3.249 \(\int x^5 (d+e x)^2 (d^2-e^2 x^2)^p \, dx\)
Optimal. Leaf size=178 \[ \frac{2}{7} d e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^6 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac{5 d^4 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)}-\frac{2 d^2 \left (d^2-e^2 x^2\right )^{p+3}}{e^6 (p+3)}+\frac{\left (d^2-e^2 x^2\right )^{p+4}}{2 e^6 (p+4)} \]
[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)
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Rubi [A] time = 0.145467, antiderivative size = 178, 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, Rules used =
{1652, 446, 77, 12, 365, 364} \[ \frac{2}{7} d e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )-\frac{d^6 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac{5 d^4 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^6 (p+2)}-\frac{2 d^2 \left (d^2-e^2 x^2\right )^{p+3}}{e^6 (p+3)}+\frac{\left (d^2-e^2 x^2\right )^{p+4}}{2 e^6 (p+4)} \]
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 1652
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]
Rule 446
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 77
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 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 365
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 364
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])
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} \operatorname{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} \operatorname{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*}
Mathematica [A] time = 0.140939, size = 159, normalized size = 0.89 \[ \frac{\left (d^2-e^2 x^2\right )^p \left (4 d e^7 x^7 \left (1-\frac{e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac{7}{2},-p;\frac{9}{2};\frac{e^2 x^2}{d^2}\right )+\frac{7 \left (d^2-e^2 x^2\right )^4}{p+4}-\frac{28 d^2 \left (d^2-e^2 x^2\right )^3}{p+3}+\frac{35 d^4 \left (d^2-e^2 x^2\right )^2}{p+2}-\frac{14 d^6 \left (d^2-e^2 x^2\right )}{p+1}\right )}{14 e^6} \]
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 = 0.67, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( ex+d \right ) ^{2} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{p}\, dx \end{align*}
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., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (p^{2} + 3 \, p + 2\right )} e^{6} x^{6} -{\left (p^{2} + p\right )} d^{2} e^{4} x^{4} - 2 \, d^{4} e^{2} p x^{2} - 2 \, d^{6}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} d^{2}}{2 \,{\left (p^{3} + 6 \, p^{2} + 11 \, p + 6\right )} e^{6}} + \int{\left (e^{2} x^{7} + 2 \, d e x^{6}\right )} e^{\left (p \log \left (e x + d\right ) + p \log \left (-e x + d\right )\right )}\,{d x} \end{align*}
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)*e^6*x^6 - (p^2 + p)*d^2*e^4*x^4 - 2*d^4*e^2*p*x^2 - 2*d^6)*(-e^2*x^2 + d^2)^p*d^2/((p^3 +
6*p^2 + 11*p + 6)*e^6) + integrate((e^2*x^7 + 2*d*e*x^6)*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. \begin{align*}{\rm integral}\left ({\left (e^{2} x^{7} + 2 \, d e x^{6} + d^{2} x^{5}\right )}{\left (-e^{2} x^{2} + d^{2}\right )}^{p}, x\right ) \end{align*}
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((e^2*x^7 + 2*d*e*x^6 + d^2*x^5)*(-e^2*x^2 + d^2)^p, x)
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Sympy [B] time = 15.0089, size = 2916, normalized size = 16.38 \begin{align*} \text{result too large to display} \end{align*}
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) - 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) - 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) + 2*e**4*x**4/(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 + 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 - 3
6*d**2*e**12*x**4 + 12*e**14*x**6) - 2*d**6/(-12*d**6*e**8 + 36*d**4*e**10*x**2 - 36*d**2*e**12*x**4 + 12*e**1
4*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) - 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) +
9*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) - 9*e**6*x**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), 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) - 3*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) - 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) + 6*d**2*e**4*x**4/(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), E
q(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 + 10
0*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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x + d\right )}^{2}{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5}\,{d x} \end{align*}
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((e*x + d)^2*(-e^2*x^2 + d^2)^p*x^5, x)