Optimal. Leaf size=63 \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]
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Rubi [A] time = 0.0560249, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[x^m*Sqrt[a + b*x^3],x]
[Out]
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Rubi in Sympy [A] time = 6.91581, size = 53, normalized size = 0.84 \[ \frac{x^{m + 1} \sqrt{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{\sqrt{1 + \frac{b x^{3}}{a}} \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(b*x**3+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0301965, size = 63, normalized size = 1. \[ \frac{x^{m+1} \sqrt{a+b x^3} \, _2F_1\left (-\frac{1}{2},\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{(m+1) \sqrt{\frac{b x^3}{a}+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x^m*Sqrt[a + b*x^3],x]
[Out]
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Maple [F] time = 0.029, size = 0, normalized size = 0. \[ \int{x}^{m}\sqrt{b{x}^{3}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(b*x^3+a)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{3} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^3 + a)*x^m,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x^{3} + a} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^3 + a)*x^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.70184, size = 54, normalized size = 0.86 \[ \frac{\sqrt{a} x x^{m} \Gamma \left (\frac{m}{3} + \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{m}{3} + \frac{1}{3} \\ \frac{m}{3} + \frac{4}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac{m}{3} + \frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(b*x**3+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{3} + a} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^3 + a)*x^m,x, algorithm="giac")
[Out]