Optimal. Leaf size=74 \[ \frac{x^6}{8 a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x^6}{24 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.0606992, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{x^6}{8 a \left (a+b x^2\right ) \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}}+\frac{x^6}{24 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a^2 + 2*a*b*x^2 + b^2*x^4)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 8.62612, size = 68, normalized size = 0.92 \[ \frac{x^{6} \left (2 a + 2 b x^{2}\right )}{16 a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{5}{2}}} + \frac{x^{6}}{24 a^{2} \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0341559, size = 50, normalized size = 0.68 \[ \frac{-a^2-4 a b x^2-6 b^2 x^4}{24 b^3 \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(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 = 43, normalized size = 0.6 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( 6\,{b}^{2}{x}^{4}+4\,ab{x}^{2}+{a}^{2} \right ) }{24\,{b}^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(b^2*x^4+2*a*b*x^2+a^2)^(5/2),x)
[Out]
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Maxima [A] time = 0.691088, size = 85, normalized size = 1.15 \[ -\frac{1}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{2}} + \frac{a b}{3 \,{\left (b^{2}\right )}^{\frac{7}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{3}} - \frac{a^{2} b^{2}}{8 \,{\left (b^{2}\right )}^{\frac{9}{2}}{\left (x^{2} + \frac{a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260328, size = 93, normalized size = 1.26 \[ -\frac{6 \, b^{2} x^{4} + 4 \, a b x^{2} + a^{2}}{24 \,{\left (b^{7} x^{8} + 4 \, a b^{6} x^{6} + 6 \, a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(b**2*x**4+2*a*b*x**2+a**2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.620702, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(b^2*x^4 + 2*a*b*x^2 + a^2)^(5/2),x, algorithm="giac")
[Out]