\(\int (e x)^{-4-2 p} (c+d x)^4 (a+b x^2)^p \, dx\) [333]

Optimal result

Integrand size = 26, antiderivative size = 302 \[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {d^4 (e x)^{-1-2 p} \left (a+b x^2\right )^{1+p}}{b e^3}-\frac {2 c^3 d (e x)^{-2 (1+p)} \left (a+b x^2\right )^{1+p}}{a e^2 (1+p)}-\frac {c^4 (e x)^{-3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-3-2 p),-p,\frac {1}{2} (-1-2 p),-\frac {b x^2}{a}\right )}{e (3+2 p)}-\frac {d^2 \left (a d^2+\frac {6 b c^2}{1+2 p}\right ) (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{b e^3}-\frac {2 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p} \] Output:

d^4*(e*x)^(-1-2*p)*(b*x^2+a)^(p+1)/b/e^3-2*c^3*d*(b*x^2+a)^(p+1)/a/e^2/(p+ 
1)/((e*x)^(2*p+2))-c^4*(e*x)^(-3-2*p)*(b*x^2+a)^p*hypergeom([-p, -3/2-p],[ 
-1/2-p],-b*x^2/a)/e/(3+2*p)/((1+b*x^2/a)^p)-d^2*(a*d^2+6*b*c^2/(1+2*p))*(e 
*x)^(-1-2*p)*(b*x^2+a)^p*hypergeom([-p, -1/2-p],[1/2-p],-b*x^2/a)/b/e^3/(( 
1+b*x^2/a)^p)-2*c*d^3*(b*x^2+a)^p*hypergeom([-p, -p],[1-p],-b*x^2/a)/e^4/p 
/((e*x)^(2*p))/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.85 \[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (-\frac {c^4 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2}-p,-p,-\frac {1}{2}-p,-\frac {b x^2}{a}\right )}{(3+2 p) x^3}+d \left (-\frac {2 b c^3}{a+a p}-\frac {2 c^3}{(1+p) x^2}-\frac {6 c^2 d \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )}{x+2 p x}+\frac {d^3 x \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )}{1-2 p}-\frac {2 c d^2 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{p}\right )\right )}{e^4} \] Input:

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

Output:

((a + b*x^2)^p*(-((c^4*Hypergeometric2F1[-3/2 - p, -p, -1/2 - p, -((b*x^2) 
/a)])/((3 + 2*p)*x^3*(1 + (b*x^2)/a)^p)) + d*((-2*b*c^3)/(a + a*p) - (2*c^ 
3)/((1 + p)*x^2) - (6*c^2*d*Hypergeometric2F1[-1/2 - p, -p, 1/2 - p, -((b* 
x^2)/a)])/((x + 2*p*x)*(1 + (b*x^2)/a)^p) + (d^3*x*Hypergeometric2F1[1/2 - 
 p, -p, 3/2 - p, -((b*x^2)/a)])/((1 - 2*p)*(1 + (b*x^2)/a)^p) - (2*c*d^2*H 
ypergeometric2F1[-p, -p, 1 - p, -((b*x^2)/a)])/(p*(1 + (b*x^2)/a)^p))))/(e 
^4*(e*x)^(2*p))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {559, 2339, 279, 278, 2340, 25, 27, 557, 242, 279, 278}

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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 2339

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {-\frac {\int -e^3 (e x)^{-2 (p+2)} \left (b^2 c^4+4 b^2 d x c^3-6 a b d^2 (2 p+3) c^2-a^2 d^4 \left (4 p^2+8 p+3\right )\right ) \left (b x^2+a\right )^pdx}{b}-\frac {d^2 e^2 (e x)^{-2 p-3} \left (a+b x^2\right )^{p+1} \left (a d^2 (2 p+1)+6 b c^2\right )}{b}}{e^3}-\frac {2 b c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p}}{b}+\frac {d^4 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1}}{b e^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int e^3 (e x)^{-2 (p+2)} \left (b^2 c^4+4 b^2 d x c^3-6 a b d^2 (2 p+3) c^2-a^2 d^4 \left (4 p^2+8 p+3\right )\right ) \left (b x^2+a\right )^pdx}{b}-\frac {d^2 e^2 (e x)^{-2 p-3} \left (a+b x^2\right )^{p+1} \left (a d^2 (2 p+1)+6 b c^2\right )}{b}}{e^3}-\frac {2 b c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p}}{b}+\frac {d^4 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1}}{b e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {e^3 \int (e x)^{-2 (p+2)} \left (b^2 c^4+4 b^2 d x c^3-6 a b d^2 (2 p+3) c^2-a^2 d^4 \left (4 p^2+8 p+3\right )\right ) \left (b x^2+a\right )^pdx}{b}-\frac {d^2 e^2 (e x)^{-2 p-3} \left (a+b x^2\right )^{p+1} \left (a d^2 (2 p+1)+6 b c^2\right )}{b}}{e^3}-\frac {2 b c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p}}{b}+\frac {d^4 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1}}{b e^3}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {\frac {\frac {e^3 \left (\left (-a^2 d^4 \left (4 p^2+8 p+3\right )-6 a b c^2 d^2 (2 p+3)+b^2 c^4\right ) \int (e x)^{-2 (p+2)} \left (b x^2+a\right )^pdx+\frac {4 b^2 c^3 d \int (e x)^{-2 p-3} \left (b x^2+a\right )^pdx}{e}\right )}{b}-\frac {d^2 e^2 (e x)^{-2 p-3} \left (a+b x^2\right )^{p+1} \left (a d^2 (2 p+1)+6 b c^2\right )}{b}}{e^3}-\frac {2 b c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p}}{b}+\frac {d^4 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1}}{b e^3}\)

