\(\int (e x)^{-2-2 p} (c+d x)^4 (a+b x^2)^p \, dx\) [317]

Optimal result

Integrand size = 26, antiderivative size = 310 \[ \int (e x)^{-2-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {d^2 \left (18 b c^2-a d^2 (1-2 p)\right ) (e x)^{-1-2 p} \left (a+b x^2\right )^{1+p}}{3 b^2 e}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{1+p}}{3 b e^3}+\frac {2 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{b e^2}-\frac {\left (3 b^2 c^4+18 a b c^2 d^2 (1+2 p)-a^2 d^4 \left (1-4 p^2\right )\right ) (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-1-2 p),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{3 b^2 e (1+2 p)}-\frac {2 c d \left (b c^2+a d^2 p\right ) (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{b e^2 p} \] Output:

1/3*d^2*(18*b*c^2-a*d^2*(1-2*p))*(e*x)^(-1-2*p)*(b*x^2+a)^(p+1)/b^2/e+1/3* 
d^4*(e*x)^(1-2*p)*(b*x^2+a)^(p+1)/b/e^3+2*c*d^3*(b*x^2+a)^(p+1)/b/e^2/((e* 
x)^(2*p))-1/3*(3*b^2*c^4+18*a*b*c^2*d^2*(1+2*p)-a^2*d^4*(-4*p^2+1))*(e*x)^ 
(-1-2*p)*(b*x^2+a)^p*hypergeom([-p, -1/2-p],[1/2-p],-b*x^2/a)/b^2/e/(1+2*p 
)/((1+b*x^2/a)^p)-2*c*d*(a*d^2*p+b*c^2)*(b*x^2+a)^p*hypergeom([-p, -p],[1- 
p],-b*x^2/a)/b/e^2/p/((e*x)^(2*p))/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.73 \[ \int (e x)^{-2-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (-\frac {c^4 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )}{x+2 p x}+d \left (\frac {6 c^2 d x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )}{1-2 p}-\frac {2 c d^2 x^2 \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{-1+p}+\frac {d^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,-p,\frac {5}{2}-p,-\frac {b x^2}{a}\right )}{3-2 p}-\frac {2 c^3 \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{p}\right )\right )}{e^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {559, 2340, 27, 2340, 27, 557, 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.

