3.3.58 \(\int x^5 (d+e x)^3 (d^2-e^2 x^2)^p \, dx\) [258]

3.3.58.1 Optimal result

Integrand size = 25, antiderivative size = 222 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=-\frac {2 d^7 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}-\frac {e x^7 \left (d^2-e^2 x^2\right )^{1+p}}{9+2 p}+\frac {11 d^5 \left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {5 d^3 \left (d^2-e^2 x^2\right )^{3+p}}{e^6 (3+p)}+\frac {3 d \left (d^2-e^2 x^2\right )^{4+p}}{2 e^6 (4+p)}+\frac {2 d^2 e (17+3 p) x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 (9+2 p)} \]

output

-2*d^7*(-e^2*x^2+d^2)^(p+1)/e^6/(p+1)-e*x^7*(-e^2*x^2+d^2)^(p+1)/(9+2*p)+1 
1/2*d^5*(-e^2*x^2+d^2)^(2+p)/e^6/(2+p)-5*d^3*(-e^2*x^2+d^2)^(3+p)/e^6/(3+p 
)+3/2*d*(-e^2*x^2+d^2)^(4+p)/e^6/(4+p)+2/7*d^2*e*(17+3*p)*x^7*(-e^2*x^2+d^ 
2)^p*hypergeom([7/2, -p],[9/2],e^2*x^2/d^2)/(9+2*p)/((1-e^2*x^2/d^2)^p)
 

3.3.58.2 Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.92 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\frac {\left (d^2-e^2 x^2\right )^p \left (-\frac {252 d^7 \left (d^2-e^2 x^2\right )}{1+p}+\frac {693 d^5 \left (d^2-e^2 x^2\right )^2}{2+p}+\frac {189 d \left (d^2-e^2 x^2\right )^4}{4+p}-\frac {630 \left (d^3-d e^2 x^2\right )^3}{3+p}+54 d^2 e^7 x^7 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )+14 e^9 x^9 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-p,\frac {11}{2},\frac {e^2 x^2}{d^2}\right )\right )}{126 e^6} \]

input

Integrate[x^5*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]
 

output

((d^2 - e^2*x^2)^p*((-252*d^7*(d^2 - e^2*x^2))/(1 + p) + (693*d^5*(d^2 - e 
^2*x^2)^2)/(2 + p) + (189*d*(d^2 - e^2*x^2)^4)/(4 + p) - (630*(d^3 - d*e^2 
*x^2)^3)/(3 + p) + (54*d^2*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^9*x^9*Hypergeometric2F1[9/2, -p, 11 
/2, (e^2*x^2)/d^2])/(1 - (e^2*x^2)/d^2)^p))/(126*e^6)
 

3.3.58.3 Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, 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.

\(\displaystyle \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx\)

\(\Big \downarrow \) 543

\(\displaystyle \int x^6 \left (d^2-e^2 x^2\right )^p \left (x^2 e^3+3 d^2 e\right )dx+\int x^5 \left (d^2-e^2 x^2\right )^p \left (d^3+3 e^2 x^2 d\right )dx\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int d x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+3 e^2 x^2\right )dx^2+\int x^6 \left (d^2-e^2 x^2\right )^p \left (x^2 e^3+3 d^2 e\right )dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} d \int x^4 \left (d^2-e^2 x^2\right )^p \left (d^2+3 e^2 x^2\right )dx^2+\int x^6 \left (d^2-e^2 x^2\right )^p \left (x^2 e^3+3 d^2 e\right )dx\)

