3.3.86 \(\int \frac {x^4 (d^2-e^2 x^2)^p}{(d+e x)^3} \, dx\) [286]

3.3.86.1 Optimal result

Integrand size = 25, antiderivative size = 220 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=-\frac {2 d^6 \left (d^2-e^2 x^2\right )^{-2+p}}{e^5 (2-p)}-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{-2+p}}{1+2 p}+\frac {9 d^4 \left (d^2-e^2 x^2\right )^{-1+p}}{2 e^5 (1-p)}+\frac {3 d^2 \left (d^2-e^2 x^2\right )^p}{e^5 p}-\frac {\left (d^2-e^2 x^2\right )^{1+p}}{2 e^5 (1+p)}+\frac {2 (8+p) x^5 \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},3-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^3 (1+2 p)} \]

output

-2*d^6*(-e^2*x^2+d^2)^(-2+p)/e^5/(2-p)-3*d*x^5*(-e^2*x^2+d^2)^(-2+p)/(1+2* 
p)+9/2*d^4*(-e^2*x^2+d^2)^(-1+p)/e^5/(1-p)+3*d^2*(-e^2*x^2+d^2)^p/e^5/p-1/ 
2*(-e^2*x^2+d^2)^(p+1)/e^5/(p+1)+2/5*(8+p)*x^5*(-e^2*x^2+d^2)^p*hypergeom( 
[5/2, 3-p],[7/2],e^2*x^2/d^2)/d^3/(1+2*p)/((1-e^2*x^2/d^2)^p)
 

3.3.86.2 Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=-\frac {2^{-3+p} \left (1+\frac {e x}{d}\right )^{-p} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (24 d e (1+p) x \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {e^2 x^2}{d^2}\right )+(d-e x) \left (1-\frac {e^2 x^2}{d^2}\right )^p \left (4 d \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p+4 e x \left (\frac {1}{2}+\frac {e x}{2 d}\right )^p+24 d \operatorname {Hypergeometric2F1}\left (1-p,1+p,2+p,\frac {d-e x}{2 d}\right )-8 d \operatorname {Hypergeometric2F1}\left (2-p,1+p,2+p,\frac {d-e x}{2 d}\right )+d \operatorname {Hypergeometric2F1}\left (3-p,1+p,2+p,\frac {d-e x}{2 d}\right )\right )\right )}{e^5 (1+p)} \]

input

Integrate[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]
 

output

-((2^(-3 + p)*(d^2 - e^2*x^2)^p*(24*d*e*(1 + p)*x*(1/2 + (e*x)/(2*d))^p*Hy 
pergeometric2F1[1/2, -p, 3/2, (e^2*x^2)/d^2] + (d - e*x)*(1 - (e^2*x^2)/d^ 
2)^p*(4*d*(1/2 + (e*x)/(2*d))^p + 4*e*x*(1/2 + (e*x)/(2*d))^p + 24*d*Hyper 
geometric2F1[1 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] - 8*d*Hypergeometric2F1 
[2 - p, 1 + p, 2 + p, (d - e*x)/(2*d)] + d*Hypergeometric2F1[3 - p, 1 + p, 
 2 + p, (d - e*x)/(2*d)])))/(e^5*(1 + p)*(1 + (e*x)/d)^p*(1 - (e^2*x^2)/d^ 
2)^p))
 

3.3.86.3 Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {570, 543, 354, 25, 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.

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 543

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 86

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

\(\Big \downarrow \) 363

\(\displaystyle \frac {2 d^3 (p+8) \int x^4 \left (d^2-e^2 x^2\right )^{p-3}dx}{2 p+1}-\frac {1}{2} e \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^{p-3}}{e^4}-\frac {9 d^4 \left (d^2-e^2 x^2\right )^{p-2}}{e^4}+\frac {6 d^2 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}-\frac {\left (d^2-e^2 x^2\right )^p}{e^4}\right )dx^2-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {2 (p+8) \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \int x^4 \left (1-\frac {e^2 x^2}{d^2}\right )^{p-3}dx}{d^3 (2 p+1)}-\frac {1}{2} e \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^{p-3}}{e^4}-\frac {9 d^4 \left (d^2-e^2 x^2\right )^{p-2}}{e^4}+\frac {6 d^2 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}-\frac {\left (d^2-e^2 x^2\right )^p}{e^4}\right )dx^2-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}\)

