\(\int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx\) [726]

Optimal result

Integrand size = 15, antiderivative size = 49 \[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=2^{1+m} 3^{-1-m} (-x)^m x^{-m} \sqrt {-2+3 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\frac {3 x}{2}\right ) \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {69, 12, 67} \[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=2^{m+1} 3^{-m-1} \sqrt {3 x-2} (-x)^m x^{-m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\frac {3 x}{2}\right ) \]

[In]

Int[(-x)^m/Sqrt[-2 + 3*x],x]

[Out]

(2^(1 + m)*3^(-1 - m)*(-x)^m*Sqrt[-2 + 3*x]*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*} \text {integral}& = \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} (-x)^m x^{-m} \sqrt {-2+3 x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=2^{1+m} 3^{-1-m} (-x)^m x^{-m} \sqrt {-2+3 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\frac {3 x}{2}\right ) \]

[In]

Integrate[(-x)^m/Sqrt[-2 + 3*x],x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 5.

Time = 0.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90

method	result	size
meijerg	\(\frac {\sqrt {2}\, \left (-x \right )^{m} \sqrt {-\operatorname {signum}\left (-\frac {2}{3}+x \right )}\, x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},1+m ;2+m ;\frac {3 x}{2}\right )}{2 \sqrt {\operatorname {signum}\left (-\frac {2}{3}+x \right )}\, \left (1+m \right )}\)	\(44\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=\int { \frac {\left (-x\right )^{m}}{\sqrt {3 \, x - 2}} \,d x } \]

[In]

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

[Out]

integral((-x)^m/sqrt(3*x - 2), x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=- \frac {\sqrt {2} i x^{m + 1} e^{i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=\int { \frac {\left (-x\right )^{m}}{\sqrt {3 \, x - 2}} \,d x } \]

[In]

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

[Out]

integrate((-x)^m/sqrt(3*x - 2), x)

Giac [F]

\[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=\int { \frac {\left (-x\right )^{m}}{\sqrt {3 \, x - 2}} \,d x } \]

[In]

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

[Out]

integrate((-x)^m/sqrt(3*x - 2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx=\int \frac {{\left (-x\right )}^m}{\sqrt {3\,x-2}} \,d x \]

[In]

int((-x)^m/(3*x - 2)^(1/2),x)

[Out]

int((-x)^m/(3*x - 2)^(1/2), x)