\(\Big \downarrow \) 242

\(\displaystyle \frac {\frac {\frac {e^3 \left (\left (-a^2 d^4 \left (4 p^2+8 p+3\right )-6 a b c^2 d^2 (2 p+3)+b^2 c^4\right ) \int (e x)^{-2 (p+2)} \left (b x^2+a\right )^pdx-\frac {2 b^2 c^3 d (e x)^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{a e^2 (p+1)}\right )}{b}-\frac {d^2 e^2 (e x)^{-2 p-3} \left (a+b x^2\right )^{p+1} \left (a d^2 (2 p+1)+6 b c^2\right )}{b}}{e^3}-\frac {2 b c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p}}{b}+\frac {d^4 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1}}{b e^3}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\frac {e^3 \left (\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^2 d^4 \left (4 p^2+8 p+3\right )-6 a b c^2 d^2 (2 p+3)+b^2 c^4\right ) \int (e x)^{-2 (p+2)} \left (\frac {b x^2}{a}+1\right )^pdx-\frac {2 b^2 c^3 d (e x)^{-2 (p+1)} \left (a+b x^2\right )^{p+1}}{a e^2 (p+1)}\right )}{b}-\frac {d^2 e^2 (e x)^{-2 p-3} \left (a+b x^2\right )^{p+1} \left (a d^2 (2 p+1)+6 b c^2\right )}{b}}{e^3}-\frac {2 b c d^3 (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^4 p}}{b}+\frac {d^4 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1}}{b e^3}\)

\(\Big \downarrow \) 278

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

Input:

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

Output:

(d^4*(e*x)^(-1 - 2*p)*(a + b*x^2)^(1 + p))/(b*e^3) + ((-((d^2*e^2*(6*b*c^2 
 + a*d^2*(1 + 2*p))*(e*x)^(-3 - 2*p)*(a + b*x^2)^(1 + p))/b) + (e^3*((-2*b 
^2*c^3*d*(a + b*x^2)^(1 + p))/(a*e^2*(1 + p)*(e*x)^(2*(1 + p))) - ((b^2*c^ 
4 - 6*a*b*c^2*d^2*(3 + 2*p) - a^2*d^4*(3 + 8*p + 4*p^2))*(e*x)^(-3 - 2*p)* 
(a + b*x^2)^p*Hypergeometric2F1[(-3 - 2*p)/2, -p, (-1 - 2*p)/2, -((b*x^2)/ 
a)])/(e*(3 + 2*p)*(1 + (b*x^2)/a)^p)))/b)/e^3 - (2*b*c*d^3*(a + b*x^2)^p*H 
ypergeometric2F1[-p, -p, 1 - p, -((b*x^2)/a)])/(e^4*p*(e*x)^(2*p)*(1 + (b* 
x^2)/a)^p))/b
 

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 242

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

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 557

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

rule 559

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

rule 2339

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With 
[{q = Expon[Pq, x]}, Simp[Coeff[Pq, x, q]/c^q   Int[(c*x)^(m + q)*(a + b*x^ 
2)^p, x], x] + Simp[1/c^q   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[c^q*Pq - 
Coeff[Pq, x, q]*(c*x)^q, x], x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0] 
] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] &&  !(IGtQ[m, 0] && ILtQ[p + 
 1/2, 0])
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

Maple [F]

\[\int \left (e x \right )^{-4-2 p} \left (d x +c \right )^{4} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(-4-2*p)*(d*x+c)^4*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(-4-2*p)*(d*x+c)^4*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{4} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 4} \,d x } \] Input:

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

Output:

integral((d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4)*(b*x^2 
+ a)^p*(e*x)^(-2*p - 4), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 106.84 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.99 \[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{4} e^{- 2 p - 4} x^{- 2 p - 3} \Gamma \left (- p - \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {3}{2} \\ - p - \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (- p - \frac {1}{2}\right )} + \frac {2 a^{p} c^{3} d e^{- 2 p - 4} x^{- 2 p - 2} \left (1 + \frac {b x^{2}}{a}\right )^{p + 1} \Gamma \left (- p - 1\right )}{\Gamma \left (- p\right )} + \frac {3 a^{p} c^{2} d^{2} e^{- 2 p - 4} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {1}{2} - p\right )} + \frac {2 a^{p} c d^{3} e^{- 2 p - 4} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (1 - p\right )} + \frac {a^{p} d^{4} e^{- 2 p - 4} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right )} \] Input:

integrate((e*x)**(-4-2*p)*(d*x+c)**4*(b*x**2+a)**p,x)
 

