3.1962 \(\int \frac{(c x)^m}{(1+\frac{b}{x^2})^{3/2}} \, dx\)

Optimal. Leaf size=44 \[ \frac{(c x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.017609, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {339, 364} \[ \frac{(c x)^{m+1} \, _2F_1\left (\frac{3}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m/(1 + b/x^2)^(3/2),x]

[Out]

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

Rule 339

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Dist[((c*x)^(m + 1)*(1/x)^(m + 1))/c, Subst
[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x], x] /; FreeQ[{a, b, c, m, p}, x] && ILtQ[n, 0] &&  !RationalQ[m]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

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

Mathematica [A]  time = 0.0238654, size = 67, normalized size = 1.52 \[ \frac{x^3 \sqrt{\frac{b+x^2}{b}} (c x)^m \, _2F_1\left (\frac{3}{2},\frac{m}{2}+2;\frac{m}{2}+3;-\frac{x^2}{b}\right )}{b (m+4) \sqrt{\frac{b}{x^2}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m/(1 + b/x^2)^(3/2),x]

[Out]

(x^3*(c*x)^m*Sqrt[(b + x^2)/b]*Hypergeometric2F1[3/2, 2 + m/2, 3 + m/2, -(x^2/b)])/(b*(4 + m)*Sqrt[1 + b/x^2])

________________________________________________________________________________________

Maple [F]  time = 0.011, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{m} \left ( 1+{\frac{b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x)^m/(1+1/x^2*b)^(3/2),x)

[Out]

int((c*x)^m/(1+1/x^2*b)^(3/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m}}{{\left (\frac{b}{x^{2}} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\left (c x\right )^{m} x^{4} \sqrt{\frac{x^{2} + b}{x^{2}}}}{x^{4} + 2 \, b x^{2} + b^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*x)^m*x^4*sqrt((x^2 + b)/x^2)/(x^4 + 2*b*x^2 + b^2), x)

________________________________________________________________________________________

Sympy [C]  time = 3.21632, size = 54, normalized size = 1.23 \begin{align*} - \frac{c^{m} x x^{m} \Gamma \left (- \frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - \frac{m}{2} - \frac{1}{2} \\ \frac{1}{2} - \frac{m}{2} \end{matrix}\middle |{\frac{b e^{i \pi }}{x^{2}}} \right )}}{2 \Gamma \left (\frac{1}{2} - \frac{m}{2}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c x\right )^{m}}{{\left (\frac{b}{x^{2}} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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