Optimal. Leaf size=59 \[ \frac{x^{2 (n+2)} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (5+\frac{8}{n}\right );3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 a (n+2)} \]
[Out]
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Rubi [A] time = 0.076905, antiderivative size = 70, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{x^{2 (n+2)} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},2 \left (1+\frac{2}{n}\right );3+\frac{4}{n};-\frac{b x^n}{a}\right )}{2 (n+2) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[x^(3 + 2*n)/Sqrt[a + b*x^n],x]
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Rubi in Sympy [A] time = 8.03668, size = 53, normalized size = 0.9 \[ \frac{x^{2 n + 4} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, 2 + \frac{4}{n} \\ 3 + \frac{4}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 a \sqrt{1 + \frac{b x^{n}}{a}} \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3+2*n)/(a+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.20525, size = 104, normalized size = 1.76 \[ \frac{2 x^4 \left (2 a^2 (n+4) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{4}{n};\frac{n+4}{n};-\frac{b x^n}{a}\right )-\left (a+b x^n\right ) \left (2 a (n+4)-b (n+8) x^n\right )\right )}{b^2 (n+8) (3 n+8) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3 + 2*n)/Sqrt[a + b*x^n],x]
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Maple [F] time = 0.081, size = 0, normalized size = 0. \[ \int{{x}^{3+2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3+2*n)/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n + 3)/sqrt(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n + 3)/sqrt(b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3+2*n)/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n + 3}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n + 3)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]