\(\int (e x)^{-5-4 p} (a+b x^2+c x^4)^p \, dx\) [1106]

Optimal result

Integrand size = 24, antiderivative size = 269 \[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=-\frac {(e x)^{-4 (1+p)} \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right ) \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right ) \left (a+b x^2+c x^4\right )^p}{4 e (1+p)}+\frac {b (e x)^{-2-4 p} \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right ) \left (\frac {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}{1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {\sqrt {b^2-4 a c} x^2}{a \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}\right )}{4 a e^3 (1+2 p)} \] Output:

-1/4*(1+2*c*x^2/(b-(-4*a*c+b^2)^(1/2)))*(1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2))) 
*(c*x^4+b*x^2+a)^p/e/(p+1)/((e*x)^(4*p+4))+1/4*b*(e*x)^(-2-4*p)*(1+2*c*x^2 
/(b-(-4*a*c+b^2)^(1/2)))*(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/(1+2*c*x^2/(b-(-4*a*c+b^2)^(1/2))))/a/e^3/(1+2*p)/ 
(((1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/(1+2*c*x^2/(b-(-4*a*c+b^2)^(1/2))))^p 
)
 

Mathematica [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx \] Input:

Integrate[(e*x)^(-5 - 4*p)*(a + b*x^2 + c*x^4)^p,x]
 

Output:

Integrate[(e*x)^(-5 - 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.

\(\displaystyle \int (e x)^{-4 p-5} \left (a+b x^2+c x^4\right )^p \, dx\)

\(\Big \downarrow \) 1461

\(\displaystyle \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \int (e x)^{-4 p-5} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

\(\Big \downarrow \) 1014

\(\displaystyle x^{4 p+5} (e x)^{-4 p-5} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \int x^{-4 p-5} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

\(\Big \downarrow \) 7299

\(\displaystyle x^{4 p+5} (e x)^{-4 p-5} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \int x^{-4 p-5} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

Input:

Int[(e*x)^(-5 - 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]

\[\int \left (e x \right )^{-5-4 p} \left (c \,x^{4}+b \,x^{2}+a \right )^{p}d x\]

Input:

int((e*x)^(-5-4*p)*(c*x^4+b*x^2+a)^p,x)
 

Output:

int((e*x)^(-5-4*p)*(c*x^4+b*x^2+a)^p,x)
 

Fricas [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} \left (e x\right )^{-4 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-4*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 - 5), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-5-4*p)*(c*x**4+b*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} \left (e x\right )^{-4 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-4*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 - 5), x)
 

Giac [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int { {\left (c x^{4} + b x^{2} + a\right )}^{p} \left (e x\right )^{-4 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-4*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 - 5), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^p}{{\left (e\,x\right )}^{4\,p+5}} \,d x \] Input:

int((a + b*x^2 + c*x^4)^p/(e*x)^(4*p + 5),x)
 

Output:

int((a + b*x^2 + c*x^4)^p/(e*x)^(4*p + 5), x)
 

