\(\int (e x)^{1-2 p} (c+d x)^4 (a+b x^2)^p \, dx\) [293]

Optimal result

Integrand size = 26, antiderivative size = 324 \[ \int (e x)^{1-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {d^2 \left (18 b c^2-a d^2 (2-p)\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{1+p}}{12 b^2 e}+\frac {4 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{1+p}}{5 b e^2}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{1+p}}{6 b e^3}-\frac {4 c d \left (a d^2-\frac {5 b c^2}{3-2 p}\right ) (e x)^{3-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),-p,\frac {1}{2} (5-2 p),-\frac {b x^2}{a}\right )}{5 b e^2}+\frac {\left (6 b^2 c^4-18 a b c^2 d^2 (1-p)+a^2 d^4 \left (2-3 p+p^2\right )\right ) (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{12 b^2 e (1-p)} \] Output:

1/12*d^2*(18*b*c^2-a*d^2*(2-p))*(e*x)^(2-2*p)*(b*x^2+a)^(p+1)/b^2/e+4/5*c* 
d^3*(e*x)^(3-2*p)*(b*x^2+a)^(p+1)/b/e^2+1/6*d^4*(e*x)^(4-2*p)*(b*x^2+a)^(p 
+1)/b/e^3-4/5*c*d*(a*d^2-5*b*c^2/(3-2*p))*(e*x)^(3-2*p)*(b*x^2+a)^p*hyperg 
eom([-p, 3/2-p],[5/2-p],-b*x^2/a)/b/e^2/((1+b*x^2/a)^p)+1/12*(6*b^2*c^4-18 
*a*b*c^2*d^2*(1-p)+a^2*d^4*(p^2-3*p+2))*(e*x)^(2-2*p)*(b*x^2+a)^p*hypergeo 
m([-p, 1-p],[2-p],-b*x^2/a)/b^2/e/(1-p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.70 \[ \int (e x)^{1-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {1}{2} e x^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {c^4 \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{1-p}+d x \left (\frac {8 c^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,-p,\frac {5}{2}-p,-\frac {b x^2}{a}\right )}{3-2 p}+d x \left (-\frac {6 c^2 \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )}{-2+p}+d x \left (\frac {8 c \operatorname {Hypergeometric2F1}\left (\frac {5}{2}-p,-p,\frac {7}{2}-p,-\frac {b x^2}{a}\right )}{5-2 p}+\frac {d x \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^2}{a}\right )}{3-p}\right )\right )\right )\right ) \] Input:

Integrate[(e*x)^(1 - 2*p)*(c + d*x)^4*(a + b*x^2)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {559, 27, 2340, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\int (e x)^{1-2 p} \left (b x^2+a\right )^p \left (15 b^2 c^4+12 b d \left (5 b c^2-a d^2 (3-2 p)\right ) x c+5 b d^2 \left (18 b c^2-a d^2 (2-p)\right ) x^2\right )dx}{5 b}+\frac {12 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{p+1}}{5 e^2}}{3 b}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 2340

\(\displaystyle \frac {\frac {\frac {\int 2 b (e x)^{1-2 p} \left (5 \left (6 b^2 c^4-18 a b d^2 (1-p) c^2+a^2 d^4 \left (p^2-3 p+2\right )\right )+24 b c d \left (5 b c^2-a d^2 (3-2 p)\right ) x\right ) \left (b x^2+a\right )^pdx}{4 b}+\frac {5 d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (2-p)\right )}{4 e}}{5 b}+\frac {12 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{p+1}}{5 e^2}}{3 b}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {1}{2} \int (e x)^{1-2 p} \left (5 \left (6 b^2 c^4-18 a b d^2 (1-p) c^2+a^2 d^4 \left (p^2-3 p+2\right )\right )+24 b c d \left (5 b c^2-a d^2 (3-2 p)\right ) x\right ) \left (b x^2+a\right )^pdx+\frac {5 d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (2-p)\right )}{4 e}}{5 b}+\frac {12 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{p+1}}{5 e^2}}{3 b}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 557

