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

Optimal. Leaf size=193 \[ \frac{2 d^2 e (3 p+13) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac{e x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac{3 d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^4 (p+3)}-\frac{2 d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}+\frac{7 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)} \]

[Out]

(-2*d^5*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 + p)) - (e*x^5*(d^2 - e^2*x^2)^(1 + p))
/(7 + 2*p) + (7*d^3*(d^2 - e^2*x^2)^(2 + p))/(2*e^4*(2 + p)) - (3*d*(d^2 - e^2*x
^2)^(3 + p))/(2*e^4*(3 + p)) + (2*d^2*e*(13 + 3*p)*x^5*(d^2 - e^2*x^2)^p*Hyperge
ometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(5*(7 + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi [A]  time = 0.36716, antiderivative size = 193, 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 \[ \frac{2 d^2 e (3 p+13) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 (2 p+7)}-\frac{e x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac{3 d \left (d^2-e^2 x^2\right )^{p+3}}{2 e^4 (p+3)}-\frac{2 d^5 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}+\frac{7 d^3 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^4 (p+2)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*d^5*(d^2 - e^2*x^2)^(1 + p))/(e^4*(1 + p)) - (e*x^5*(d^2 - e^2*x^2)^(1 + p))
/(7 + 2*p) + (7*d^3*(d^2 - e^2*x^2)^(2 + p))/(2*e^4*(2 + p)) - (3*d*(d^2 - e^2*x
^2)^(3 + p))/(2*e^4*(3 + p)) + (2*d^2*e*(13 + 3*p)*x^5*(d^2 - e^2*x^2)^p*Hyperge
ometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(5*(7 + 2*p)*(1 - (e^2*x^2)/d^2)^p)

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Rubi in Sympy [A]  time = 82.2782, size = 182, normalized size = 0.94 \[ - \frac{2 d^{5} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{e^{4} \left (p + 1\right )} + \frac{7 d^{3} \left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{2 e^{4} \left (p + 2\right )} + \frac{3 d^{2} e x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5} - \frac{3 d \left (d^{2} - e^{2} x^{2}\right )^{p + 3}}{2 e^{4} \left (p + 3\right )} + \frac{e^{3} x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*d**5*(d**2 - e**2*x**2)**(p + 1)/(e**4*(p + 1)) + 7*d**3*(d**2 - e**2*x**2)**
(p + 2)/(2*e**4*(p + 2)) + 3*d**2*e*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 - 3*d*(d**2 - e**2*x**2)**(
p + 3)/(2*e**4*(p + 3)) + e**3*x**7*(1 - e**2*x**2/d**2)**(-p)*(d**2 - e**2*x**2
)**p*hyper((-p, 7/2), (9/2,), e**2*x**2/d**2)/7

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Mathematica [A]  time = 0.360297, size = 262, normalized size = 1.36 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (10 e^7 \left (p^3+6 p^2+11 p+6\right ) x^7 \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )+42 d^2 e^5 \left (p^3+6 p^2+11 p+6\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-35 d \left (-3 e^6 \left (p^2+3 p+2\right ) x^6 \left (1-\frac{e^2 x^2}{d^2}\right )^p+d^2 e^4 \left (2 p^2-p-3\right ) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p+d^6 (p+9) \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )+d^4 e^2 p (p+9) x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )\right )}{70 e^4 (p+1) (p+2) (p+3)} \]

Antiderivative was successfully verified.

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

[Out]

((d^2 - e^2*x^2)^p*(-35*d*(d^4*e^2*p*(9 + p)*x^2*(1 - (e^2*x^2)/d^2)^p + d^2*e^4
*(-3 - p + 2*p^2)*x^4*(1 - (e^2*x^2)/d^2)^p - 3*e^6*(2 + 3*p + p^2)*x^6*(1 - (e^
2*x^2)/d^2)^p + d^6*(9 + p)*(-1 + (1 - (e^2*x^2)/d^2)^p)) + 42*d^2*e^5*(6 + 11*p
+ 6*p^2 + p^3)*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2] + 10*e^7*(6 +
11*p + 6*p^2 + p^3)*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2]))/(70*e^
4*(1 + p)*(2 + p)*(3 + p)*(1 - (e^2*x^2)/d^2)^p)

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{x}^{3} \left ( ex+d \right ) ^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^3*(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. \[ \frac{{\left (e^{4}{\left (p + 1\right )} x^{4} - d^{2} e^{2} p x^{2} - d^{4}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} d^{3}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} e^{4}} + \int{\left (e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4}\right )} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(e^4*(p + 1)*x^4 - d^2*e^2*p*x^2 - d^4)*(-e^2*x^2 + d^2)^p*d^3/((p^2 + 3*p +
2)*e^4) + integrate((e^3*x^6 + 3*d*e^2*x^5 + 3*d^2*e*x^4)*e^(p*log(e*x + d) + p
*log(-e*x + d)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} x^{3}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e^3*x^6 + 3*d*e^2*x^5 + 3*d^2*e*x^4 + d^3*x^3)*(-e^2*x^2 + d^2)^p, x)

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Sympy [A]  time = 26.9542, size = 1370, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*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**2*d**(2*p)*e*x**5*hyper((5/2, -p), (7/2
,), e**2*x**2*exp_polar(2*I*pi)/d**2)/5 + 3*d*e**2*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*log(
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)) + d**(2*p)*e**3
*x**7*hyper((7/2, -p), (9/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/7

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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