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

Optimal. Leaf size=118 \[ \frac{2}{3} d e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^2 (p+2)} \]

[Out]

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

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Rubi [A]  time = 0.182293, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ \frac{2}{3} d e x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2 (p+1)}+\frac{\left (d^2-e^2 x^2\right )^{p+2}}{2 e^2 (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 34.0773, size = 102, normalized size = 0.86 \[ - \frac{4 d^{3} \left (\frac{\frac{d}{2} + \frac{e x}{2}}{d}\right )^{- p} \left (d - e x\right )^{- p} \left (d - e x\right )^{p + 1} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - 2, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{d}{2} - \frac{e x}{2}}{d}} \right )}}{e^{2} \left (p + 1\right ) \left (p + 2\right )} - \frac{\left (d + e x\right )^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{2 e^{2} \left (p + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*d**3*((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**2*(p + 1)*(p + 2)
) - (d + e*x)**2*(d**2 - e**2*x**2)**(p + 1)/(2*e**2*(p + 2))

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Mathematica [A]  time = 0.156604, size = 163, normalized size = 1.38 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (6 d^2 e^2 x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+3 e^4 (p+1) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+4 d e^3 \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-3 d^4 (p+3) \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )\right )}{6 e^2 (p+1) (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \[ \int x \left ( ex+d \right ) ^{2} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x*(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. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.7053, size = 440, normalized size = 3.73 \[ d^{2} \left (\begin{cases} \frac{x^{2} \left (d^{2}\right )^{p}}{2} & \text{for}\: e^{2} = 0 \\- \frac{\begin{cases} \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (d^{2} - e^{2} x^{2} \right )} & \text{otherwise} \end{cases}}{2 e^{2}} & \text{otherwise} \end{cases}\right ) + \frac{2 d d^{2 p} e x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{3} + e^{2} \left (\begin{cases} \frac{x^{4} \left (d^{2}\right )^{p}}{4} & \text{for}\: e = 0 \\- \frac{d^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} - \frac{d^{2}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{e^{2} x^{2} \log{\left (- \frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} + \frac{e^{2} x^{2} \log{\left (\frac{d}{e} + x \right )}}{- 2 d^{2} e^{4} + 2 e^{6} x^{2}} & \text{for}\: p = -2 \\- \frac{d^{2} \log{\left (- \frac{d}{e} + x \right )}}{2 e^{4}} - \frac{d^{2} \log{\left (\frac{d}{e} + x \right )}}{2 e^{4}} - \frac{x^{2}}{2 e^{2}} & \text{for}\: p = -1 \\- \frac{d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} - \frac{d^{2} e^{2} p x^{2} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} p x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} + \frac{e^{4} x^{4} \left (d^{2} - e^{2} x^{2}\right )^{p}}{2 e^{4} p^{2} + 6 e^{4} p + 4 e^{4}} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*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)) +
2*d*d**(2*p)*e*x**3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/3
+ e**2*Piecewise((x**4*(d**2)**p/4, Eq(e, 0)), (-d**2*log(-d/e + x)/(-2*d**2*e*
*4 + 2*e**6*x**2) - d**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) - d**2/(-2*d*
*2*e**4 + 2*e**6*x**2) + e**2*x**2*log(-d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2) +
e**2*x**2*log(d/e + x)/(-2*d**2*e**4 + 2*e**6*x**2), Eq(p, -2)), (-d**2*log(-d/e
+ x)/(2*e**4) - d**2*log(d/e + x)/(2*e**4) - x**2/(2*e**2), Eq(p, -1)), (-d**4*
(d**2 - e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) - d**2*e**2*p*x**2*(d**2
- e**2*x**2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*p*x**4*(d**2 - e**2*x*
*2)**p/(2*e**4*p**2 + 6*e**4*p + 4*e**4) + e**4*x**4*(d**2 - e**2*x**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. \[ \int{\left (e x + d\right )}^{2}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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