\(\int x^2 (d+e x)^3 (d^2-e^2 x^2)^p \, dx\) [261]

Optimal result

Integrand size = 25, antiderivative size = 189 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {2 d^4 \left (d^2-e^2 x^2\right )^{1+p}}{e^3 (1+p)}-\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{1+p}}{5+2 p}+\frac {5 d^2 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^3 (2+p)}-\frac {\left (d^2-e^2 x^2\right )^{3+p}}{2 e^3 (3+p)}+\frac {2 d^3 (7+p) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )}{3 (5+2 p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1666, 470, 372, 371, 457, 78} \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}+\frac {5 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{2 e^3 (p+2)}-\frac {\left (d^2-e^2 x^2\right )^{p+3}}{2 e^3 (p+3)}-\frac {2 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^3 (p+1)}+\frac {2 d^3 (p+7) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )}{3 (2 p+5)} \]

[In]

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

[Out]

(-2*d^4*(d^2 - e^2*x^2)^(1 + p))/(e^3*(1 + p)) - (3*d*x^3*(d^2 - e^2*x^2)^(1 + p))/(5 + 2*p) + (5*d^2*(d^2 - e
^2*x^2)^(2 + p))/(2*e^3*(2 + p)) - (d^2 - e^2*x^2)^(3 + p)/(2*e^3*(3 + p)) + (2*d^3*(7 + p)*x^3*(d^2 - e^2*x^2
)^p*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(3*(5 + 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*} \text {integral}& = \int x^2 \left (d^2-e^2 x^2\right )^p \left (d^3+3 d e^2 x^2\right ) \, dx+\int x^3 \left (d^2-e^2 x^2\right )^p \left (3 d^2 e+e^3 x^2\right ) \, dx \\ & = -\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{1+p}}{5+2 p}+\frac {1}{2} \text {Subst}\left (\int x \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 (7+p)\right ) \int x^2 \left (d^2-e^2 x^2\right )^p \, dx}{5+2 p} \\ & = -\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{1+p}}{5+2 p}+\frac {1}{2} \text {Subst}\left (\int \left (\frac {4 d^4 \left (d^2-e^2 x\right )^p}{e}-\frac {5 d^2 \left (d^2-e^2 x\right )^{1+p}}{e}+\frac {\left (d^2-e^2 x\right )^{2+p}}{e}\right ) \, dx,x,x^2\right )+\frac {\left (2 d^3 (7+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{5+2 p} \\ & = -\frac {2 d^4 \left (d^2-e^2 x^2\right )^{1+p}}{e^3 (1+p)}-\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{1+p}}{5+2 p}+\frac {5 d^2 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^3 (2+p)}-\frac {\left (d^2-e^2 x^2\right )^{3+p}}{2 e^3 (3+p)}+\frac {2 d^3 (7+p) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};\frac {e^2 x^2}{d^2}\right )}{3 (5+2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.99 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\frac {1}{30} \left (d^2-e^2 x^2\right )^p \left (-\frac {15 \left (d^2-e^2 x^2\right ) \left (d^4 (11+3 p)+d^2 e^2 \left (11+14 p+3 p^2\right ) x^2+e^4 \left (2+3 p+p^2\right ) x^4\right )}{e^3 (1+p) (2+p) (3+p)}+10 d^3 x^3 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )+18 d e^2 x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )\right ) \]

[In]

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

[Out]

((d^2 - e^2*x^2)^p*((-15*(d^2 - e^2*x^2)*(d^4*(11 + 3*p) + d^2*e^2*(11 + 14*p + 3*p^2)*x^2 + e^4*(2 + 3*p + p^
2)*x^4))/(e^3*(1 + p)*(2 + p)*(3 + p)) + (10*d^3*x^3*Hypergeometric2F1[3/2, -p, 5/2, (e^2*x^2)/d^2])/(1 - (e^2
*x^2)/d^2)^p + (18*d*e^2*x^5*Hypergeometric2F1[5/2, -p, 7/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/30

Maple [F]

\[\int x^{2} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (160) = 320\).

Time = 2.55 (sec) , antiderivative size = 1370, normalized size of antiderivative = 7.25 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int x^2\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \]

[In]

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

[Out]

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