3.1961 \(\int \frac{(c x)^m}{\sqrt{1+\frac{b}{x^2}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0173229, 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} \[ \frac{(c x)^{m+1} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-m-1);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(c*x)^m/Sqrt[1 + b/x^2],x]

[Out]

((c*x)^(1 + m)*Hypergeometric2F1[1/2, (-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}{\sqrt{1+\frac{b}{x^2}}} \, dx &=-\frac{\left (\left (\frac{1}{x}\right )^{1+m} (c x)^{1+m}\right ) \operatorname{Subst}\left (\int \frac{x^{-2-m}}{\sqrt{1+b x^2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{(c x)^{1+m} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-1-m);\frac{1-m}{2};-\frac{b}{x^2}\right )}{c (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0211419, size = 64, normalized size = 1.45 \[ \frac{x \sqrt{\frac{x^2}{b}+1} (c x)^m \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+2}{2}+1;-\frac{x^2}{b}\right )}{(m+2) \sqrt{\frac{b}{x^2}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c*x)^m/Sqrt[1 + b/x^2],x]

[Out]

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

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Maple [F]  time = 0.009, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{m}{\frac{1}{\sqrt{1+{\frac{b}{{x}^{2}}}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m/sqrt(b/x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.36886, 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{1}{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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c*x)^m/sqrt(b/x^2 + 1), x)