3.11.69 \(\int x^{1-2 p} (a+b x^2)^p \, dx\) [1069]

Optimal. Leaf size=49 \[ \frac {x^{2-2 p} \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,2;2-p;-\frac {b x^2}{a}\right )}{2 a (1-p)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 64, normalized size of antiderivative = 1.31, 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 = {372, 371} \begin {gather*} \frac {x^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {b x^2}{a}\right )}{2 (1-p)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(1 - 2*p)*(a + b*x^2)^p,x]

[Out]

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

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 61, normalized size = 1.24 \begin {gather*} \frac {x^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {b x^2}{a}\right )}{2-2 p} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(1 - 2*p)*(a + b*x^2)^p,x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{1-2 p} \left (b \,x^{2}+a \right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^p*x^(-2*p + 1), x)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2 + a)^p*x^(-2*p + 1), x)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 111.87, size = 41, normalized size = 0.84 \begin {gather*} \frac {a^{p} x^{2} x^{- 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^2 + a)^p*x^(-2*p + 1), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^{1-2\,p}\,{\left (b\,x^2+a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1 - 2*p)*(a + b*x^2)^p,x)

[Out]

int(x^(1 - 2*p)*(a + b*x^2)^p, x)

________________________________________________________________________________________