3.950 \(\int (d+e x)^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx\)

Optimal. Leaf size=57 \[ -\frac{d^3 2^{p+2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0989112, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{d^3 2^{p+2} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-2,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((2^(2 + p)*d^3*((d - e*x)/d)^(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 = 32.9692, size = 80, normalized size = 1.4 \[ - \frac{4 d^{4} \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{- p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p - 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{1}{2} - \frac{e x}{2 d}} \right )}}{e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2*(1-e**2*x**2/d**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.243436, size = 112, normalized size = 1.96 \[ d^2 x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )+\frac{1}{3} e^2 x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )+\frac{d \left (-d^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+d^2\right )}{e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(d*(d^2 - d^2*(1 - (e^2*x^2)/d^2)^p + e^2*x^2*(1 - (e^2*x^2)/d^2)^p))/(e*(1 + p)
) + d^2*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (e^2*x^3*Hypergeometr
ic2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/3

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Maple [A]  time = 0.071, size = 75, normalized size = 1.3 \[{\frac{{e}^{2}{x}^{3}}{3}{\mbox{$_2$F$_1$}({\frac{3}{2}},-p;\,{\frac{5}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+ed{x}^{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}+{d}^{2}x{\mbox{$_2$F$_1$}({\frac{1}{2}},-p;\,{\frac{3}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{2}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*(-e^2*x^2/d^2 + 1)^p,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*(-e^2*x^2/d^2 + 1)^p,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.8553, size = 116, normalized size = 2.04 \[ d^{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 2 d e \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: e^{2} = 0 \\- \frac{d^{2} \left (\begin{cases} \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )} & \text{otherwise} \end{cases}\right )}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2*(1-e**2*x**2/d**2)**p,x)

[Out]

d**2*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + 2*d*e*Piecew
ise((x**2/2, Eq(e**2, 0)), (-d**2*Piecewise(((1 - e**2*x**2/d**2)**(p + 1)/(p +
1), Ne(p, -1)), (log(1 - e**2*x**2/d**2), True))/(2*e**2), True)) + e**2*x**3*hy
per((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. \[ \int{\left (e x + d\right )}^{2}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2*(-e^2*x^2/d^2 + 1)^p,x, algorithm="giac")

[Out]

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