\(\displaystyle \frac {\frac {\frac {1}{2} \left (5 \left (a^2 d^4 \left (p^2-3 p+2\right )-18 a b c^2 d^2 (1-p)+6 b^2 c^4\right ) \int (e x)^{1-2 p} \left (b x^2+a\right )^pdx+\frac {24 b c d \left (5 b c^2-a d^2 (3-2 p)\right ) \int (e x)^{2-2 p} \left (b x^2+a\right )^pdx}{e}\right )+\frac {5 d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (2-p)\right )}{4 e}}{5 b}+\frac {12 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{p+1}}{5 e^2}}{3 b}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\frac {1}{2} \left (5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a^2 d^4 \left (p^2-3 p+2\right )-18 a b c^2 d^2 (1-p)+6 b^2 c^4\right ) \int (e x)^{1-2 p} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {24 b c d \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (5 b c^2-a d^2 (3-2 p)\right ) \int (e x)^{2-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{e}\right )+\frac {5 d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (2-p)\right )}{4 e}}{5 b}+\frac {12 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{p+1}}{5 e^2}}{3 b}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {\frac {1}{2} \left (\frac {5 (e x)^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (a^2 d^4 \left (p^2-3 p+2\right )-18 a b c^2 d^2 (1-p)+6 b^2 c^4\right ) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{2 e (1-p)}+\frac {24 b c d (e x)^{3-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (5 b c^2-a d^2 (3-2 p)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),-p,\frac {1}{2} (5-2 p),-\frac {b x^2}{a}\right )}{e^2 (3-2 p)}\right )+\frac {5 d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (18 b c^2-a d^2 (2-p)\right )}{4 e}}{5 b}+\frac {12 c d^3 (e x)^{3-2 p} \left (a+b x^2\right )^{p+1}}{5 e^2}}{3 b}+\frac {d^4 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

Input:

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

Output:

(d^4*(e*x)^(4 - 2*p)*(a + b*x^2)^(1 + p))/(6*b*e^3) + ((12*c*d^3*(e*x)^(3 
- 2*p)*(a + b*x^2)^(1 + p))/(5*e^2) + ((5*d^2*(18*b*c^2 - a*d^2*(2 - p))*( 
e*x)^(2 - 2*p)*(a + b*x^2)^(1 + p))/(4*e) + ((24*b*c*d*(5*b*c^2 - a*d^2*(3 
 - 2*p))*(e*x)^(3 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[(3 - 2*p)/2, -p, 
(5 - 2*p)/2, -((b*x^2)/a)])/(e^2*(3 - 2*p)*(1 + (b*x^2)/a)^p) + (5*(6*b^2* 
c^4 - 18*a*b*c^2*d^2*(1 - p) + a^2*d^4*(2 - 3*p + p^2))*(e*x)^(2 - 2*p)*(a 
 + b*x^2)^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x^2)/a)])/(2*e*(1 - p 
)*(1 + (b*x^2)/a)^p))/2)/(5*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]

\[\int \left (e x \right )^{1-2 p} \left (d x +c \right )^{4} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(1-2*p)*(d*x+c)^4*(b*x^2+a)^p,x)
 

Output:

int((e*x)^(1-2*p)*(d*x+c)^4*(b*x^2+a)^p,x)
 

Fricas [F]

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

integrate((e*x)^(1-2*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 + 1), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 155.04 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.85 \[ \int (e x)^{1-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{4} e^{1 - 2 p} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} + \frac {2 a^{p} c^{3} d e^{1 - 2 p} x^{3 - 2 p} \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {5}{2} - p\right )} + \frac {3 a^{p} c^{2} d^{2} e^{1 - 2 p} x^{4 - 2 p} \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 - p \\ 3 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (3 - p\right )} + \frac {2 a^{p} c d^{3} e^{1 - 2 p} x^{5 - 2 p} \Gamma \left (\frac {5}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {5}{2} - p \\ \frac {7}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {7}{2} - p\right )} + \frac {a^{p} d^{4} e^{1 - 2 p} x^{6 - 2 p} \Gamma \left (3 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 3 - p \\ 4 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (4 - p\right )} \] Input:

integrate((e*x)**(1-2*p)*(d*x+c)**4*(b*x**2+a)**p,x)
 

