3.3.75 \(\int \frac {x^5 (d^2-e^2 x^2)^p}{(d+e x)^2} \, dx\) [275]

3.3.75.1 Optimal result

Integrand size = 25, antiderivative size = 179 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {d^6 \left (d^2-e^2 x^2\right )^{-1+p}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{2 e^6 p}-\frac {2 d^2 \left (d^2-e^2 x^2\right )^{1+p}}{e^6 (1+p)}+\frac {\left (d^2-e^2 x^2\right )^{2+p}}{2 e^6 (2+p)}-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},2-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 d^3} \]

output

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

3.3.75.2 Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.37 \[ \int \frac {x^5 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {x^6 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {AppellF1}\left (6,-p,2-p,7,\frac {e x}{d},-\frac {e x}{d}\right )}{6 d^2} \]

input

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

output

(x^6*(d - e*x)^p*(d + e*x)^p*AppellF1[6, -p, 2 - p, 7, (e*x)/d, -((e*x)/d) 
])/(6*d^2*(1 - (e^2*x^2)/d^2)^p)
 

3.3.75.3 Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 180, 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 = {570, 543, 27, 279, 278, 354, 86, 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.

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

\(\Big \downarrow \) 570

\(\displaystyle \int x^5 (d-e x)^2 \left (d^2-e^2 x^2\right )^{p-2}dx\)

\(\Big \downarrow \) 543

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 279

\(\displaystyle \int x^5 \left (d^2-e^2 x^2\right )^{p-2} \left (d^2+e^2 x^2\right )dx-\frac {2 e \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \int x^6 \left (1-\frac {e^2 x^2}{d^2}\right )^{p-2}dx}{d^3}\)

\(\Big \downarrow \) 278

\(\displaystyle \int x^5 \left (d^2-e^2 x^2\right )^{p-2} \left (d^2+e^2 x^2\right )dx-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},2-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 d^3}\)

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 86

\(\displaystyle \frac {1}{2} \int \left (\frac {2 d^6 \left (d^2-e^2 x^2\right )^{p-2}}{e^4}-\frac {5 d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}+\frac {4 d^2 \left (d^2-e^2 x^2\right )^p}{e^4}-\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^4}\right )dx^2-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},2-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {4 d^2 \left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac {\left (d^2-e^2 x^2\right )^{p+2}}{e^6 (p+2)}+\frac {2 d^6 \left (d^2-e^2 x^2\right )^{p-1}}{e^6 (1-p)}+\frac {5 d^4 \left (d^2-e^2 x^2\right )^p}{e^6 p}\right )-\frac {2 e x^7 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},2-p,\frac {9}{2},\frac {e^2 x^2}{d^2}\right )}{7 d^3}\)

input

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

output

((2*d^6*(d^2 - e^2*x^2)^(-1 + p))/(e^6*(1 - p)) + (5*d^4*(d^2 - e^2*x^2)^p 
)/(e^6*p) - (4*d^2*(d^2 - e^2*x^2)^(1 + p))/(e^6*(1 + p)) + (d^2 - e^2*x^2 
)^(2 + p)/(e^6*(2 + p)))/2 - (2*e*x^7*(d^2 - e^2*x^2)^p*Hypergeometric2F1[ 
7/2, 2 - p, 9/2, (e^2*x^2)/d^2])/(7*d^3*(1 - (e^2*x^2)/d^2)^p)
 

3.3.75.3.1 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 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 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 2009

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

3.3.75.4 Maple [F]

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

input

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

output

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

3.3.75.5 Fricas [F]

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

input

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

output

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

3.3.75.6 Sympy [F]

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

input

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

output

Integral(x**5*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**2, x)
 

3.3.75.7 Maxima [F]

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

input

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

output

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

3.3.75.8 Giac [F]

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

input

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

output

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

3.3.75.9 Mupad [F(-1)]

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

input

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

output

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