\(\int x^4 (a+b x^2+c x^4)^p \, dx\) [1099]

Optimal result

Integrand size = 18, antiderivative size = 138 \[ \int x^4 \left (a+b x^2+c x^4\right )^p \, dx=\frac {1}{5} x^5 \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right ) \] Output:

1/5*x^5*(c*x^4+b*x^2+a)^p*AppellF1(5/2,-p,-p,7/2,-2*c*x^2/(b-(-4*a*c+b^2)^ 
(1/2)),-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)/((1+2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))^p)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.20 \[ \int x^4 \left (a+b x^2+c x^4\right )^p \, dx=\frac {1}{5} x^5 \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right ) \] Input:

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

Output:

(x^5*(a + b*x^2 + c*x^4)^p*AppellF1[5/2, -p, -p, 7/2, (-2*c*x^2)/(b + Sqrt 
[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/(5*((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 = 0.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

\(\displaystyle \int x^4 \left (a+b x^2+c x^4\right )^p \, dx\)

\(\Big \downarrow \) 1461

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

\(\Big \downarrow \) 394

\(\displaystyle \frac {1}{5} x^5 \left (\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (\frac {5}{2},-p,-p,\frac {7}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )\)

Input:

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

Output:

(x^5*(a + b*x^2 + c*x^4)^p*AppellF1[5/2, -p, -p, 7/2, (-2*c*x^2)/(b - Sqrt 
[b^2 - 4*a*c]), (-2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])])/(5*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(16*(a + b*x**2 + c*x**4)**p*a*c*p**2*x + 12*(a + b*x**2 + c*x**4)**p*a*c* 
p*x - 4*(a + b*x**2 + c*x**4)**p*b**2*p**2*x - 6*(a + b*x**2 + c*x**4)**p* 
b**2*p*x + 8*(a + b*x**2 + c*x**4)**p*b*c*p**2*x**3 + 2*(a + b*x**2 + c*x* 
*4)**p*b*c*p*x**3 + 16*(a + b*x**2 + c*x**4)**p*c**2*p**2*x**5 + 16*(a + b 
*x**2 + c*x**4)**p*c**2*p*x**5 + 3*(a + b*x**2 + c*x**4)**p*c**2*x**5 - 10 
24*int((a + b*x**2 + c*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 
64*b*p**3*x**2 + 144*b*p**2*x**2 + 92*b*p*x**2 + 15*b*x**2 + 64*c*p**3*x** 
4 + 144*c*p**2*x**4 + 92*c*p*x**4 + 15*c*x**4),x)*a**2*c*p**5 - 3072*int(( 
a + b*x**2 + c*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p** 
3*x**2 + 144*b*p**2*x**2 + 92*b*p*x**2 + 15*b*x**2 + 64*c*p**3*x**4 + 144* 
c*p**2*x**4 + 92*c*p*x**4 + 15*c*x**4),x)*a**2*c*p**4 - 3200*int((a + b*x* 
*2 + c*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**2 + 
 144*b*p**2*x**2 + 92*b*p*x**2 + 15*b*x**2 + 64*c*p**3*x**4 + 144*c*p**2*x 
**4 + 92*c*p*x**4 + 15*c*x**4),x)*a**2*c*p**3 - 1344*int((a + b*x**2 + c*x 
**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**2 + 144*b*p 
**2*x**2 + 92*b*p*x**2 + 15*b*x**2 + 64*c*p**3*x**4 + 144*c*p**2*x**4 + 92 
*c*p*x**4 + 15*c*x**4),x)*a**2*c*p**2 - 180*int((a + b*x**2 + c*x**4)**p/( 
64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**2 + 144*b*p**2*x**2 
+ 92*b*p*x**2 + 15*b*x**2 + 64*c*p**3*x**4 + 144*c*p**2*x**4 + 92*c*p*x**4 
 + 15*c*x**4),x)*a**2*c*p + 256*int((a + b*x**2 + c*x**4)**p/(64*a*p**3...