3.949 \(\int (d+e x)^3 \left (1-\frac{e^2 x^2}{d^2}\right )^p \, dx\)

Optimal. Leaf size=57 \[ -\frac{d^4 2^{p+3} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-3,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

[Out]

-((2^(3 + p)*d^4*((d - e*x)/d)^(1 + p)*Hypergeometric2F1[-3 - p, 1 + p, 2 + p, (
d - e*x)/(2*d)])/(e*(1 + p)))

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Rubi [A]  time = 0.10693, 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 \[ -\frac{d^4 2^{p+3} \left (\frac{d-e x}{d}\right )^{p+1} \, _2F_1\left (-p-3,p+1;p+2;\frac{d-e x}{2 d}\right )}{e (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((2^(3 + p)*d^4*((d - e*x)/d)^(1 + p)*Hypergeometric2F1[-3 - p, 1 + p, 2 + p, (
d - e*x)/(2*d)])/(e*(1 + p)))

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Rubi in Sympy [A]  time = 34.1974, size = 80, normalized size = 1.4 \[ - \frac{8 d^{5} \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{- p} \left (\frac{1}{d} - \frac{e x}{d^{2}}\right )^{p + 1}{{}_{2}F_{1}\left (\begin{matrix} - p - 3, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{1}{2} - \frac{e x}{2 d}} \right )}}{e \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*d**5*((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 - 3, p + 1), (p + 2,), 1/2 - e*x/(2*d))/(e*(p
+ 1))

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Mathematica [B]  time = 0.204271, size = 240, normalized size = 4.21 \[ \frac{3 d^4 p+7 d^4+6 d^2 e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+2 d^2 e^2 p x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+e^4 x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+e^4 p x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+2 d e^3 \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-7 d^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p-3 d^4 p \left (1-\frac{e^2 x^2}{d^2}\right )^p+2 d^3 e \left (p^2+3 p+2\right ) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};\frac{e^2 x^2}{d^2}\right )}{2 e (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

(7*d^4 + 3*d^4*p - 7*d^4*(1 - (e^2*x^2)/d^2)^p - 3*d^4*p*(1 - (e^2*x^2)/d^2)^p +
6*d^2*e^2*x^2*(1 - (e^2*x^2)/d^2)^p + 2*d^2*e^2*p*x^2*(1 - (e^2*x^2)/d^2)^p + e
^4*x^4*(1 - (e^2*x^2)/d^2)^p + e^4*p*x^4*(1 - (e^2*x^2)/d^2)^p + 2*d^3*e*(2 + 3*
p + p^2)*x*Hypergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + 2*d*e^3*(2 + 3*p + p
^2)*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(2*e*(1 + p)*(2 + p))

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Maple [A]  time = 0.101, size = 104, normalized size = 1.8 \[{\frac{{e}^{3}{x}^{4}}{4}{\mbox{$_2$F$_1$}(2,-p;\,3;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+d{e}^{2}{x}^{3}{\mbox{$_2$F$_1$}({\frac{3}{2}},-p;\,{\frac{5}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}+{\frac{3\,{d}^{2}e{x}^{2}}{2}{\mbox{$_2$F$_1$}(1,-p;\,2;\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})}}+{d}^{3}x{\mbox{$_2$F$_1$}({\frac{1}{2}},-p;\,{\frac{3}{2}};\,{\frac{{e}^{2}{x}^{2}}{{d}^{2}}})} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*e^3*x^4*hypergeom([2,-p],[3],e^2*x^2/d^2)+d*e^2*x^3*hypergeom([3/2,-p],[5/2]
,e^2*x^2/d^2)+3/2*e*d^2*x^2*hypergeom([1,-p],[2],e^2*x^2/d^2)+d^3*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. \[ \int{\left (e x + d\right )}^{3}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(-e^2*x^2/d^2 + 1)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)^3*(-e^2*x^2/d^2 + 1)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \left (-\frac{e^{2} x^{2} - d^{2}}{d^{2}}\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(-e^2*x^2/d^2 + 1)^p,x, algorithm="fricas")

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(-(e^2*x^2 - d^2)/d^2)^p, x)

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Sympy [A]  time = 13.7988, size = 479, normalized size = 8.4 \[ d^{3} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + 3 d^{2} e \left (\begin{cases} \frac{x^{2}}{2} & \text{for}\: e^{2} = 0 \\- \frac{d^{2} \left (\begin{cases} \frac{\left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (1 - \frac{e^{2} x^{2}}{d^{2}} \right )} & \text{otherwise} \end{cases}\right )}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + d e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )} + e^{3} \left (\begin{cases} \frac{x^{4}}{4} & \text{for}\: e = 0 \\- \frac{d^{6} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{6} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{6}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{d^{4} e^{2} x^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{d^{4} e^{2} x^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text{for}\: p = -2 \\- \frac{d^{4} \log{\left (- \frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{4} \log{\left (\frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{2} x^{2}}{2 e^{2}} & \text{for}\: p = -1 \\- \frac{d^{4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac{d^{2} e^{2} p x^{2} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} p x^{4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} x^{4} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*x*hyper((1/2, -p), (3/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + 3*d**2*e*Pie
cewise((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)) + d*e**2*x*
*3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2) + e**3*Piecewise((
x**4/4, Eq(e, 0)), (-d**6*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**6*log(
d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**6/(-2*d**2*e**4 + 2*e**6*x**2) + d**4
*e**2*x**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) + d**4*e**2*x**2*log(d/e +
x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2)), (-d**4*log(-d/e + x)/(2*e**4) - d*
*4*log(d/e + x)/(2*e**4) - d**2*x**2/(2*e**2), Eq(p, -1)), (-d**4*(1 - e**2*x**2
/d**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) - d**2*e**2*p*x**2*(1 - e**2*x**2/d*
*2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*p*x**4*(1 - e**2*x**2/d**2)**p/(
2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*x**4*(1 - e**2*x**2/d**2)**p/(2*e**4*p**
2 + 6*e**4*p + 4*e**4), True))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (-\frac{e^{2} x^{2}}{d^{2}} + 1\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^3*(-e^2*x^2/d^2 + 1)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)^3*(-e^2*x^2/d^2 + 1)^p, x)