\(\int (e x)^{-1-4 p} (a+b x^2+c x^4)^p \, dx\) [1108]

Optimal result

Integrand size = 24, antiderivative size = 150 \[ \int (e x)^{-1-4 p} \left (a+b x^2+c x^4\right )^p \, dx=-\frac {(e x)^{-4 p} \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{4 e p} \] Output:

-1/4*(c*x^4+b*x^2+a)^p*AppellF1(-2*p,-p,-p,1-2*p,-2*c*x^2/(b-(-4*a*c+b^2)^ 
(1/2)),-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/e/p/((e*x)^(4*p))/((1+2*c*x^2/(b-( 
-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^p)
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.19 \[ \int (e x)^{-1-4 p} \left (a+b x^2+c x^4\right )^p \, dx=-\frac {(e x)^{-4 p} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )}{4 e p} \] Input:

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

Output:

-1/4*((a + b*x^2 + c*x^4)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (-2*c*x^2)/(b 
+ Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(e*p*(e*x)^(4*p 
)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt 
[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1461, 394}

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.

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

\(\Big \downarrow \) 1461

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

\(\Big \downarrow \) 394

\(\displaystyle -\frac {(e x)^{-4 p} \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{4 e p}\)

Input:

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

Output:

-1/4*((a + b*x^2 + c*x^4)^p*AppellF1[-2*p, -p, -p, 1 - 2*p, (-2*c*x^2)/(b 
- Sqrt[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(e*p*(e*x)^(4*p 
)*(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)
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((e*x)^(-1-4*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 - 1), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((e*x)^(-1-4*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 - 1), x)
 

Giac [F]

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

integrate((e*x)^(-1-4*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 - 1), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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