\(\Big \downarrow \) 86

\(\displaystyle \int x^6 \left (d^2-e^2 x^2\right )^p \left (x^2 e^3+3 d^2 e\right )dx+\frac {1}{2} d \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {11 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {10 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^4}-\frac {3 \left (d^2-e^2 x^2\right )^{p+3}}{e^4}\right )dx^2\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {2 d^2 e (3 p+17) \int x^6 \left (d^2-e^2 x^2\right )^pdx}{2 p+9}+\frac {1}{2} d \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {11 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {10 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^4}-\frac {3 \left (d^2-e^2 x^2\right )^{p+3}}{e^4}\right )dx^2-\frac {e x^7 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+9}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {2 d^2 e (3 p+17) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int x^6 \left (1-\frac {e^2 x^2}{d^2}\right )^pdx}{2 p+9}+\frac {1}{2} d \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {11 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {10 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^4}-\frac {3 \left (d^2-e^2 x^2\right )^{p+3}}{e^4}\right )dx^2-\frac {e x^7 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+9}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {1}{2} d \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {11 d^4 \left (d^2-e^2 x^2\right )^{p+1}}{e^4}+\frac {10 d^2 \left (d^2-e^2 x^2\right )^{p+2}}{e^4}-\frac {3 \left (d^2-e^2 x^2\right )^{p+3}}{e^4}\right )dx^2+\frac {2 d^2 e (3 p+17) x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 (2 p+9)}-\frac {e x^7 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+9}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d^2 e (3 p+17) x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 (2 p+9)}-\frac {e x^7 \left (d^2-e^2 x^2\right )^{p+1}}{2 p+9}+\frac {1}{2} d \left (-\frac {10 d^2 \left (d^2-e^2 x^2\right )^{p+3}}{e^6 (p+3)}+\frac {3 \left (d^2-e^2 x^2\right )^{p+4}}{e^6 (p+4)}-\frac {4 d^6 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac {11 d^4 \left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)}\right )\)

input

Int[x^5*(d + e*x)^3*(d^2 - e^2*x^2)^p,x]
 

output

-((e*x^7*(d^2 - e^2*x^2)^(1 + p))/(9 + 2*p)) + (d*((-4*d^6*(d^2 - e^2*x^2) 
^(1 + p))/(e^6*(1 + p)) + (11*d^4*(d^2 - e^2*x^2)^(2 + p))/(e^6*(2 + p)) - 
 (10*d^2*(d^2 - e^2*x^2)^(3 + p))/(e^6*(3 + p)) + (3*(d^2 - e^2*x^2)^(4 + 
p))/(e^6*(4 + p))))/2 + (2*d^2*e*(17 + 3*p)*x^7*(d^2 - e^2*x^2)^p*Hypergeo 
metric2F1[7/2, -p, 9/2, (e^2*x^2)/d^2])/(7*(9 + 2*p)*(1 - (e^2*x^2)/d^2)^p 
)
 

3.3.58.3.1 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]
 

3.3.58.4 Maple [F]

\[\int x^{5} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}d x\]

input

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

output

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

3.3.58.5 Fricas [F]

\[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5} \,d x } \]

input

integrate(x^5*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="fricas")
 

output

integral((e^3*x^8 + 3*d*e^2*x^7 + 3*d^2*e*x^6 + d^3*x^5)*(-e^2*x^2 + d^2)^ 
p, x)
 

3.3.58.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (190) = 380\).

Time = 3.68 (sec) , antiderivative size = 2966, normalized size of antiderivative = 13.36 \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\text {Too large to display} \]

input

integrate(x**5*(e*x+d)**3*(-e**2*x**2+d**2)**p,x)
 

output

d**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*lo 
g(-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 ...
 

3.3.58.7 Maxima [F]

\[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5} \,d x } \]

input

integrate(x^5*(e*x+d)^3*(-e^2*x^2+d^2)^p,x, algorithm="maxima")
 

output

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^3/((p^3 + 6*p^2 + 11*p + 6)*e^6) + integrate((e 
^3*x^8 + 3*d*e^2*x^7 + 3*d^2*e*x^6)*e^(p*log(e*x + d) + p*log(-e*x + d)), 
x)
 

3.3.58.8 Giac [F]

\[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int { {\left (e x + d\right )}^{3} {\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{5} \,d x } \]

input

integrate(x^5*(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^5, x)
 

3.3.58.9 Mupad [F(-1)]

Timed out. \[ \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx=\int x^5\,{\left (d^2-e^2\,x^2\right )}^p\,{\left (d+e\,x\right )}^3 \,d x \]

input

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

output

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