3.259 \(\int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx\)
Optimal. Leaf size=218 \[ -\frac{3 d x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac{3 d^2 \left (d^2-e^2 x^2\right )^{p+3}}{e^5 (p+3)}+\frac{\left (d^2-e^2 x^2\right )^{p+4}}{2 e^5 (p+4)}-\frac{2 d^6 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac{9 d^4 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)}+\frac{2 d^3 (p+11) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 (2 p+7)} \]
[Out]
(-2*d^6*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p)) - (3*d*x^5*(d^2 - e^2*x^2)^(1 + p
))/(7 + 2*p) + (9*d^4*(d^2 - e^2*x^2)^(2 + p))/(2*e^5*(2 + p)) - (3*d^2*(d^2 - e
^2*x^2)^(3 + p))/(e^5*(3 + p)) + (d^2 - e^2*x^2)^(4 + p)/(2*e^5*(4 + p)) + (2*d^
3*(11 + p)*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[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 = 0.399901, antiderivative size = 218, 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
\[ -\frac{3 d x^5 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+7}-\frac{3 d^2 \left (d^2-e^2 x^2\right )^{p+3}}{e^5 (p+3)}+\frac{\left (d^2-e^2 x^2\right )^{p+4}}{2 e^5 (p+4)}-\frac{2 d^6 \left (d^2-e^2 x^2\right )^{p+1}}{e^5 (p+1)}+\frac{9 d^4 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^5 (p+2)}+\frac{2 d^3 (p+11) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )}{5 (2 p+7)} \]
Antiderivative was successfully verified.
[In] Int[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]
[Out]
(-2*d^6*(d^2 - e^2*x^2)^(1 + p))/(e^5*(1 + p)) - (3*d*x^5*(d^2 - e^2*x^2)^(1 + p
))/(7 + 2*p) + (9*d^4*(d^2 - e^2*x^2)^(2 + p))/(2*e^5*(2 + p)) - (3*d^2*(d^2 - e
^2*x^2)^(3 + p))/(e^5*(3 + p)) + (d^2 - e^2*x^2)^(4 + p)/(2*e^5*(4 + p)) + (2*d^
3*(11 + p)*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[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 = 94.6582, size = 204, normalized size = 0.94 \[ - \frac{2 d^{6} \left (d^{2} - e^{2} x^{2}\right )^{p + 1}}{e^{5} \left (p + 1\right )} + \frac{9 d^{4} \left (d^{2} - e^{2} x^{2}\right )^{p + 2}}{2 e^{5} \left (p + 2\right )} + \frac{d^{3} x^{5} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{5} - \frac{3 d^{2} \left (d^{2} - e^{2} x^{2}\right )^{p + 3}}{e^{5} \left (p + 3\right )} + \frac{3 d e^{2} x^{7} \left (1 - \frac{e^{2} x^{2}}{d^{2}}\right )^{- p} \left (d^{2} - e^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{e^{2} x^{2}}{d^{2}}} \right )}}{7} + \frac{\left (d^{2} - e^{2} x^{2}\right )^{p + 4}}{2 e^{5} \left (p + 4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(e*x+d)**3*(-e**2*x**2+d**2)**p,x)
[Out]
-2*d**6*(d**2 - e**2*x**2)**(p + 1)/(e**5*(p + 1)) + 9*d**4*(d**2 - e**2*x**2)**
(p + 2)/(2*e**5*(p + 2)) + d**3*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**2*(d**2 - e**2*x**2)**(p
+ 3)/(e**5*(p + 3)) + 3*d*e**2*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 + (d**2 - e**2*x**2)**(p + 4)/(
2*e**5*(p + 4))
