3.3.76 \(\int \frac {x^4 (d^2-e^2 x^2)^p}{(d+e x)^2} \, dx\) [276]

3.3.76.1 Optimal result

Integrand size = 25, antiderivative size = 184 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=-\frac {d^5 \left (d^2-e^2 x^2\right )^{-1+p}}{e^5 (1-p)}-\frac {x^5 \left (d^2-e^2 x^2\right )^{-1+p}}{3+2 p}-\frac {2 d^3 \left (d^2-e^2 x^2\right )^p}{e^5 p}+\frac {d \left (d^2-e^2 x^2\right )^{1+p}}{e^5 (1+p)}+\frac {2 (4+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},2-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^2 (3+2 p)} \]

output

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

3.3.76.2 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 = 0.36 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\frac {x^5 (d-e x)^p (d+e x)^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {AppellF1}\left (5,-p,2-p,6,\frac {e x}{d},-\frac {e x}{d}\right )}{5 d^2} \]

input

Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]
 

output

(x^5*(d - e*x)^p*(d + e*x)^p*AppellF1[5, -p, 2 - p, 6, (e*x)/d, -((e*x)/d) 
])/(5*d^2*(1 - (e^2*x^2)/d^2)^p)
 

3.3.76.3 Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {570, 543, 27, 243, 53, 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.

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

\(\Big \downarrow \) 570

\(\displaystyle \int x^4 (d-e x)^2 \left (d^2-e^2 x^2\right )^{p-2}dx\)

\(\Big \downarrow \) 543

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 2009

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

input

Int[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x]
 

output

-((x^5*(d^2 - e^2*x^2)^(-1 + p))/(3 + 2*p)) - d*e*((d^4*(d^2 - e^2*x^2)^(- 
1 + p))/(e^6*(1 - p)) + (2*d^2*(d^2 - e^2*x^2)^p)/(e^6*p) - (d^2 - e^2*x^2 
)^(1 + p)/(e^6*(1 + p))) + (2*(4 + p)*x^5*(d^2 - e^2*x^2)^p*Hypergeometric 
2F1[5/2, 2 - p, 7/2, (e^2*x^2)/d^2])/(5*d^2*(3 + 2*p)*(1 - (e^2*x^2)/d^2)^ 
p)
 

3.3.76.3.1 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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 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 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]
 

3.3.76.4 Maple [F]

\[\int \frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{2}}d x\]

input

int(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^2,x)
 

output

int(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^2,x)
 

3.3.76.5 Fricas [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{2}} \,d x } \]

input

integrate(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^2,x, algorithm="fricas")
 

output

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

3.3.76.6 Sympy [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{2}}\, dx \]

input

integrate(x**4*(-e**2*x**2+d**2)**p/(e*x+d)**2,x)
 

output

Integral(x**4*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**2, x)
 

3.3.76.7 Maxima [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{2}} \,d x } \]

input

integrate(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^2,x, algorithm="maxima")
 

output

integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d)^2, x)
 

3.3.76.8 Giac [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{2}} \,d x } \]

input

integrate(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^2,x, algorithm="giac")
 

output

integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d)^2, x)
 

3.3.76.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]

input

int((x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^2,x)
 

output

int((x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^2, x)