3.2696 \(\int \frac{x^m}{\sqrt{a+b x^{2-m}}} \, dx\)

Optimal. Leaf size=67 \[ \frac{x^{m+1} \sqrt{a+b x^{2-m}} \, _2F_1\left (1,\frac{m+4}{2 (2-m)};\frac{3}{2-m};-\frac{b x^{2-m}}{a}\right )}{a (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0341092, antiderivative size = 81, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ \frac{x^{m+1} \sqrt{\frac{b x^{2-m}}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2-m};\frac{3}{2-m};-\frac{b x^{2-m}}{a}\right )}{(m+1) \sqrt{a+b x^{2-m}}} \]

Antiderivative was successfully verified.

[In]

Int[x^m/Sqrt[a + b*x^(2 - m)],x]

[Out]

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

Rule 365

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

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{x^m}{\sqrt{a+b x^{2-m}}} \, dx &=\frac{\sqrt{1+\frac{b x^{2-m}}{a}} \int \frac{x^m}{\sqrt{1+\frac{b x^{2-m}}{a}}} \, dx}{\sqrt{a+b x^{2-m}}}\\ &=\frac{x^{1+m} \sqrt{1+\frac{b x^{2-m}}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m}{2-m};\frac{3}{2-m};-\frac{b x^{2-m}}{a}\right )}{(1+m) \sqrt{a+b x^{2-m}}}\\ \end{align*}

Mathematica [A]  time = 0.0449311, size = 79, normalized size = 1.18 \[ \frac{x^{m+1} \sqrt{\frac{b x^{2-m}}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+1}{2-m};-\frac{3}{m-2};-\frac{b x^{2-m}}{a}\right )}{(m+1) \sqrt{a+b x^{2-m}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m/Sqrt[a + b*x^(2 - m)],x]

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\sqrt{a+b{x}^{2-m}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{b x^{-m + 2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/sqrt(b*x^(-m + 2) + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 102.598, size = 95, normalized size = 1.42 \begin{align*} - \frac{x x^{m} \Gamma \left (\frac{m}{2 - m} + \frac{1}{2 - m}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2 - m} + \frac{1}{2 - m} \\ \frac{m}{2 - m} + 1 + \frac{1}{2 - m} \end{matrix}\middle |{\frac{b x^{2} x^{- m} e^{i \pi }}{a}} \right )}}{\sqrt{a} m \Gamma \left (\frac{m}{2 - m} + 1 + \frac{1}{2 - m}\right ) - 2 \sqrt{a} \Gamma \left (\frac{m}{2 - m} + 1 + \frac{1}{2 - m}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m/(a+b*x**(2-m))**(1/2),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{b x^{-m + 2} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m/sqrt(b*x^(-m + 2) + a), x)