3.261 \(\int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=189 \[ -\frac{3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}+\frac{5 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^3 (p+2)}-\frac{\left (d^2-e^2 x^2\right )^{p+3}}{2 e^3 (p+3)}-\frac{2 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^3 (p+1)}+\frac{2 d^3 (p+7) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 (2 p+5)} \]

[Out]

(-2*d^4*(d^2 - e^2*x^2)^(1 + p))/(e^3*(1 + p)) - (3*d*x^3*(d^2 - e^2*x^2)^(1 + p
))/(5 + 2*p) + (5*d^2*(d^2 - e^2*x^2)^(2 + p))/(2*e^3*(2 + p)) - (d^2 - e^2*x^2)
^(3 + p)/(2*e^3*(3 + p)) + (2*d^3*(7 + p)*x^3*(d^2 - e^2*x^2)^p*Hypergeometric2F
1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(3*(5 + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.361741, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}+\frac{5 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^3 (p+2)}-\frac{\left (d^2-e^2 x^2\right )^{p+3}}{2 e^3 (p+3)}-\frac{2 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^3 (p+1)}+\frac{2 d^3 (p+7) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )}{3 (2 p+5)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d^4*(d^2 - e^2*x^2)^(1 + p))/(e^3*(1 + p)) - (3*d*x^3*(d^2 - e^2*x^2)^(1 + p
))/(5 + 2*p) + (5*d^2*(d^2 - e^2*x^2)^(2 + p))/(2*e^3*(2 + p)) - (d^2 - e^2*x^2)
^(3 + p)/(2*e^3*(3 + p)) + (2*d^3*(7 + p)*x^3*(d^2 - e^2*x^2)^p*Hypergeometric2F
1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(3*(5 + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 77.5815, size = 178, normalized size = 0.94 \[ - \frac{2 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{e^{3} \left (p + 1\right )} + \frac{d^{3} x^{3} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{3} + \frac{5 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{2 e^{3} \left (p + 2\right )} + \frac{3 d e^{2} x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5} - \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 3}}{2 e^{3} \left (p + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d**4*(d**2 - e**2*x**2)**(p + 1)/(e**3*(p + 1)) + d**3*x**3*(1 - e**2*x**2/d*
*2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p, 3/2), (5/2,), e**2*x**2/d**2)/3 + 5*d
**2*(d**2 - e**2*x**2)**(p + 2)/(2*e**3*(p + 2)) + 3*d*e**2*x**5*(1 - e**2*x**2/
d**2)**(-p)*(d**2 - e**2*x**2)**p*hyper((-p, 5/2), (7/2,), e**2*x**2/d**2)/5 - (
d**2 - e**2*x**2)**(p + 3)/(2*e**3*(p + 3))

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Mathematica [A]  time = 0.344381, size = 269, normalized size = 1.42 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (10 d^3 e^3 \left (p^3+6 p^2+11 p+6\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};\frac{e^2 x^2}{d^2}\right )-3 \left (-5 e^6 \left (p^2+3 p+2\right ) x^6 \left (1-\frac{e^2 x^2}{d^2}\right )^p-6 d e^5 \left (p^3+6 p^2+11 p+6\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-5 d^2 e^4 \left (2 p^2+11 p+9\right ) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+5 d^6 (3 p+11) \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )+5 d^4 e^2 p (3 p+11) x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )\right )}{30 e^3 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

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

[Out]

((d^2 - e^2*x^2)^p*(10*d^3*e^3*(6 + 11*p + 6*p^2 + p^3)*x^3*Hypergeometric2F1[3/
2, -p, 5/2, (e^2*x^2)/d^2] - 3*(5*d^4*e^2*p*(11 + 3*p)*x^2*(1 - (e^2*x^2)/d^2)^p
- 5*d^2*e^4*(9 + 11*p + 2*p^2)*x^4*(1 - (e^2*x^2)/d^2)^p - 5*e^6*(2 + 3*p + p^2
)*x^6*(1 - (e^2*x^2)/d^2)^p + 5*d^6*(11 + 3*p)*(-1 + (1 - (e^2*x^2)/d^2)^p) - 6*
d*e^5*(6 + 11*p + 6*p^2 + p^3)*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2
])))/(30*e^3*(1 + p)*(2 + p)*(3 + p)*(1 - (e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( ex+d \right ) ^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{5} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{3} + d^{3} x^{2}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^5 + 3*d*e^2*x^4 + 3*d^2*e*x^3 + d^3*x^2)*(-e^2*x^2 + d^2)^p, x)

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Sympy [A]  time = 23.9146, size = 1370, normalized size = 7.25 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*d**(2*p)*x**3*hyper((3/2, -p), (5/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/3
+ 3*d**2*e*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)) + 3*d*d**(2*p)*e**2*x**5*hyper((5/2, -p)
, (7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/5 + e**3*Piecewise((x**6*(d**2)**p/6
, Eq(e, 0)), (-2*d**4*log(-d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x*
*4) - 2*d**4*log(d/e + x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) - 3*d*
*4/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) + 4*d**2*e**2*x**2*log(-d/e +
x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) + 4*d**2*e**2*x**2*log(d/e +
x)/(4*d**4*e**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) + 4*d**2*e**2*x**2/(4*d**4*e
**6 - 8*d**2*e**8*x**2 + 4*e**10*x**4) - 2*e**4*x**4*log(-d/e + x)/(4*d**4*e**6
- 8*d**2*e**8*x**2 + 4*e**10*x**4) - 2*e**4*x**4*log(d/e + x)/(4*d**4*e**6 - 8*d
**2*e**8*x**2 + 4*e**10*x**4), Eq(p, -3)), (-2*d**4*log(-d/e + x)/(-2*d**2*e**6
+ 2*e**8*x**2) - 2*d**4*log(d/e + x)/(-2*d**2*e**6 + 2*e**8*x**2) - 2*d**4/(-2*d
**2*e**6 + 2*e**8*x**2) + 2*d**2*e**2*x**2*log(-d/e + x)/(-2*d**2*e**6 + 2*e**8*
x**2) + 2*d**2*e**2*x**2*log(d/e + x)/(-2*d**2*e**6 + 2*e**8*x**2) + e**4*x**4/(
-2*d**2*e**6 + 2*e**8*x**2), Eq(p, -2)), (-d**4*log(-d/e + x)/(2*e**6) - d**4*lo
g(d/e + x)/(2*e**6) - d**2*x**2/(2*e**4) - x**4/(4*e**2), Eq(p, -1)), (-2*d**6*(
d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) - 2*d**4
*e**2*p*x**2*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*
e**6) - d**2*e**4*p**2*x**4*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*e**6*p**2 +
22*e**6*p + 12*e**6) - d**2*e**4*p*x**4*(d**2 - e**2*x**2)**p/(2*e**6*p**3 + 12*
e**6*p**2 + 22*e**6*p + 12*e**6) + e**6*p**2*x**6*(d**2 - e**2*x**2)**p/(2*e**6*
p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + 3*e**6*p*x**6*(d**2 - e**2*x**2)**p
/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6) + 2*e**6*x**6*(d**2 - e**2*x
**2)**p/(2*e**6*p**3 + 12*e**6*p**2 + 22*e**6*p + 12*e**6), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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