}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a**2+2*a*b*x**(1/3)+b**2*x**(2/3))**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.2843, size = 138, normalized size = 1.01 \[ \frac{3}{8} \, b^{5} x^{\frac{8}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{15}{7} \, a b^{4} x^{\frac{7}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + 5 \, a^{2} b^{3} x^{2}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + 6 \, a^{3} b^{2} x^{\frac{5}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + \frac{15}{4} \, a^{4} b x^{\frac{4}{3}}{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) + a^{5} x{\rm sign}\left (b x^{\frac{1}{3}} + a\right ) \]
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)^(5/2),x, algorithm="giac")
[Out]