Reduce [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2+c x^4\right )^p \, dx=\frac {-2 \left (c \,x^{4}+b \,x^{2}+a \right )^{p} a p -\left (c \,x^{4}+b \,x^{2}+a \right )^{p} a -\left (c \,x^{4}+b \,x^{2}+a \right )^{p} b p \,x^{2}+\left (c \,x^{4}+b \,x^{2}+a \right )^{p} c \,x^{4}+16 x^{4 p} \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 x^{4 p} a p x +x^{4 p} a x +2 x^{4 p} b p \,x^{3}+x^{4 p} b \,x^{3}+2 x^{4 p} c p \,x^{5}+x^{4 p} c \,x^{5}}d x \right ) a c \,p^{3} x^{4}+24 x^{4 p} \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 x^{4 p} a p x +x^{4 p} a x +2 x^{4 p} b p \,x^{3}+x^{4 p} b \,x^{3}+2 x^{4 p} c p \,x^{5}+x^{4 p} c \,x^{5}}d x \right ) a c \,p^{2} x^{4}+8 x^{4 p} \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 x^{4 p} a p x +x^{4 p} a x +2 x^{4 p} b p \,x^{3}+x^{4 p} b \,x^{3}+2 x^{4 p} c p \,x^{5}+x^{4 p} c \,x^{5}}d x \right ) a c p \,x^{4}-4 x^{4 p} \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 x^{4 p} a p x +x^{4 p} a x +2 x^{4 p} b p \,x^{3}+x^{4 p} b \,x^{3}+2 x^{4 p} c p \,x^{5}+x^{4 p} c \,x^{5}}d x \right ) b^{2} p^{3} x^{4}-6 x^{4 p} \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 x^{4 p} a p x +x^{4 p} a x +2 x^{4 p} b p \,x^{3}+x^{4 p} b \,x^{3}+2 x^{4 p} c p \,x^{5}+x^{4 p} c \,x^{5}}d x \right ) b^{2} p^{2} x^{4}-2 x^{4 p} \left (\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{p}}{2 x^{4 p} a p x +x^{4 p} a x +2 x^{4 p} b p \,x^{3}+x^{4 p} b \,x^{3}+2 x^{4 p} c p \,x^{5}+x^{4 p} c \,x^{5}}d x \right ) b^{2} p \,x^{4}}{4 x^{4 p} e^{4 p} a \,e^{5} x^{4} \left (2 p^{2}+3 p +1\right )} \] Input:

int((e*x)^(-5-4*p)*(c*x^4+b*x^2+a)^p,x)
 

Output:

( - 2*(a + b*x**2 + c*x**4)**p*a*p - (a + b*x**2 + c*x**4)**p*a - (a + b*x 
**2 + c*x**4)**p*b*p*x**2 + (a + b*x**2 + c*x**4)**p*c*x**4 + 16*x**(4*p)* 
int((a + b*x**2 + c*x**4)**p/(2*x**(4*p)*a*p*x + x**(4*p)*a*x + 2*x**(4*p) 
*b*p*x**3 + x**(4*p)*b*x**3 + 2*x**(4*p)*c*p*x**5 + x**(4*p)*c*x**5),x)*a* 
c*p**3*x**4 + 24*x**(4*p)*int((a + b*x**2 + c*x**4)**p/(2*x**(4*p)*a*p*x + 
 x**(4*p)*a*x + 2*x**(4*p)*b*p*x**3 + x**(4*p)*b*x**3 + 2*x**(4*p)*c*p*x** 
5 + x**(4*p)*c*x**5),x)*a*c*p**2*x**4 + 8*x**(4*p)*int((a + b*x**2 + c*x** 
4)**p/(2*x**(4*p)*a*p*x + x**(4*p)*a*x + 2*x**(4*p)*b*p*x**3 + x**(4*p)*b* 
x**3 + 2*x**(4*p)*c*p*x**5 + x**(4*p)*c*x**5),x)*a*c*p*x**4 - 4*x**(4*p)*i 
nt((a + b*x**2 + c*x**4)**p/(2*x**(4*p)*a*p*x + x**(4*p)*a*x + 2*x**(4*p)* 
b*p*x**3 + x**(4*p)*b*x**3 + 2*x**(4*p)*c*p*x**5 + x**(4*p)*c*x**5),x)*b** 
2*p**3*x**4 - 6*x**(4*p)*int((a + b*x**2 + c*x**4)**p/(2*x**(4*p)*a*p*x + 
x**(4*p)*a*x + 2*x**(4*p)*b*p*x**3 + x**(4*p)*b*x**3 + 2*x**(4*p)*c*p*x**5 
 + x**(4*p)*c*x**5),x)*b**2*p**2*x**4 - 2*x**(4*p)*int((a + b*x**2 + c*x** 
4)**p/(2*x**(4*p)*a*p*x + x**(4*p)*a*x + 2*x**(4*p)*b*p*x**3 + x**(4*p)*b* 
x**3 + 2*x**(4*p)*c*p*x**5 + x**(4*p)*c*x**5),x)*b**2*p*x**4)/(4*x**(4*p)* 
e**(4*p)*a*e**5*x**4*(2*p**2 + 3*p + 1))