\(\int (e x)^{-7-4 p} (a+b x^2+c x^4)^p \, dx\) [1105]

Optimal result

Integrand size = 24, antiderivative size = 267 \[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\frac {(e x)^{-4 p} \left (-2 a (1+p)+b (2+p) x^2\right ) \left (a+b x^2+c x^4\right )^{1+p}}{4 a^2 e^7 \left (3+5 p+2 p^2\right ) x^6}-\frac {\left (-2 a c+b^2 (2+p)\right ) (e x)^{-4 p} \left (b+\sqrt {b^2-4 a c}+2 c x^2\right ) \left (\frac {2 a+\left (b+\sqrt {b^2-4 a c}\right ) x^2}{2 a+\left (b-\sqrt {b^2-4 a c}\right ) x^2}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,-\frac {\sqrt {b^2-4 a c} x^2}{a+\frac {2 a c x^2}{b+\sqrt {b^2-4 a c}}}\right )}{4 a^2 \left (b+\sqrt {b^2-4 a c}\right ) e^7 \left (3+8 p+4 p^2\right ) x^2} \] Output:

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

Mathematica [F]

\[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx \] Input:

Integrate[(e*x)^(-7 - 4*p)*(a + b*x^2 + c*x^4)^p,x]
 

Output:

Integrate[(e*x)^(-7 - 4*p)*(a + b*x^2 + c*x^4)^p, x]
 

Rubi [F]

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.

\(\displaystyle \int (e x)^{-4 p-7} \left (a+b x^2+c x^4\right )^p \, dx\)

\(\Big \downarrow \) 1461

\(\displaystyle \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \int (e x)^{-4 p-7} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

\(\Big \downarrow \) 1014

\(\displaystyle x^{4 p+7} (e x)^{-4 p-7} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \int x^{-4 p-7} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

\(\Big \downarrow \) 7299

\(\displaystyle x^{4 p+7} (e x)^{-4 p-7} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \int x^{-4 p-7} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

Input:

Int[(e*x)^(-7 - 4*p)*(a + b*x^2 + c*x^4)^p,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1014

Int[(u_)^(m_.)*((a_.) + (b_.)*(v_)^(n_))^(p_.)*((c_.) + (d_.)*(v_)^(n_))^(q 
_.), x_Symbol] :> Simp[u^m/(Coefficient[v, x, 1]*v^m)   Subst[Int[x^m*(a + 
b*x^n)^p*(c + d*x^n)^q, x], x, v], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] 
 && LinearPairQ[u, v, x]
 

rule 1461

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^2 + c*x^4)^FracPart[p]/((1 + 2*c*(x^2/(b + R 
t[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^2/(b - Rt[b^2 - 4*a*c, 2])))^F 
racPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^2/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2 
*c*(x^2/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x]
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int \left (e x \right )^{-7-4 p} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(-7-4*p)*(c*x^4+b*x^2+a)^p,x)
 

Output:

int((e*x)^(-7-4*p)*(c*x^4+b*x^2+a)^p,x)
 

Fricas [F]

\[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} \left (e x\right )^{-4 \, p - 7} \,d x } \] Input:

integrate((e*x)^(-7-4*p)*(c*x^4+b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((c*x^4 + b*x^2 + a)^p*(e*x)^(-4*p - 7), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-7-4*p)*(c*x**4+b*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} \left (e x\right )^{-4 \, p - 7} \,d x } \] Input:

integrate((e*x)^(-7-4*p)*(c*x^4+b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((c*x^4 + b*x^2 + a)^p*(e*x)^(-4*p - 7), x)
 

Giac [F]

\[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} \left (e x\right )^{-4 \, p - 7} \,d x } \] Input:

integrate((e*x)^(-7-4*p)*(c*x^4+b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((c*x^4 + b*x^2 + a)^p*(e*x)^(-4*p - 7), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^p}{{\left (e\,x\right )}^{4\,p+7}} \,d x \] Input:

int((a + b*x^2 + c*x^4)^p/(e*x)^(4*p + 7),x)
 

Output:

int((a + b*x^2 + c*x^4)^p/(e*x)^(4*p + 7), x)
 

Reduce [F]

\[ \int (e x)^{-7-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\text {too large to display} \] Input:

int((e*x)^(-7-4*p)*(c*x^4+b*x^2+a)^p,x)
 

Output:

( - 8*(a + b*x**2 + c*x**4)**p*a**3*c*p**2 - 12*(a + b*x**2 + c*x**4)**p*a 
**3*c*p - 4*(a + b*x**2 + c*x**4)**p*a**3*c + 4*(a + b*x**2 + c*x**4)**p*a 
**2*b**2*p**3 + 10*(a + b*x**2 + c*x**4)**p*a**2*b**2*p**2 + 8*(a + b*x**2 
 + c*x**4)**p*a**2*b**2*p + 2*(a + b*x**2 + c*x**4)**p*a**2*b**2 - 4*(a + 
b*x**2 + c*x**4)**p*a**2*b*c*p**2*x**2 - 2*(a + b*x**2 + c*x**4)**p*a**2*b 
*c*p*x**2 - 8*(a + b*x**2 + c*x**4)**p*a**2*c**2*p**2*x**4 - 8*(a + b*x**2 
 + c*x**4)**p*a**2*c**2*p*x**4 + 2*(a + b*x**2 + c*x**4)**p*a*b**3*p**3*x* 
*2 + 3*(a + b*x**2 + c*x**4)**p*a*b**3*p**2*x**2 + (a + b*x**2 + c*x**4)** 
p*a*b**3*p*x**2 + 4*(a + b*x**2 + c*x**4)**p*a*b**2*c*p**3*x**4 + 10*(a + 
b*x**2 + c*x**4)**p*a*b**2*c*p**2*x**4 + 8*(a + b*x**2 + c*x**4)**p*a*b**2 
*c*p*x**4 + 4*(a + b*x**2 + c*x**4)**p*a*b*c**2*p*x**6 + 2*(a + b*x**2 + c 
*x**4)**p*a*b*c**2*x**6 - (a + b*x**2 + c*x**4)**p*b**4*p**3*x**4 - 3*(a + 
 b*x**2 + c*x**4)**p*b**4*p**2*x**4 - 2*(a + b*x**2 + c*x**4)**p*b**4*p*x* 
*4 - 128*x**(4*p)*int((a + b*x**2 + c*x**4)**p/(8*x**(4*p)*a**2*c*p**2*x + 
 16*x**(4*p)*a**2*c*p*x + 6*x**(4*p)*a**2*c*x - 4*x**(4*p)*a*b**2*p**3*x - 
 12*x**(4*p)*a*b**2*p**2*x - 11*x**(4*p)*a*b**2*p*x - 3*x**(4*p)*a*b**2*x 
+ 8*x**(4*p)*a*b*c*p**2*x**3 + 16*x**(4*p)*a*b*c*p*x**3 + 6*x**(4*p)*a*b*c 
*x**3 + 8*x**(4*p)*a*c**2*p**2*x**5 + 16*x**(4*p)*a*c**2*p*x**5 + 6*x**(4* 
p)*a*c**2*x**5 - 4*x**(4*p)*b**3*p**3*x**3 - 12*x**(4*p)*b**3*p**2*x**3 - 
11*x**(4*p)*b**3*p*x**3 - 3*x**(4*p)*b**3*x**3 - 4*x**(4*p)*b**2*c*p**3...