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

\(\Big \downarrow \) 559

\(\displaystyle \frac {\int (e x)^{-2 (p+1)} \left (b x^2+a\right )^p \left (3 b c^4+12 b d x c^3+12 b d^3 x^3 c+d^2 \left (18 b c^2-a d^2 (1-2 p)\right ) x^2\right )dx}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int 2 (e x)^{-2 (p+1)} \left (b x^2+a\right )^p \left (3 b^2 c^4+12 b d \left (b c^2+a d^2 p\right ) x c+b d^2 \left (18 b c^2-a d^2 (1-2 p)\right ) x^2\right )dx}{2 b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int (e x)^{-2 (p+1)} \left (b x^2+a\right )^p \left (3 b^2 c^4+12 b d \left (b c^2+a d^2 p\right ) x c+b d^2 \left (18 b c^2-a d^2 (1-2 p)\right ) x^2\right )dx}{b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\frac {\int b (e x)^{-2 (p+1)} \left (3 b^2 c^4+18 a b d^2 (2 p+1) c^2+12 b d \left (b c^2+a d^2 p\right ) x c-a^2 d^4 \left (1-4 p^2\right )\right ) \left (b x^2+a\right )^pdx}{b}+\frac {d^2 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (1-2 p)\right )}{e}}{b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int (e x)^{-2 (p+1)} \left (3 b^2 c^4+18 a b d^2 (2 p+1) c^2+12 b d \left (b c^2+a d^2 p\right ) x c-a^2 d^4 \left (1-4 p^2\right )\right ) \left (b x^2+a\right )^pdx+\frac {d^2 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (1-2 p)\right )}{e}}{b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {\frac {\left (-a^2 d^4 \left (1-4 p^2\right )+18 a b c^2 d^2 (2 p+1)+3 b^2 c^4\right ) \int (e x)^{-2 (p+1)} \left (b x^2+a\right )^pdx+\frac {12 b c d \left (a d^2 p+b c^2\right ) \int (e x)^{-2 p-1} \left (b x^2+a\right )^pdx}{e}+\frac {d^2 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (1-2 p)\right )}{e}}{b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^2 d^4 \left (1-4 p^2\right )+18 a b c^2 d^2 (2 p+1)+3 b^2 c^4\right ) \int (e x)^{-2 (p+1)} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {12 b c d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a d^2 p+b c^2\right ) \int (e x)^{-2 p-1} \left (\frac {b x^2}{a}+1\right )^pdx}{e}+\frac {d^2 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (1-2 p)\right )}{e}}{b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {-\frac {(e x)^{-2 p-1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^2 d^4 \left (1-4 p^2\right )+18 a b c^2 d^2 (2 p+1)+3 b^2 c^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-2 p-1),-p,\frac {1}{2} (1-2 p),-\frac {b x^2}{a}\right )}{e (2 p+1)}-\frac {6 b c d (e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a d^2 p+b c^2\right ) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{e^2 p}+\frac {d^2 (e x)^{-2 p-1} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (1-2 p)\right )}{e}}{b}+\frac {6 c d^3 (e x)^{-2 p} \left (a+b x^2\right )^{p+1}}{e^2}}{3 b}+\frac {d^4 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{3 b e^3}\)

Input:

Int[(e*x)^(-2 - 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))/(3*b*e^3) + ((6*c*d^3*(a + b*x^2 
)^(1 + p))/(e^2*(e*x)^(2*p)) + ((d^2*(18*b*c^2 - a*d^2*(1 - 2*p))*(e*x)^(- 
1 - 2*p)*(a + b*x^2)^(1 + p))/e - ((3*b^2*c^4 + 18*a*b*c^2*d^2*(1 + 2*p) - 
 a^2*d^4*(1 - 4*p^2))*(e*x)^(-1 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[(-1 
 - 2*p)/2, -p, (1 - 2*p)/2, -((b*x^2)/a)])/(e*(1 + 2*p)*(1 + (b*x^2)/a)^p) 
 - (6*b*c*d*(b*c^2 + a*d^2*p)*(a + b*x^2)^p*Hypergeometric2F1[-p, -p, 1 - 
p, -((b*x^2)/a)])/(e^2*p*(e*x)^(2*p)*(1 + (b*x^2)/a)^p))/b)/(3*b)
 

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 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 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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(-2-2*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 - 2), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 136.61 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.94 \[ \int (e x)^{-2-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{4} e^{- 2 p - 2} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \frac {2 a^{p} c^{3} d e^{- 2 p - 2} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (1 - p\right )} + \frac {3 a^{p} c^{2} d^{2} e^{- 2 p - 2} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {3}{2} - p\right )} + \frac {2 a^{p} c d^{3} e^{- 2 p - 2} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (2 - p\right )} + \frac {a^{p} d^{4} e^{- 2 p - 2} x^{3 - 2 p} \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {5}{2} - p\right )} \] Input:

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

Output:

a**p*c**4*e**(-2*p - 2)*x**(-2*p - 1)*gamma(-p - 1/2)*hyper((-p, -p - 1/2) 
, (1/2 - p,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(1/2 - p)) + 2*a**p*c**3*d 
*e**(-2*p - 2)*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_polar(I*pi)/ 
a)/(x**(2*p)*gamma(1 - p)) + 3*a**p*c**2*d**2*e**(-2*p - 2)*x**(1 - 2*p)*g 
amma(1/2 - p)*hyper((-p, 1/2 - p), (3/2 - p,), b*x**2*exp_polar(I*pi)/a)/g 
amma(3/2 - p) + 2*a**p*c*d**3*e**(-2*p - 2)*x**(2 - 2*p)*gamma(1 - p)*hype 
r((-p, 1 - p), (2 - p,), b*x**2*exp_polar(I*pi)/a)/gamma(2 - p) + a**p*d** 
4*e**(-2*p - 2)*x**(3 - 2*p)*gamma(3/2 - p)*hyper((-p, 3/2 - p), (5/2 - p, 
), b*x**2*exp_polar(I*pi)/a)/(2*gamma(5/2 - p))
 

