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

Optimal. Leaf size=71 \[ -\frac{d 2^{p+2} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0775095, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{d 2^{p+2} \left (\frac{e x}{d}+1\right )^{-p-1} \left (d^2-e^2 x^2\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.7078, size = 68, normalized size = 0.96 \[ - \frac{4 d^{2} \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 1} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*d**2*((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 - 2, p + 1), (p + 2,), (d/2 - e*x/2)/d)/(e*(p + 1))

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Mathematica [B]  time = 0.182647, size = 150, normalized size = 2.11 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (3 d^2 e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+3 d e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+e^3 (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-3 d^3 \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )\right )}{3 e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

((d^2 - e^2*x^2)^p*(3*d*e^2*x^2*(1 - (e^2*x^2)/d^2)^p - 3*d^3*(-1 + (1 - (e^2*x^
2)/d^2)^p) + 3*d^2*e*(1 + p)*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] +
e^3*(1 + p)*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2]))/(3*e*(1 + p)*(1
- (e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.68044, size = 124, normalized size = 1.75 \[ d^{2} d^{2 p} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 2 d e \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{d^{2 p} e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*d**(2*p)*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + 2*d
*e*Piecewise((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)) + d*
*(2*p)*e**2*x**3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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