Optimal. Leaf size=64 \[ \frac{x^3 \left (a+b x^n\right ) \, _2F_1\left (3,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )}{3 a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
[Out]
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Rubi [A] time = 0.0770577, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{x^3 \left (a+b x^n\right ) \, _2F_1\left (3,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )}{3 a^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 14.6689, size = 58, normalized size = 0.91 \[ \frac{b x^{3} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}{{}_{2}F_{1}\left (\begin{matrix} 3, \frac{3}{n} \\ \frac{n + 3}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{3 a^{3} \left (a b + b^{2} x^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)
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Mathematica [A] time = 0.102641, size = 98, normalized size = 1.53 \[ \frac{x^3 \left (a+b x^n\right ) \left (3 a^2 n+\left (2 n^2-9 n+9\right ) \left (a+b x^n\right )^2 \, _2F_1\left (1,\frac{3}{n};\frac{n+3}{n};-\frac{b x^n}{a}\right )+3 a (2 n-3) \left (a+b x^n\right )\right )}{6 a^3 n^2 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]
[Out]
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Maple [F] time = 0.088, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[{\left (2 \, n^{2} - 9 \, n + 9\right )} \int \frac{x^{2}}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n - 3\right )} x^{3} x^{n} + 3 \, a{\left (n - 1\right )} x^{3}}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (\left (a + b x^{n}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="giac")
[Out]