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

3.3.61.1 Optimal result

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

output

-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)
 

3.3.61.2 Mathematica [A] (verified)

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

input

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

output

((d^2 - e^2*x^2)^p*((-15*(d^2 - e^2*x^2)*(d^4*(11 + 3*p) + d^2*e^2*(11 + 1 
4*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
 

3.3.61.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {543, 354, 27, 86, 363, 279, 278, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx\)

\(\Big \downarrow \) 543

\(\displaystyle \int x^3 \left (d^2-e^2 x^2\right )^p \left (x^2 e^3+3 d^2 e\right )dx+\int x^2 \left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int e x^2 \left (d^2-e^2 x^2\right )^p \left (3 d^2+e^2 x^2\right )dx^2+\int x^2 \left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} e \int x^2 \left (d^2-e^2 x^2\right )^p \left (3 d^2+e^2 x^2\right )dx^2+\int x^2 \left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )dx\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {1}{2} e \int \left (\frac {4 d^4 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {5 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^2}\right )dx^2+\int x^2 \left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )dx\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {1}{2} e \int \left (\frac {4 d^4 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {5 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^2}\right )dx^2+\frac {2 d^3 (p+7) \int x^2 \left (d^2-e^2 x^2\right )^pdx}{2 p+5}-\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {1}{2} e \int \left (\frac {4 d^4 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {5 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^2}\right )dx^2+\frac {2 d^3 (p+7) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int x^2 \left (1-\frac {e^2 x^2}{d^2}\right )^pdx}{2 p+5}-\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {1}{2} e \int \left (\frac {4 d^4 \left (d^2-e^2 x^2\right )^p}{e^2}-\frac {5 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^2}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^2}\right )dx^2-\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}+\frac {2 d^3 (p+7) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )}{3 (2 p+5)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 d x^3 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+5}+\frac {1}{2} e \left (\frac {5 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^4 (p+2)}-\frac {\left (d^2-e^2 x^2\right )^{p+3}}{e^4 (p+3)}-\frac {4 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^4 (p+1)}\right )+\frac {2 d^3 (p+7) x^3 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {e^2 x^2}{d^2}\right )}{3 (2 p+5)}\)

input

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

output

(-3*d*x^3*(d^2 - e^2*x^2)^(1 + p))/(5 + 2*p) + (e*((-4*d^4*(d^2 - e^2*x^2) 
^(1 + p))/(e^4*(1 + p)) + (5*d^2*(d^2 - e^2*x^2)^(2 + p))/(e^4*(2 + p)) - 
(d^2 - e^2*x^2)^(3 + p)/(e^4*(3 + p))))/2 + (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)
 

3.3.61.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((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 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 543

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Module[{k}, Int[x^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2*k)*x^(2*k), 
 {k, 0, n/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Binomial[n, 2*k + 1]*c^ 
(n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2)^p, x]] /; 
 FreeQ[{a, b, c, d, p}, x] && IGtQ[n, 1] && IntegerQ[m] &&  !IntegerQ[2*p] 
&&  !(EqQ[m, 1] && EqQ[b*c^2 + a*d^2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

3.3.61.4 Maple [F]

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

input

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

output

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

3.3.61.5 Fricas [F]

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

input

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

output

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)
 

3.3.61.6 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 \[ \int x^2 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\text {Too large to display} \]

input

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

output

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, ...
 

3.3.61.7 Maxima [F]

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

input

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

output

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

3.3.61.8 Giac [F]

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

input

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

output

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

3.3.61.9 Mupad [F(-1)]

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

input

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

output

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