Optimal. Leaf size=255 \[ \frac{b^5 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}+\frac{5 a b^4 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{5 a^4 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \]
[Out]
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Rubi [A] time = 0.163892, antiderivative size = 255, 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 \[ \frac{b^5 x^{20} \sqrt{a^2+2 a b x^3+b^2 x^6}}{20 \left (a+b x^3\right )}+\frac{5 a b^4 x^{17} \sqrt{a^2+2 a b x^3+b^2 x^6}}{17 \left (a+b x^3\right )}+\frac{5 a^2 b^3 x^{14} \sqrt{a^2+2 a b x^3+b^2 x^6}}{7 \left (a+b x^3\right )}+\frac{a^5 x^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}{5 \left (a+b x^3\right )}+\frac{5 a^4 b x^8 \sqrt{a^2+2 a b x^3+b^2 x^6}}{8 \left (a+b x^3\right )}+\frac{10 a^3 b^2 x^{11} \sqrt{a^2+2 a b x^3+b^2 x^6}}{11 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Int[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 27.0897, size = 207, normalized size = 0.81 \[ \frac{729 a^{5} x^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{52360 \left (a + b x^{3}\right )} + \frac{243 a^{4} x^{5} \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{10472} + \frac{81 a^{3} x^{5} \left (a + b x^{3}\right ) \sqrt{a^{2} + 2 a b x^{3} + b^{2} x^{6}}}{2618} + \frac{9 a^{2} x^{5} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{238} + \frac{3 a x^{5} \left (a + b x^{3}\right ) \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{3}{2}}}{68} + \frac{x^{5} \left (a^{2} + 2 a b x^{3} + b^{2} x^{6}\right )^{\frac{5}{2}}}{20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
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Mathematica [A] time = 0.0369222, size = 83, normalized size = 0.33 \[ \frac{x^5 \sqrt{\left (a+b x^3\right )^2} \left (10472 a^5+32725 a^4 b x^3+47600 a^3 b^2 x^6+37400 a^2 b^3 x^9+15400 a b^4 x^{12}+2618 b^5 x^{15}\right )}{52360 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2),x]
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Maple [A] time = 0.01, size = 80, normalized size = 0.3 \[{\frac{{x}^{5} \left ( 2618\,{b}^{5}{x}^{15}+15400\,a{b}^{4}{x}^{12}+37400\,{a}^{2}{b}^{3}{x}^{9}+47600\,{a}^{3}{b}^{2}{x}^{6}+32725\,{a}^{4}b{x}^{3}+10472\,{a}^{5} \right ) }{52360\, \left ( b{x}^{3}+a \right ) ^{5}} \left ( \left ( b{x}^{3}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.822102, size = 77, normalized size = 0.3 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{17} \, a b^{4} x^{17} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{5} \, a^{5} x^{5} \]
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^4,x, algorithm="maxima")
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Fricas [A] time = 0.25404, size = 77, normalized size = 0.3 \[ \frac{1}{20} \, b^{5} x^{20} + \frac{5}{17} \, a b^{4} x^{17} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{8} \, a^{4} b x^{8} + \frac{1}{5} \, a^{5} x^{5} \]
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^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{4} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.265001, size = 142, normalized size = 0.56 \[ \frac{1}{20} \, b^{5} x^{20}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{17} \, a b^{4} x^{17}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{7} \, a^{2} b^{3} x^{14}{\rm sign}\left (b x^{3} + a\right ) + \frac{10}{11} \, a^{3} b^{2} x^{11}{\rm sign}\left (b x^{3} + a\right ) + \frac{5}{8} \, a^{4} b x^{8}{\rm sign}\left (b x^{3} + a\right ) + \frac{1}{5} \, a^{5} x^{5}{\rm sign}\left (b x^{3} + a\right ) \]
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^4,x, algorithm="giac")
[Out]