3.1068 \(\int x^{-2 p} (a+b x^2)^p \, dx\)

Optimal. Leaf size=52 \[ \frac{x^{1-2 p} \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,\frac{3}{2};\frac{1}{2} (3-2 p);-\frac{b x^2}{a}\right )}{a (1-2 p)} \]

[Out]

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

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Rubi [A]  time = 0.0191409, antiderivative size = 69, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac{x^{1-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2} (1-2 p),-p;\frac{1}{2} (3-2 p);-\frac{b x^2}{a}\right )}{1-2 p} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^p/x^(2*p),x]

[Out]

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

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

Mathematica [A]  time = 0.0161678, size = 65, normalized size = 1.25 \[ \frac{x^{1-2 p} \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2}-p,-p;\frac{3}{2}-p;-\frac{b x^2}{a}\right )}{1-2 p} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^p/x^(2*p),x]

[Out]

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

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( b{x}^{2}+a \right ) ^{p}}{{x}^{2\,p}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^p/(x^(2*p)),x)

[Out]

int((b*x^2+a)^p/(x^(2*p)),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^p/x^(2*p), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2 + a)^p/x^(2*p), x)

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Sympy [C]  time = 24.1166, size = 24, normalized size = 0.46 \begin{align*} b^{p} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - p \\ \frac{1}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{2}}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**p/(x**(2*p)),x)

[Out]

b**p*x*hyper((-1/2, -p), (1/2,), a*exp_polar(I*pi)/(b*x**2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^p/x^(2*p), x)