Optimal. Leaf size=49 \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]
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Rubi [A] time = 0.0332155, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]
Antiderivative was successfully verified.
[In] Int[(-x)^m/Sqrt[-2 + 3*x],x]
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Rubi in Sympy [A] time = 3.72673, size = 37, normalized size = 0.76 \[ \frac{2 \left (- x\right )^{m} \left (\frac{3 x}{2}\right )^{- m} \sqrt{3 x - 2}{{}_{2}F_{1}\left (\begin{matrix} - m, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{3 x}{2} + 1} \right )}}{3} \]
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 = 0.0165527, size = 49, normalized size = 1. \[ 2^{m+1} 3^{-m-1} \sqrt{3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1-\frac{3 x}{2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(-x)^m/Sqrt[-2 + 3*x],x]
[Out]
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Maple [C] time = 0.047, size = 44, normalized size = 0.9 \[{\frac{\sqrt{2} \left ( -x \right ) ^{m}x}{2+2\,m}\sqrt{-{\it signum} \left ( x-{\frac{2}{3}} \right ) }{\mbox{$_2$F$_1$}({\frac{1}{2}},1+m;\,2+m;\,{\frac{3\,x}{2}})}{\frac{1}{\sqrt{{\it signum} \left ( x-{\frac{2}{3}} \right ) }}}} \]
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 = 3.45889, size = 42, normalized size = 0.86 \[ - \frac{\sqrt{2} i x x^{m} e^{i \pi m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]
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]