Optimal. Leaf size=20 \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]
[Out]
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Rubi [A] time = 0.0201132, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
[In] Int[x^n*Sqrt[1 + x^(1 + n)],x]
[Out]
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Rubi in Sympy [A] time = 1.98794, size = 15, normalized size = 0.75 \[ \frac{2 \left (x^{n + 1} + 1\right )^{\frac{3}{2}}}{3 \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**n*(1+x**(1+n))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0197861, size = 20, normalized size = 1. \[ \frac{2 \left (x^{n+1}+1\right )^{3/2}}{3 (n+1)} \]
Antiderivative was successfully verified.
[In] Integrate[x^n*Sqrt[1 + x^(1 + n)],x]
[Out]
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Maple [A] time = 0.043, size = 17, normalized size = 0.9 \[{\frac{2}{3+3\,n} \left ( 1+x{x}^{n} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^n*(1+x^(1+n))^(1/2),x)
[Out]
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Maxima [A] time = 1.33615, size = 22, normalized size = 1.1 \[ \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^(n + 1) + 1)*x^n,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221292, size = 22, normalized size = 1.1 \[ \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^(n + 1) + 1)*x^n,x, algorithm="fricas")
[Out]
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Sympy [A] time = 16.5285, size = 48, normalized size = 2.4 \[ \begin{cases} \frac{2 x x^{n} \sqrt{x x^{n} + 1}}{3 n + 3} + \frac{2 \sqrt{x x^{n} + 1}}{3 n + 3} & \text{for}\: n \neq -1 \\\sqrt{2} \log{\left (x \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**n*(1+x**(1+n))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214726, size = 22, normalized size = 1.1 \[ \frac{2 \,{\left (x^{n + 1} + 1\right )}^{\frac{3}{2}}}{3 \,{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^(n + 1) + 1)*x^n,x, algorithm="giac")
[Out]