Optimal. Leaf size=160 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]
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Rubi [A] time = 0.292674, antiderivative size = 160, 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 \[ \frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^8}{18 b^4}-\frac{3 a \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^7}{16 b^4}+\frac{3 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^6}{14 b^4}-\frac{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^5}{12 b^4} \]
Antiderivative was successfully verified.
[In] Int[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 28.4076, size = 151, normalized size = 0.94 \[ - \frac{a^{3} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{288 b^{4}} + \frac{a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{7}{2}}}{168 b^{4}} - \frac{a x^{4} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{96 b^{2}} + \frac{x^{6} \left (2 a + 2 b x^{2}\right ) \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}}{36 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
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Mathematica [A] time = 0.0348257, size = 83, normalized size = 0.52 \[ \frac{x^8 \sqrt{\left (a+b x^2\right )^2} \left (126 a^5+504 a^4 b x^2+840 a^3 b^2 x^4+720 a^2 b^3 x^6+315 a b^4 x^8+56 b^5 x^{10}\right )}{1008 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^7*(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Maple [A] time = 0.009, size = 80, normalized size = 0.5 \[{\frac{{x}^{8} \left ( 56\,{b}^{5}{x}^{10}+315\,a{b}^{4}{x}^{8}+720\,{a}^{2}{b}^{3}{x}^{6}+840\,{a}^{3}{b}^{2}{x}^{4}+504\,{a}^{4}b{x}^{2}+126\,{a}^{5} \right ) }{1008\, \left ( b{x}^{2}+a \right ) ^{5}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7*(b^2*x^4+2*a*b*x^2+a^2)^(5/2),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^4 + 2*a*b*x^2 + a^2)^(5/2)*x^7,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.258676, size = 77, normalized size = 0.48 \[ \frac{1}{18} \, b^{5} x^{18} + \frac{5}{16} \, a b^{4} x^{16} + \frac{5}{7} \, a^{2} b^{3} x^{14} + \frac{5}{6} \, a^{3} b^{2} x^{12} + \frac{1}{2} \, a^{4} b x^{10} + \frac{1}{8} \, a^{5} x^{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^7,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x^{7} \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7*(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.270583, size = 142, normalized size = 0.89 \[ \frac{1}{18} \, b^{5} x^{18}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{16} \, a b^{4} x^{16}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{7} \, a^{2} b^{3} x^{14}{\rm sign}\left (b x^{2} + a\right ) + \frac{5}{6} \, a^{3} b^{2} x^{12}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{2} \, a^{4} b x^{10}{\rm sign}\left (b x^{2} + a\right ) + \frac{1}{8} \, a^{5} x^{8}{\rm sign}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2)*x^7,x, algorithm="giac")
[Out]