Output:

a**p*c**4*e**(-2*p - 4)*x**(-2*p - 3)*gamma(-p - 3/2)*hyper((-p, -p - 3/2) 
, (-p - 1/2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(-p - 1/2)) + 2*a**p*c**3 
*d*e**(-2*p - 4)*x**(-2*p - 2)*(1 + b*x**2/a)**(p + 1)*gamma(-p - 1)/gamma 
(-p) + 3*a**p*c**2*d**2*e**(-2*p - 4)*x**(-2*p - 1)*gamma(-p - 1/2)*hyper( 
(-p, -p - 1/2), (1/2 - p,), b*x**2*exp_polar(I*pi)/a)/gamma(1/2 - p) + 2*a 
**p*c*d**3*e**(-2*p - 4)*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_po 
lar(I*pi)/a)/(x**(2*p)*gamma(1 - p)) + a**p*d**4*e**(-2*p - 4)*x**(1 - 2*p 
)*gamma(1/2 - p)*hyper((-p, 1/2 - p), (3/2 - p,), b*x**2*exp_polar(I*pi)/a 
)/(2*gamma(3/2 - p))
 

Maxima [F]

\[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{4} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 4} \,d x } \] Input:

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

Output:

integrate((d*x + c)^4*(b*x^2 + a)^p*(e*x)^(-2*p - 4), x)
 

Giac [F]

\[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\int { {\left (d x + c\right )}^{4} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p - 4} \,d x } \] Input:

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

Output:

integrate((d*x + c)^4*(b*x^2 + a)^p*(e*x)^(-2*p - 4), x)
 

Mupad [F(-1)]

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

int(((a + b*x^2)^p*(c + d*x)^4)/(e*x)^(2*p + 4),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x)^4)/(e*x)^(2*p + 4), x)
 

Reduce [F]

\[ \int (e x)^{-4-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {-\left (b \,x^{2}+a \right )^{p} a \,c^{4} p -\left (b \,x^{2}+a \right )^{p} a \,c^{4}-6 \left (b \,x^{2}+a \right )^{p} a \,c^{3} d x -18 \left (b \,x^{2}+a \right )^{p} a \,c^{2} d^{2} p \,x^{2}-18 \left (b \,x^{2}+a \right )^{p} a \,c^{2} d^{2} x^{2}+3 \left (b \,x^{2}+a \right )^{p} a \,d^{4} p \,x^{4}+3 \left (b \,x^{2}+a \right )^{p} a \,d^{4} x^{4}-6 \left (b \,x^{2}+a \right )^{p} b \,c^{3} d \,x^{3}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{2} c^{4} p^{2} x^{3}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{4}+x^{2 p} b \,x^{6}}d x \right ) a^{2} c^{4} p \,x^{3}-36 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c^{2} d^{2} p^{2} x^{3}-36 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} c^{2} d^{2} p \,x^{3}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{4} p^{2} x^{3}+6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{4} p \,x^{3}+12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a c \,d^{3} p \,x^{3}+12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} x}d x \right ) a c \,d^{3} x^{3}}{3 x^{2 p} e^{2 p} a \,e^{4} x^{3} \left (p +1\right )} \] Input:

int((e*x)^(-4-2*p)*(d*x+c)^4*(b*x^2+a)^p,x)
 

Output:

( - (a + b*x**2)**p*a*c**4*p - (a + b*x**2)**p*a*c**4 - 6*(a + b*x**2)**p* 
a*c**3*d*x - 18*(a + b*x**2)**p*a*c**2*d**2*p*x**2 - 18*(a + b*x**2)**p*a* 
c**2*d**2*x**2 + 3*(a + b*x**2)**p*a*d**4*p*x**4 + 3*(a + b*x**2)**p*a*d** 
4*x**4 - 6*(a + b*x**2)**p*b*c**3*d*x**3 - 2*x**(2*p)*int((a + b*x**2)**p/ 
(x**(2*p)*a*x**4 + x**(2*p)*b*x**6),x)*a**2*c**4*p**2*x**3 - 2*x**(2*p)*in 
t((a + b*x**2)**p/(x**(2*p)*a*x**4 + x**(2*p)*b*x**6),x)*a**2*c**4*p*x**3 
- 36*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**2 + x**(2*p)*b*x**4),x)*a 
**2*c**2*d**2*p**2*x**3 - 36*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**2 
 + x**(2*p)*b*x**4),x)*a**2*c**2*d**2*p*x**3 + 6*x**(2*p)*int((a + b*x**2) 
**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*d**4*p**2*x**3 + 6*x**(2*p)*int 
((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*d**4*p*x**3 + 12*x 
**(2*p)*int((a + b*x**2)**p/(x**(2*p)*x),x)*a*c*d**3*p*x**3 + 12*x**(2*p)* 
int((a + b*x**2)**p/(x**(2*p)*x),x)*a*c*d**3*x**3)/(3*x**(2*p)*e**(2*p)*a* 
e**4*x**3*(p + 1))