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Mathematica [A] time = 0.447252, size = 323, normalized size = 1.48 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \left (30 d e^7 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^7 \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )+14 d^3 e^5 \left (p^4+10 p^3+35 p^2+50 p+24\right ) x^5 \, _2F_1\left (\frac{5}{2},-p;\frac{7}{2};\frac{e^2 x^2}{d^2}\right )-35 \left (-e^8 \left (p^3+6 p^2+11 p+6\right ) x^8 \left (1-\frac{e^2 x^2}{d^2}\right )^p-2 d^2 e^6 \left (p^3+9 p^2+20 p+12\right ) x^6 \left (1-\frac{e^2 x^2}{d^2}\right )^p+6 d^8 (p+5) \left (\left (1-\frac{e^2 x^2}{d^2}\right )^p-1\right )+6 d^6 e^2 p (p+5) x^2 \left (1-\frac{e^2 x^2}{d^2}\right )^p+3 d^4 e^4 p \left (p^2+6 p+5\right ) x^4 \left (1-\frac{e^2 x^2}{d^2}\right )^p\right )\right )}{70 e^5 (p+1) (p+2) (p+3) (p+4)} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]
[Out]
((d^2 - e^2*x^2)^p*(-35*(6*d^6*e^2*p*(5 + p)*x^2*(1 - (e^2*x^2)/d^2)^p + 3*d^4*e
^4*p*(5 + 6*p + p^2)*x^4*(1 - (e^2*x^2)/d^2)^p - 2*d^2*e^6*(12 + 20*p + 9*p^2 +
p^3)*x^6*(1 - (e^2*x^2)/d^2)^p - e^8*(6 + 11*p + 6*p^2 + p^3)*x^8*(1 - (e^2*x^2)
/d^2)^p + 6*d^8*(5 + p)*(-1 + (1 - (e^2*x^2)/d^2)^p)) + 14*d^3*e^5*(24 + 50*p +
35*p^2 + 10*p^3 + p^4)*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2] + 30*d
*e^7*(24 + 50*p + 35*p^2 + 10*p^3 + p^4)*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^
2*x^2)/d^2]))/(70*e^5*(1 + p)*(2 + p)*(3 + p)*(4 + p)*(1 - (e^2*x^2)/d^2)^p)
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Maple [F] time = 0.092, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( ex+d \right ) ^{3} \left ( -{e}^{2}{x}^{2}+{d}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(e*x+d)^3*(-e^2*x^2+d^2)^p,x)
[Out]
int(x^4*(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. \[ \int{\left (e x + d\right )}^{3}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*x^4,x, algorithm="maxima")
[Out]
integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*x^4, x)
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{3} x^{7} + 3 \, d e^{2} x^{6} + 3 \, d^{2} e x^{5} + d^{3} x^{4}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*x^4,x, algorithm="fricas")
[Out]
integral((e^3*x^7 + 3*d*e^2*x^6 + 3*d^2*e*x^5 + d^3*x^4)*(-e^2*x^2 + d^2)^p, x)
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Sympy [A] time = 39.7092, size = 2966, normalized size = 13.61 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(e*x+d)**3*(-e**2*x**2+d**2)**p,x)
[Out]
d**3*d**(2*p)*x**5*hyper((5/2, -p), (7/2,), e**2*x**2*exp_polar(2*I*pi)/d**2)/5
+ 3*d**2*e*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)) + 3*d*d**(2*p)*e**2*x**7*hyper((7/2, -p), (9/2,), e**2*
x**2*exp_polar(2*I*pi)/d**2)/7 + e**3*Piecewise((x**8*(d**2)**p/8, Eq(e, 0)), (-
6*d**6*log(-d/e + x)/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d**2*e**12*x**4 +
12*e**14*x**6) - 6*d**6*log(d/e + x)/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d*
*2*e**12*x**4 + 12*e**14*x**6) - 11*d**6/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 3
6*d**2*e**12*x**4 + 12*e**14*x**6) + 18*d**4*e**2*x**2*log(-d/e + x)/(-12*d**6*e
**8 + 36*d**4*e**10*x**2 - 36*d**2*e**12*x**4 + 12*e**14*x**6) + 18*d**4*e**2*x*
*2*log(d/e + x)/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d**2*e**12*x**4 + 12*e*
*14*x**6) + 27*d**4*e**2*x**2/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d**2*e**1
