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 = 0.0060438, 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, Rules used = {64} \[ -\frac{(-x)^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]

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)))

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-(d/(b*c)), 0])))

Rubi steps

\begin{align*} \int \frac{(-x)^m}{\sqrt{2+3 x}} \, dx &=-\frac{(-x)^{1+m} \, _2F_1\left (\frac{1}{2},1+m;2+m;-\frac{3 x}{2}\right )}{\sqrt{2} (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0050011, size = 32, normalized size = 0.94 \[ \frac{x (-x)^m \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(-x)^m/Sqrt[2 + 3*x],x]

[Out]

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

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Maple [A]  time = 0.017, size = 30, normalized size = 0.9 \begin{align*}{\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}})}} \end{align*}

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. \begin{align*} \int \frac{\left (-x\right )^{m}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^m/(2+3*x)^(1/2),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. \begin{align*}{\rm integral}\left (\frac{\left (-x\right )^{m}}{\sqrt{3 \, x + 2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^m/(2+3*x)^(1/2),x, algorithm="fricas")

[Out]

integral((-x)^m/sqrt(3*x + 2), x)

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Sympy [C]  time = 0.989137, size = 44, normalized size = 1.29 \begin{align*} \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} \end{align*}

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_polar(2*I*pi)/2)/3

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (-x\right )^{m}}{\sqrt{3 \, x + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x)^m/(2+3*x)^(1/2),x, algorithm="giac")

[Out]

integrate((-x)^m/sqrt(3*x + 2), x)