Maxima [F]

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

integrate((e*x)^(-2-2*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 - 2), x)
 

Giac [F]

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

integrate((e*x)^(-2-2*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 - 2), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^{-2-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {-4 \left (b \,x^{2}+a \right )^{p} a^{2} d^{4} p^{3}+2 \left (b \,x^{2}+a \right )^{p} a^{2} d^{4} p^{2}-36 \left (b \,x^{2}+a \right )^{p} a b \,c^{2} d^{2} p^{2}+2 \left (b \,x^{2}+a \right )^{p} a b \,d^{4} p^{2} x^{2}-3 \left (b \,x^{2}+a \right )^{p} b^{2} c^{4} p -6 \left (b \,x^{2}+a \right )^{p} b^{2} c^{3} d x +18 \left (b \,x^{2}+a \right )^{p} b^{2} c^{2} d^{2} p \,x^{2}+6 \left (b \,x^{2}+a \right )^{p} b^{2} c \,d^{3} p \,x^{3}+\left (b \,x^{2}+a \right )^{p} b^{2} d^{4} p \,x^{4}-8 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{3} d^{4} p^{4} x +2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{3} d^{4} p^{2} x -72 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} b \,c^{2} d^{2} p^{3} x -36 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a^{2} b \,c^{2} d^{2} p^{2} x -6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a \,x^{2}+x^{2 p} b \,x^{4}}d x \right ) a \,b^{2} c^{4} p^{2} x +12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a \,b^{2} c \,d^{3} p^{2} x +12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) b^{3} c^{3} d p x}{3 x^{2 p} e^{2 p} b^{2} e^{2} p x} \] Input:

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

Output:

( - 4*(a + b*x**2)**p*a**2*d**4*p**3 + 2*(a + b*x**2)**p*a**2*d**4*p**2 - 
36*(a + b*x**2)**p*a*b*c**2*d**2*p**2 + 2*(a + b*x**2)**p*a*b*d**4*p**2*x* 
*2 - 3*(a + b*x**2)**p*b**2*c**4*p - 6*(a + b*x**2)**p*b**2*c**3*d*x + 18* 
(a + b*x**2)**p*b**2*c**2*d**2*p*x**2 + 6*(a + b*x**2)**p*b**2*c*d**3*p*x* 
*3 + (a + b*x**2)**p*b**2*d**4*p*x**4 - 8*x**(2*p)*int((a + b*x**2)**p/(x* 
*(2*p)*a*x**2 + x**(2*p)*b*x**4),x)*a**3*d**4*p**4*x + 2*x**(2*p)*int((a + 
 b*x**2)**p/(x**(2*p)*a*x**2 + x**(2*p)*b*x**4),x)*a**3*d**4*p**2*x - 72*x 
**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*x**2 + x**(2*p)*b*x**4),x)*a**2*b* 
c**2*d**2*p**3*x - 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*b*c**2*d**2*p**2*x - 6*x**(2*p)*int((a + b*x**2)**p/( 
x**(2*p)*a*x**2 + x**(2*p)*b*x**4),x)*a*b**2*c**4*p**2*x + 12*x**(2*p)*int 
(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*b**2*c*d**3*p**2* 
x + 12*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)* 
b**3*c**3*d*p*x)/(3*x**(2*p)*e**(2*p)*b**2*e**2*p*x)