3.3.59 \(\int x^4 (d+e x)^3 (d^2-e^2 x^2)^p \, dx\) [259]
Optimal. Leaf size=218 \[ -\frac {2 d^6 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {9 d^4 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^5 (2+p)}-\frac {3 d^2 \left (d^2-e^2 x^2\right )^{3+p}}{e^5 (3+p)}+\frac {\left (d^2-e^2 x^2\right )^{4+p}}{2 e^5 (4+p)}+\frac {2 d^3 (11+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {5}{2},-p;\frac {7}{2};\frac {e^2 x^2}{d^2}\right )}{5 (7+2 p)} \]
[Out]
-2*d^6*(-e^2*x^2+d^2)^(1+p)/e^5/(1+p)-3*d*x^5*(-e^2*x^2+d^2)^(1+p)/(7+2*p)+9/2*d^4*(-e^2*x^2+d^2)^(2+p)/e^5/(2
+p)-3*d^2*(-e^2*x^2+d^2)^(3+p)/e^5/(3+p)+1/2*(-e^2*x^2+d^2)^(4+p)/e^5/(4+p)+2/5*d^3*(11+p)*x^5*(-e^2*x^2+d^2)^
p*hypergeom([5/2, -p],[7/2],e^2*x^2/d^2)/(7+2*p)/((1-e^2*x^2/d^2)^p)
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Rubi [A]
time = 0.12, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1666, 470, 372,
371, 457, 78} \begin {gather*} -\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)} \end {gather*}
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)
Rule 78
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Rule 371
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
Rule 372
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/
(1 + b*(x^n/a))^FracPart[p]), Int[(c*x)^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[
p, 0] && !(ILtQ[p, 0] || GtQ[a, 0])
Rule 457
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Rule 470
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m +
n*(p + 1) + 1)), Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0]
&& NeQ[m + n*(p + 1) + 1, 0]
Rule 1666
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], k}, Int[x^m*Sum[Coeff[
Pq, x, 2*k]*x^(2*k), {k, 0, q/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0,
(q - 1)/2}]*(a + b*x^2)^p, x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !PolyQ[Pq, x^2] && IGtQ[m, -2] && !
IntegerQ[2*p]
Rubi steps
\begin {align*} \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx &=\int x^4 \left (d^2-e^2 x^2\right )^p \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^5 \left (d^2-e^2 x^2\right )^p \left (3 d^2 e+e^3 x^2\right ) \, dx\\ &=-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {1}{2} \text {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \left (3 d^2 e+e^3 x\right ) \, dx,x,x^2\right )+\frac {\left (2 d^3 (11+p)\right ) \int x^4 \left (d^2-e^2 x^2\right )^p \, dx}{7+2 p}\\ &=-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {4 d^6 \left (d^2-e^2 x\right )^p}{e^3}-\frac {9 d^4 \left (d^2-e^2 x\right )^{1+p}}{e^3}+\frac {6 d^2 \left (d^2-e^2 x\right )^{2+p}}{e^3}-\frac {\left (d^2-e^2 x\right )^{3+p}}{e^3}\right ) \, dx,x,x^2\right )+\frac {\left (2 d^3 (11+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{7+2 p}\\ &=-\frac {2 d^6 \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{1+p}}{7+2 p}+\frac {9 d^4 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^5 (2+p)}-\frac {3 d^2 \left (d^2-e^2 x^2\right )^{3+p}}{e^5 (3+p)}+\frac {\left (d^2-e^2 x^2\right )^{4+p}}{2 e^5 (4+p)}+\frac {2 d^3 (11+p) 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 (7+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 219, normalized size = 1.00 \begin {gather*} \frac {1}{70} \left (d^2-e^2 x^2\right )^p \left (-\frac {35 \left (d^2-e^2 x^2\right ) \left (6 d^6 (5+p)+6 d^4 e^2 \left (5+6 p+p^2\right ) x^2+3 d^2 e^4 \left (10+17 p+8 p^2+p^3\right ) x^4+e^6 \left (6+11 p+6 p^2+p^3\right ) x^6\right )}{e^5 (1+p) (2+p) (3+p) (4+p)}+14 d^3 x^5 \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 )+30 d e^2 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right )\right ) \end {gather*}
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*(d^2 - e^2*x^2)*(6*d^6*(5 + p) + 6*d^4*e^2*(5 + 6*p + p^2)*x^2 + 3*d^2*e^4*(10 + 17*p
+ 8*p^2 + p^3)*x^4 + e^6*(6 + 11*p + 6*p^2 + p^3)*x^6))/(e^5*(1 + p)*(2 + p)*(3 + p)*(4 + p)) + (14*d^3*x^5*H
ypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p + (30*d*e^2*x^7*Hypergeometric2F1[7/2, -p
, 9/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/70
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{4} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]
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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
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, algorithm="maxima")
[Out]
integrate((x*e + d)^3*(-x^2*e^2 + d^2)^p*x^4, x)
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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, algorithm="fricas")
[Out]
integral((x^7*e^3 + 3*d*x^6*e^2 + 3*d^2*x^5*e + d^3*x^4)*(-x^2*e^2 + d^2)^p, x)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1945 vs.
\(2 (185) = 370\).
time = 4.17, size = 2966, normalized size = 13.61 \begin {gather*} \text {Too large to display} \end {gather*}
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 + 2
2*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 - 3
6*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 - 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) + 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**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*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 - 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 - 36*d**2*e**12*x**4 + 12*e**14*x**6), Eq(p, -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) - 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 + 1
00*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 + 1
00*e**8*p + 48*e**8) + e**8*p**3*x**8*(d**2 - e...
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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, algorithm="giac")
[Out]
integrate((x*e + d)^3*(-x^2*e^2 + d^2)^p*x^4, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(d^2 - e^2*x^2)^p*(d + e*x)^3,x)
[Out]
int(x^4*(d^2 - e^2*x^2)^p*(d + e*x)^3, x)
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