3.8.21 \(\int \frac {x^m}{\sqrt {-2+3 x}} \, dx\) [721]

Optimal. Leaf size=36 \[ \left (\frac {3}{2}\right )^{-1-m} \sqrt {-2+3 x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {67} \begin {gather*} \left (\frac {3}{2}\right )^{-m-1} \sqrt {3 x-2} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[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]

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])

Rubi steps

\begin {align*} \int \frac {x^m}{\sqrt {-2+3 x}} \, dx &=\left (\frac {3}{2}\right )^{-1-m} \sqrt {-2+3 x} \, _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 = 0.03, size = 36, normalized size = 1.00 \begin {gather*} \left (\frac {3}{2}\right )^{-1-m} \sqrt {-2+3 x} \, _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]

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 2.26, size = 30, normalized size = 0.83 \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}{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 / 2] / (2 + 2 m)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 5.
time = 0.13, size = 43, normalized size = 1.19

method	result	size
meijerg	\(\frac {\sqrt {2}\, \sqrt {-\mathrm {signum}\left (x -\frac {2}{3}\right )}\, x^{1+m} \hypergeom \left (\left [\frac {1}{2}, 1+m \right ], \left [2+m \right ], \frac {3 x}{2}\right )}{2 \sqrt {\mathrm {signum}\left (x -\frac {2}{3}\right )}\, \left (1+m \right )}\)	\(43\)

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

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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 = 0.31, 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)

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Sympy [C] Result contains complex when optimal does not.
time = 0.61, size = 36, normalized size = 1.00 \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}{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/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 = -0.03 \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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