3.245 \(\int (d+e x) \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=83 \[ d x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e (p+1)} \]

[Out]

-(d^2 - e^2*x^2)^(1 + p)/(2*e*(1 + p)) + (d*x*(d^2 - e^2*x^2)^p*Hypergeometric2F
1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p

_______________________________________________________________________________________

Rubi [A]  time = 0.0567295, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ d x \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )-\frac{\left (d^2-e^2 x^2\right )^{p+1}}{2 e (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)*(d^2 - e^2*x^2)^p,x]

[Out]

-(d^2 - e^2*x^2)^(1 + p)/(2*e*(1 + p)) + (d*x*(d^2 - e^2*x^2)^p*Hypergeometric2F
1[1/2, -p, 3/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 25.1072, size = 66, normalized size = 0.8 \[ - \frac{2 d \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 1} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 1, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(-e**2*x**2+d**2)**p,x)

[Out]

-2*d*((d/2 + e*x/2)/d)**(-p)*(d - e*x)**(-p)*(d - e*x)**(p + 1)*(d**2 - e**2*x**
2)**p*hyper((-p - 1, p + 1), (p + 2,), (d/2 - e*x/2)/d)/(e*(p + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0719559, size = 116, normalized size = 1.4 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (2 d e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p-d^2 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )\right )}{2 e (p+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)*(d^2 - e^2*x^2)^p,x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(-e^2*x^2+d^2)^p,x)

[Out]

int((e*x+d)*(-e^2*x^2+d^2)^p,x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(-e^2*x^2 + d^2)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)*(-e^2*x^2 + d^2)^p, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(-e^2*x^2 + d^2)^p,x, algorithm="fricas")

[Out]

integral((e*x + d)*(-e^2*x^2 + d^2)^p, x)

_______________________________________________________________________________________

Sympy [A]  time = 6.31458, size = 82, normalized size = 0.99 \[ d d^{2 p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(-e**2*x**2+d**2)**p,x)

[Out]

d*d**(2*p)*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + e*Piec
ewise((x**2*(d**2)**p/2, Eq(e**2, 0)), (-Piecewise(((d**2 - e**2*x**2)**(p + 1)/
(p + 1), Ne(p, -1)), (log(d**2 - e**2*x**2), True))/(2*e**2), True))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(-e^2*x^2 + d^2)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)*(-e^2*x^2 + d^2)^p, x)