2*x**4 + 12*e**14*x**6) - 18*d**2*e**4*x**4*log(-d/e + x)/(-12*d**6*e**8 + 36*d*
*4*e**10*x**2 - 36*d**2*e**12*x**4 + 12*e**14*x**6) - 18*d**2*e**4*x**4*log(d/e
+ x)/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d**2*e**12*x**4 + 12*e**14*x**6) -
18*d**2*e**4*x**4/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d**2*e**12*x**4 + 12
*e**14*x**6) + 6*e**6*x**6*log(-d/e + x)/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 3
6*d**2*e**12*x**4 + 12*e**14*x**6) + 6*e**6*x**6*log(d/e + x)/(-12*d**6*e**8 + 3
6*d**4*e**10*x**2 - 36*d**2*e**12*x**4 + 12*e**14*x**6), Eq(p, -4)), (-6*d**6*lo
g(-d/e + x)/(4*d**4*e**8 - 8*d**2*e**10*x**2 + 4*e**12*x**4) - 6*d**6*log(d/e +
x)/(4*d**4*e**8 - 8*d**2*e**10*x**2 + 4*e**12*x**4) - 9*d**6/(4*d**4*e**8 - 8*d*
*2*e**10*x**2 + 4*e**12*x**4) + 12*d**4*e**2*x**2*log(-d/e + x)/(4*d**4*e**8 - 8
*d**2*e**10*x**2 + 4*e**12*x**4) + 12*d**4*e**2*x**2*log(d/e + x)/(4*d**4*e**8 -
8*d**2*e**10*x**2 + 4*e**12*x**4) + 12*d**4*e**2*x**2/(4*d**4*e**8 - 8*d**2*e**
10*x**2 + 4*e**12*x**4) - 6*d**2*e**4*x**4*log(-d/e + x)/(4*d**4*e**8 - 8*d**2*e
**10*x**2 + 4*e**12*x**4) - 6*d**2*e**4*x**4*log(d/e + x)/(4*d**4*e**8 - 8*d**2*
e**10*x**2 + 4*e**12*x**4) - 2*e**6*x**6/(4*d**4*e**8 - 8*d**2*e**10*x**2 + 4*e*
*12*x**4), Eq(p, -3)), (-6*d**6*log(-d/e + x)/(-4*d**2*e**8 + 4*e**10*x**2) - 6*
d**6*log(d/e + x)/(-4*d**2*e**8 + 4*e**10*x**2) - 6*d**6/(-4*d**2*e**8 + 4*e**10
*x**2) + 6*d**4*e**2*x**2*log(-d/e + x)/(-4*d**2*e**8 + 4*e**10*x**2) + 6*d**4*e
**2*x**2*log(d/e + x)/(-4*d**2*e**8 + 4*e**10*x**2) + 3*d**2*e**4*x**4/(-4*d**2*
e**8 + 4*e**10*x**2) + e**6*x**6/(-4*d**2*e**8 + 4*e**10*x**2), Eq(p, -2)), (-d*
*6*log(-d/e + x)/(2*e**8) - d**6*log(d/e + x)/(2*e**8) - d**4*x**2/(2*e**6) - d*
*2*x**4/(4*e**4) - x**6/(6*e**2), Eq(p, -1)), (-6*d**8*(d**2 - e**2*x**2)**p/(2*
e**8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8*p + 48*e**8) - 6*d**6*e**2*p*
x**2*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8
*p + 48*e**8) - 3*d**4*e**4*p**2*x**4*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e*
*8*p**3 + 70*e**8*p**2 + 100*e**8*p + 48*e**8) - 3*d**4*e**4*p*x**4*(d**2 - e**2
*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8*p + 48*e**8) - d
**2*e**6*p**3*x**6*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 + 70*e**8*p
**2 + 100*e**8*p + 48*e**8) - 3*d**2*e**6*p**2*x**6*(d**2 - e**2*x**2)**p/(2*e**
8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8*p + 48*e**8) - 2*d**2*e**6*p*x**
6*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8*p
+ 48*e**8) + e**8*p**3*x**8*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 +
70*e**8*p**2 + 100*e**8*p + 48*e**8) + 6*e**8*p**2*x**8*(d**2 - e**2*x**2)**p/(2
*e**8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8*p + 48*e**8) + 11*e**8*p*x**
8*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 + 70*e**8*p**2 + 100*e**8*p
+ 48*e**8) + 6*e**8*x**8*(d**2 - e**2*x**2)**p/(2*e**8*p**4 + 20*e**8*p**3 + 70*
e**8*p**2 + 100*e**8*p + 48*e**8), True))
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{3}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*x^4,x, algorithm="giac")
[Out]
integrate((e*x + d)^3*(-e^2*x^2 + d^2)^p*x^4, x)