Optimal. Leaf size=119 \[ \frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3} \]
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Rubi [A] time = 0.207968, antiderivative size = 119, 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 \[ \frac{\sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^7}{24 b^3}-\frac{2 a \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^6}{21 b^3}+\frac{a^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^5}{18 b^3} \]
Antiderivative was successfully verified.
[In] Int[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 21.9823, size = 107, normalized size = 0.9 \[ \frac{a^{2} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{144 b^{3}} - \frac{a \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{7}{2}}}{84 b^{3}} + \frac{x^{6} \left (2 a + 2 b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{48 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
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Mathematica [A] time = 0.0367037, size = 83, normalized size = 0.7 \[ \frac{x^9 \sqrt{\left (a+b x^3\right )^2} \left (56 a^5+210 a^4 b x^3+336 a^3 b^2 x^6+280 a^2 b^3 x^9+120 a b^4 x^{12}+21 b^5 x^{15}\right )}{504 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^8*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
[Out]
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Maple [A] time = 0.009, size = 80, normalized size = 0.7 \[{\frac{{x}^{9} \left ( 21\,{b}^{5}{x}^{15}+120\,a{b}^{4}{x}^{12}+280\,{a}^{2}{b}^{3}{x}^{9}+336\,{a}^{3}{b}^{2}{x}^{6}+210\,{a}^{4}b{x}^{3}+56\,{a}^{5} \right ) }{504\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(b^2*x^6+2*a*b*x^3+a^2)^(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^6 + 2*a*b*x^3 + a^2)^(5/2)*x^8,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.250057, size = 77, normalized size = 0.65 \[ \frac{1}{24} \, b^{5} x^{24} + \frac{5}{21} \, a b^{4} x^{21} + \frac{5}{9} \, a^{2} b^{3} x^{18} + \frac{2}{3} \, a^{3} b^{2} x^{15} + \frac{5}{12} \, a^{4} b x^{12} + \frac{1}{9} \, a^{5} x^{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^8,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{8} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.288693, size = 142, normalized size = 1.19 \[ \frac{1}{24} \, b^{5} x^{24}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{21} \, a b^{4} x^{21}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{9} \, a^{2} b^{3} x^{18}{\rm sign}\left (b x^{3} + a\right ) + \frac{2}{3} \, a^{3} b^{2} x^{15}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{12} \, a^{4} b x^{12}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{9} \, a^{5} x^{9}{\rm sign}\left (b x^{3} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^6 + 2*a*b*x^3 + a^2)^(5/2)*x^8,x, algorithm="giac")
[Out]