3.724 \(\int \left (a+c x^2\right )^p \, dx\)

Optimal. Leaf size=35 \[ \frac{x \left (a+c x^2\right )^{p+1} \, _2F_1\left (1,p+\frac{3}{2};\frac{3}{2};-\frac{c x^2}{a}\right )}{a} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0301267, antiderivative size = 44, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^p,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.79566, size = 34, normalized size = 0.97 \[ x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**p,x)

[Out]

x*(1 + c*x**2/a)**(-p)*(a + c*x**2)**p*hyper((-p, 1/2), (3/2,), -c*x**2/a)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0107402, size = 44, normalized size = 1.26 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^p,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int \left ( c{x}^{2}+a \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^p,x)

[Out]

int((c*x^2+a)^p,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p,x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^p, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{2} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p,x, algorithm="fricas")

[Out]

integral((c*x^2 + a)^p, x)

_______________________________________________________________________________________

Sympy [A]  time = 8.88157, size = 22, normalized size = 0.63 \[ a^{p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**p,x)

[Out]

a**p*x*hyper((1/2, -p), (3/2,), c*x**2*exp_polar(I*pi)/a)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c x^{2} + a\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^p,x, algorithm="giac")

[Out]

integrate((c*x^2 + a)^p, x)