3.8.24 \(\int \frac {(-x)^m}{\sqrt {2+3 x}} \, dx\) [724]

Optimal. Leaf size=34 \[ -\frac {(-x)^{1+m} \, _2F_1\left (\frac {1}{2},1+m;2+m;-\frac {3 x}{2}\right )}{\sqrt {2} (1+m)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {66} \begin {gather*} -\frac {(-x)^{m+1} \, _2F_1\left (\frac {1}{2},m+1;m+2;-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \end {gather*}

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 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], 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 = 0.01, size = 32, normalized size = 0.94 \begin {gather*} \frac {(-x)^m x \, _2F_1\left (\frac {1}{2},1+m;2+m;-\frac {3 x}{2}\right )}{\sqrt {2} (1+m)} \end {gather*}

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))

________________________________________________________________________________________

Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 2.28, size = 32, normalized size = 0.94 \begin {gather*} \left (\frac {2}{3}\right )^{1+m} \sqrt {2+3 x} \text {hyper}\left [\left \{\frac {1}{2},-m\right \},\left \{\frac {3}{2}\right \},\frac {\left (2+3 x\right ) \text {exp\_polar}\left [2 I \text {Pi}\right ]}{2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 30, normalized size = 0.88

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

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

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), x)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.59, size = 44, normalized size = 1.29 \begin {gather*} \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 {gather*}

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (-x\right )}^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)

________________________________________________________________________________________