Output:

a**p*c**4*e**(1 - 2*p)*x**(2 - 2*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p 
,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(2 - p)) + 2*a**p*c**3*d*e**(1 - 2*p 
)*x**(3 - 2*p)*gamma(3/2 - p)*hyper((-p, 3/2 - p), (5/2 - p,), b*x**2*exp_ 
polar(I*pi)/a)/gamma(5/2 - p) + 3*a**p*c**2*d**2*e**(1 - 2*p)*x**(4 - 2*p) 
*gamma(2 - p)*hyper((-p, 2 - p), (3 - p,), b*x**2*exp_polar(I*pi)/a)/gamma 
(3 - p) + 2*a**p*c*d**3*e**(1 - 2*p)*x**(5 - 2*p)*gamma(5/2 - p)*hyper((-p 
, 5/2 - p), (7/2 - p,), b*x**2*exp_polar(I*pi)/a)/gamma(7/2 - p) + a**p*d* 
*4*e**(1 - 2*p)*x**(6 - 2*p)*gamma(3 - p)*hyper((-p, 3 - p), (4 - p,), b*x 
**2*exp_polar(I*pi)/a)/(2*gamma(4 - p))
 

Maxima [F]

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

integrate((e*x)^(1-2*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 + 1), x)
 

Giac [F]

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

integrate((e*x)^(1-2*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 + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{1-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx=\int {\left (e\,x\right )}^{1-2\,p}\,{\left (b\,x^2+a\right )}^p\,{\left (c+d\,x\right )}^4 \,d x \] Input:

int((e*x)^(1 - 2*p)*(a + b*x^2)^p*(c + d*x)^4,x)
 

Output:

int((e*x)^(1 - 2*p)*(a + b*x^2)^p*(c + d*x)^4, x)
 

Reduce [F]

\[ \int (e x)^{1-2 p} (c+d x)^4 \left (a+b x^2\right )^p \, dx =\text {Too large to display} \] Input:

int((e*x)^(1-2*p)*(d*x+c)^4*(b*x^2+a)^p,x)
 

Output:

(e*(64*(a + b*x**2)**p*a**2*c*d**3*p**2*x - 96*(a + b*x**2)**p*a**2*c*d**3 
*p*x + 5*(a + b*x**2)**p*a**2*d**4*p**2*x**2 - 10*(a + b*x**2)**p*a**2*d** 
4*p*x**2 + 160*(a + b*x**2)**p*a*b*c**3*d*p*x + 90*(a + b*x**2)**p*a*b*c** 
2*d**2*p*x**2 + 32*(a + b*x**2)**p*a*b*c*d**3*p*x**3 + 5*(a + b*x**2)**p*a 
*b*d**4*p*x**4 + 30*(a + b*x**2)**p*b**2*c**4*x**2 + 80*(a + b*x**2)**p*b* 
*2*c**3*d*x**3 + 90*(a + b*x**2)**p*b**2*c**2*d**2*x**4 + 48*(a + b*x**2)* 
*p*b**2*c*d**3*x**5 + 10*(a + b*x**2)**p*b**2*d**4*x**6 + 128*x**(2*p)*int 
((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*c*d**3*p**3 - 256* 
x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*c*d**3 
*p**2 + 96*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)* 
a**3*c*d**3*p + 320*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b* 
x**2),x)*a**2*b*c**3*d*p**2 - 160*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a 
 + x**(2*p)*b*x**2),x)*a**2*b*c**3*d*p + 10*x**(2*p)*int(((a + b*x**2)**p* 
x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*d**4*p**3 - 30*x**(2*p)*int(((a 
+ b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*d**4*p**2 + 20*x**( 
2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*d**4*p 
 + 180*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)* 
a**2*b*c**2*d**2*p**2 - 180*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + 
 x**(2*p)*b*x**2),x)*a**2*b*c**2*d**2*p + 60*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**4*p))/(60*x**(2*p)*e**(...