3.8.22 \(\int \frac {x^m}{\sqrt {-2-3 x}} \, dx\) [722]

Optimal. Leaf size=50 \[ -2^{1+m} 3^{-1-m} \sqrt {-2-3 x} (-x)^{-m} x^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {3 x}{2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {69, 12, 67} \begin {gather*} -2^{m+1} 3^{-m-1} \sqrt {-3 x-2} (-x)^{-m} x^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};\frac {3 x}{2}+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m/Sqrt[-2 - 3*x],x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 67

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

Rule 69

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

Rubi steps

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

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Mathematica [A]
time = 0.05, size = 48, normalized size = 0.96 \begin {gather*} -\frac {2}{3} \left (1+\frac {1}{2} (-2-3 x)\right )^{-m} \sqrt {-2-3 x} x^m \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+\frac {3 x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m/Sqrt[-2 - 3*x],x]

[Out]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 2.28, size = 34, normalized size = 0.68 \begin {gather*} \frac {-I \sqrt {2} x^{1+m} \text {hyper}\left [\left \{\frac {1}{2},1+m\right \},\left \{2+m\right \},\frac {3 x \text {exp\_polar}\left [I \text {Pi}\right ]}{2}\right ]}{2+2 m} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^m/Sqrt[-2 - 3*x],x]')

[Out]

-I Sqrt[2] x ^ (1 + m) hyper[{1 / 2, 1 + m}, {2 + m}, 3 x exp_polar[I Pi] / 2] / (2 + 2 m)

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Maple [C] Result contains complex when optimal does not.
time = 0.10, size = 30, normalized size = 0.60

method	result	size
meijerg	\(-\frac {i x^{1+m} \hypergeom \left (\left [\frac {1}{2}, 1+m \right ], \left [2+m \right ], -\frac {3 x}{2}\right ) \sqrt {2}}{2 \left (1+m \right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(-2-3*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)/(3*x + 2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.60, size = 41, normalized size = 0.82 \begin {gather*} - \frac {\sqrt {2} i x x^{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 e^{i \pi }}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(-2-3*x)**(1/2),x)

[Out]

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

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m/(-2-3*x)^(1/2),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^m}{\sqrt {-3\,x-2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m/(- 3*x - 2)^(1/2),x)

[Out]

int(x^m/(- 3*x - 2)^(1/2), x)

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