Optimal. Leaf size=72 \[ \frac{x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{2 (m+1)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{2 \sqrt{b} (m+1)} \]
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Rubi [A] time = 0.074197, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{x^{m+1} \sqrt{a+b x^{2 (m+1)}}}{2 (m+1)}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a+b x^{2 (m+1)}}}\right )}{2 \sqrt{b} (m+1)} \]
Antiderivative was successfully verified.
[In] Int[x^m*Sqrt[a + b*x^(2 + 2*m)],x]
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Rubi in Sympy [A] time = 6.85381, size = 56, normalized size = 0.78 \[ \frac{x^{m + 1} \sqrt{a + b x^{2 m + 2}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{\sqrt{1 + \frac{b x^{2 m + 2}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(a+b*x**(2+2*m))**(1/2),x)
[Out]
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Mathematica [A] time = 0.139838, size = 93, normalized size = 1.29 \[ \frac{a^{3/2} \sqrt{\frac{b x^{2 m+2}}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )+\sqrt{b} x^{m+1} \left (a+b x^{2 m+2}\right )}{2 \sqrt{b} (m+1) \sqrt{a+b x^{2 m+2}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*Sqrt[a + b*x^(2 + 2*m)],x]
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Maple [F] time = 0.087, size = 0, normalized size = 0. \[ \int{x}^{m}\sqrt{a+b{x}^{2+2\,m}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(a+b*x^(2+2*m))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2 \, m + 2} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^(2*m + 2) + a)*x^m,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(sqrt(b*x^(2*m + 2) + a)*x^m,x, algorithm="fricas")
[Out]
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Sympy [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**m*(a+b*x**(2+2*m))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2 \, m + 2} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^(2*m + 2) + a)*x^m,x, algorithm="giac")
[Out]