\(\Big \downarrow \) 278

\(\displaystyle -\frac {1}{2} e \int \left (\frac {4 d^6 \left (d^2-e^2 x^2\right )^{p-3}}{e^4}-\frac {9 d^4 \left (d^2-e^2 x^2\right )^{p-2}}{e^4}+\frac {6 d^2 \left (d^2-e^2 x^2\right )^{p-1}}{e^4}-\frac {\left (d^2-e^2 x^2\right )^p}{e^4}\right )dx^2-\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}+\frac {2 (p+8) x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {5}{2},3-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^3 (2 p+1)}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3 d x^5 \left (d^2-e^2 x^2\right )^{p-2}}{2 p+1}+\frac {2 (p+8) x^5 \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (d^2-e^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {5}{2},3-p,\frac {7}{2},\frac {e^2 x^2}{d^2}\right )}{5 d^3 (2 p+1)}-\frac {1}{2} e \left (-\frac {6 d^2 \left (d^2-e^2 x^2\right )^p}{e^6 p}+\frac {\left (d^2-e^2 x^2\right )^{p+1}}{e^6 (p+1)}+\frac {4 d^6 \left (d^2-e^2 x^2\right )^{p-2}}{e^6 (2-p)}-\frac {9 d^4 \left (d^2-e^2 x^2\right )^{p-1}}{e^6 (1-p)}\right )\)

input

Int[(x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x]
 

output

(-3*d*x^5*(d^2 - e^2*x^2)^(-2 + p))/(1 + 2*p) - (e*((4*d^6*(d^2 - e^2*x^2) 
^(-2 + p))/(e^6*(2 - p)) - (9*d^4*(d^2 - e^2*x^2)^(-1 + p))/(e^6*(1 - p)) 
- (6*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 + (2*(8 + p)*x^5*(d^2 - e^2*x^2)^p*Hypergeometric2F1[5/2, 3 - p, 7/2 
, (e^2*x^2)/d^2])/(5*d^3*(1 + 2*p)*(1 - (e^2*x^2)/d^2)^p)
 

3.3.86.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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]
 

3.3.86.4 Maple [F]

\[\int \frac {x^{4} \left (-e^{2} x^{2}+d^{2}\right )^{p}}{\left (e x +d \right )^{3}}d x\]

input

int(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^3,x)
 

output

int(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^3,x)
 

3.3.86.5 Fricas [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^3,x, algorithm="fricas")
 

output

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

3.3.86.6 Sympy [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {x^{4} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{p}}{\left (d + e x\right )^{3}}\, dx \]

input

integrate(x**4*(-e**2*x**2+d**2)**p/(e*x+d)**3,x)
 

output

Integral(x**4*(-(-d + e*x)*(d + e*x))**p/(d + e*x)**3, x)
 

3.3.86.7 Maxima [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^3,x, algorithm="maxima")
 

output

integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d)^3, x)
 

3.3.86.8 Giac [F]

\[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int { \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{p} x^{4}}{{\left (e x + d\right )}^{3}} \,d x } \]

input

integrate(x^4*(-e^2*x^2+d^2)^p/(e*x+d)^3,x, algorithm="giac")
 

output

integrate((-e^2*x^2 + d^2)^p*x^4/(e*x + d)^3, x)
 

3.3.86.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (d^2-e^2 x^2\right )^p}{(d+e x)^3} \, dx=\int \frac {x^4\,{\left (d^2-e^2\,x^2\right )}^p}{{\left (d+e\,x\right )}^3} \,d x \]

input

int((x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^3,x)
 

output

int((x^4*(d^2 - e^2*x^2)^p)/(d + e*x)^3, x)