3.724 \(\int \frac{(-x)^m}{\sqrt{2+3 x}} \, dx\)

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]

-(((-x)^(1 + m)*Hypergeometric2F1[1/2, 1 + m, 2 + m, (-3*x)/2])/(Sqrt[2]*(1 + m)
))

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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]

-(((-x)^(1 + m)*Hypergeometric2F1[1/2, 1 + m, 2 + m, (-3*x)/2])/(Sqrt[2]*(1 + m)
))

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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]

-sqrt(2)*(-x)**(m + 1)*hyper((1/2, m + 1), (m + 2,), -3*x/2)/(2*(m + 1))

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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]

(3/2)^(-1 - m)*Sqrt[2 + 3*x]*Hypergeometric2F1[1/2, -m, 3/2, 1 + (3*x)/2]

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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]

1/2*2^(1/2)*(-x)^m/(1+m)*x*hypergeom([1/2,1+m],[2+m],-3/2*x)

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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]

integrate((-x)^m/sqrt(3*x + 2), x)

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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]

integral((-x)^m/sqrt(3*x + 2), x)

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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]

2*2**m*sqrt(3)*3**(-m)*sqrt(x + 2/3)*hyper((1/2, -m), (3/2,), 3*(x + 2/3)*exp_po
lar(2*I*pi)/2)/3

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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]

integrate((-x)^m/sqrt(3*x + 2), x)