Optimal. Leaf size=34 \[ -\frac{(-x)^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]
[Out]
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Rubi [A] time = 0.0209125, antiderivative size = 34, 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 \[ -\frac{(-x)^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(-x)^m/Sqrt[2 + 3*x],x]
[Out]
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Rubi in Sympy [A] time = 2.82217, size = 31, normalized size = 0.91 \[ - \frac{\sqrt{2} \left (- x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{3 x}{2}} \right )}}{2 \left (m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x)**m/(2+3*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0149829, size = 36, normalized size = 1.06 \[ \left (\frac{3}{2}\right )^{-m-1} \sqrt{3 x+2} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{3 x}{2}+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(-x)^m/Sqrt[2 + 3*x],x]
[Out]
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Maple [A] time = 0.026, size = 30, normalized size = 0.9 \[{\frac{\sqrt{2} \left ( -x \right ) ^{m}x}{2+2\,m}{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,-{\frac{3\,x}{2}})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x)^m/(2+3*x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (-x\right )^{m}}{\sqrt{3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x)^m/sqrt(3*x + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (-x\right )^{m}}{\sqrt{3 \, x + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x)^m/sqrt(3*x + 2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.25599, size = 44, normalized size = 1.29 \[ \frac{2 \cdot 2^{m} \sqrt{3} \cdot 3^{- m} \sqrt{x + \frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - m \\ \frac{3}{2} \end{matrix}\middle |{\frac{3 \left (x + \frac{2}{3}\right ) e^{2 i \pi }}{2}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x)**m/(2+3*x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (-x\right )^{m}}{\sqrt{3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x)^m/sqrt(3*x + 2),x, algorithm="giac")
[Out]