3.3.49 \(\int x^5 (d+e x)^2 (d^2-e^2 x^2)^p \, dx\) [249]
Optimal. Leaf size=178 \[ -\frac {d^6 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{3+p}}{e^6 (3+p)}+\frac {\left (d^2-e^2 x^2\right )^{4+p}}{2 e^6 (4+p)}+\frac {2}{7} d e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {7}{2},-p;\frac {9}{2};\frac {e^2 x^2}{d^2}\right ) \]
[Out]
-d^6*(-e^2*x^2+d^2)^(1+p)/e^6/(1+p)+5/2*d^4*(-e^2*x^2+d^2)^(2+p)/e^6/(2+p)-2*d^2*(-e^2*x^2+d^2)^(3+p)/e^6/(3+p
)+1/2*(-e^2*x^2+d^2)^(4+p)/e^6/(4+p)+2/7*d*e*x^7*(-e^2*x^2+d^2)^p*hypergeom([7/2, -p],[9/2],e^2*x^2/d^2)/((1-e
^2*x^2/d^2)^p)
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Rubi [A]
time = 0.10, antiderivative size = 178, 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, 457, 78,
12, 372, 371} \begin {gather*} \frac {2}{7} d e x^7 \left (d^2-e^2 x^2\right )^p \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 )-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{p+3}}{e^6 (p+3)}+\frac {\left (d^2-e^2 x^2\right )^{p+4}}{2 e^6 (p+4)}-\frac {d^6 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[x^5*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]
[Out]
-((d^6*(d^2 - e^2*x^2)^(1 + p))/(e^6*(1 + p))) + (5*d^4*(d^2 - e^2*x^2)^(2 + p))/(2*e^6*(2 + p)) - (2*d^2*(d^2
- e^2*x^2)^(3 + p))/(e^6*(3 + p)) + (d^2 - e^2*x^2)^(4 + p)/(2*e^6*(4 + p)) + (2*d*e*x^7*(d^2 - e^2*x^2)^p*Hy
pergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(7*(1 - (e^2*x^2)/d^2)^p)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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 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^5 (d+e x)^2 \left (d^2-e^2 x^2\right )^p \, dx &=\int 2 d e x^6 \left (d^2-e^2 x^2\right )^p \, dx+\int x^5 \left (d^2-e^2 x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int x^2 \left (d^2-e^2 x\right )^p \left (d^2+e^2 x\right ) \, dx,x,x^2\right )+(2 d e) \int x^6 \left (d^2-e^2 x^2\right )^p \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {2 d^6 \left (d^2-e^2 x\right )^p}{e^4}-\frac {5 d^4 \left (d^2-e^2 x\right )^{1+p}}{e^4}+\frac {4 d^2 \left (d^2-e^2 x\right )^{2+p}}{e^4}-\frac {\left (d^2-e^2 x\right )^{3+p}}{e^4}\right ) \, dx,x,x^2\right )+\left (2 d e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^6 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx\\ &=-\frac {d^6 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{3+p}}{e^6 (3+p)}+\frac {\left (d^2-e^2 x^2\right )^{4+p}}{2 e^6 (4+p)}+\frac {2}{7} d e x^7 \left (d^2-e^2 x^2\right )^p \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 )\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 159, normalized size = 0.89 \begin {gather*} \frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {14 d^6 \left (d^2-e^2 x^2\right )}{1+p}+\frac {35 d^4 \left (d^2-e^2 x^2\right )^2}{2+p}-\frac {28 d^2 \left (d^2-e^2 x^2\right )^3}{3+p}+\frac {7 \left (d^2-e^2 x^2\right )^4}{4+p}+4 d e^7 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 )}{14 e^6} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^5*(d + e*x)^2*(d^2 - e^2*x^2)^p,x]
[Out]
((d^2 - e^2*x^2)^p*((-14*d^6*(d^2 - e^2*x^2))/(1 + p) + (35*d^4*(d^2 - e^2*x^2)^2)/(2 + p) - (28*d^2*(d^2 - e^
2*x^2)^3)/(3 + p) + (7*(d^2 - e^2*x^2)^4)/(4 + p) + (4*d*e^7*x^7*Hypergeometric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2
])/(1 - (e^2*x^2)/d^2)^p))/(14*e^6)
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{5} \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(e*x+d)^2*(-e^2*x^2+d^2)^p,x)
[Out]
int(x^5*(e*x+d)^2*(-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^5*(e*x+d)^2*(-e^2*x^2+d^2)^p,x, algorithm="maxima")
[Out]
1/2*((p^2 + 3*p + 2)*x^6*e^6 - (p^2 + p)*d^2*x^4*e^4 - 2*d^4*p*x^2*e^2 - 2*d^6)*d^2*e^(p*log(-x^2*e^2 + d^2) -
6)/(p^3 + 6*p^2 + 11*p + 6) + integrate((x^7*e^2 + 2*d*x^6*e)*e^(p*log(x*e + d) + p*log(-x*e + d)), 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^5*(e*x+d)^2*(-e^2*x^2+d^2)^p,x, algorithm="fricas")
[Out]
integral((x^7*e^2 + 2*d*x^6*e + d^2*x^5)*(-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. 938 vs.
\(2 (150) = 300\).
time = 3.57, size = 2924, normalized size = 16.43 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**5*(e*x+d)**2*(-e**2*x**2+d**2)**p,x)
[Out]
d**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)) + 2*d*d**(2*p)*e*x**7*hyper((7/2, -p), (9/2,), e**2*x**2*exp_
polar(2*I*pi)/d**2)/7 + e**2*Piecewise((x**8*(d**2)**p/8, Eq(e, 0)), (-6*d**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) - 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 + 1
2*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 + 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*...
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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^5*(e*x+d)^2*(-e^2*x^2+d^2)^p,x, algorithm="giac")
[Out]
integrate((x*e + d)^2*(-x^2*e^2 + d^2)^p*x^5, x)
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^5*(d^2 - e^2*x^2)^p*(d + e*x)^2,x)
[Out]
int(x^5*(d^2 - e^2*x^2)^p*(d + e*x)^2, x)
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