3.719 \(\int \frac{x^m}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=31 \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]

[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 = 0.0055908, antiderivative size = 31, normalized size of antiderivative = 1., 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 = {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 = 0.005287, size = 31, normalized size = 1. \[ \frac{x^{m+1} \, _2F_1\left (\frac{1}{2},m+1;m+2;-\frac{3 x}{2}\right )}{\sqrt{2} (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[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))

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Maple [A]  time = 0.023, size = 29, normalized size = 0.9 \begin{align*}{\frac{{x}^{1+m}\sqrt{2}}{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*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., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{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{x^{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 = 1.00617, size = 37, normalized size = 1.19 \begin{align*} \frac{\sqrt{2} 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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{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)