\(\int (a+c x^2+b x^4)^p \, dx\) [18]

Optimal result

Integrand size = 14, antiderivative size = 133 \[ \int \left (a+c x^2+b x^4\right )^p \, dx=x \left (1+\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (1+\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {-4 a b+c^2}},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}}\right ) \] Output:

x*(b*x^4+c*x^2+a)^p*AppellF1(1/2,-p,-p,3/2,-2*b*x^2/(c-(-4*a*b+c^2)^(1/2)) 
,-2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))/((1+2*b*x^2/(c-(-4*a*b+c^2)^(1/2)))^p)/( 
(1+2*b*x^2/(c+(-4*a*b+c^2)^(1/2)))^p)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.21 \[ \int \left (a+c x^2+b x^4\right )^p \, dx=x \left (\frac {c-\sqrt {-4 a b+c^2}+2 b x^2}{c-\sqrt {-4 a b+c^2}}\right )^{-p} \left (\frac {c+\sqrt {-4 a b+c^2}+2 b x^2}{c+\sqrt {-4 a b+c^2}}\right )^{-p} \left (a+c x^2+b x^4\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c+\sqrt {-4 a b+c^2}},\frac {2 b x^2}{-c+\sqrt {-4 a b+c^2}}\right ) \] Input:

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

Output:

(x*(a + c*x^2 + b*x^4)^p*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c + Sqrt[- 
4*a*b + c^2]), (2*b*x^2)/(-c + Sqrt[-4*a*b + c^2])])/(((c - Sqrt[-4*a*b + 
c^2] + 2*b*x^2)/(c - Sqrt[-4*a*b + c^2]))^p*((c + Sqrt[-4*a*b + c^2] + 2*b 
*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1418, 333}

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.

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

\(\Big \downarrow \) 1418

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

\(\Big \downarrow \) 333

\(\displaystyle x \left (\frac {2 b x^2}{c-\sqrt {c^2-4 a b}}+1\right )^{-p} \left (\frac {2 b x^2}{\sqrt {c^2-4 a b}+c}+1\right )^{-p} \left (a+b x^4+c x^2\right )^p \operatorname {AppellF1}\left (\frac {1}{2},-p,-p,\frac {3}{2},-\frac {2 b x^2}{c-\sqrt {c^2-4 a b}},-\frac {2 b x^2}{c+\sqrt {c^2-4 a b}}\right )\)

Input:

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

Output:

(x*(a + c*x^2 + b*x^4)^p*AppellF1[1/2, -p, -p, 3/2, (-2*b*x^2)/(c - Sqrt[- 
4*a*b + c^2]), (-2*b*x^2)/(c + Sqrt[-4*a*b + c^2])])/((1 + (2*b*x^2)/(c - 
Sqrt[-4*a*b + c^2]))^p*(1 + (2*b*x^2)/(c + Sqrt[-4*a*b + c^2]))^p)
 

Defintions of rubi rules used

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1418

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{q = Rt[b^ 
2 - 4*a*c, 2]}, Simp[a^IntPart[p]*((a + b*x^2 + c*x^4)^FracPart[p]/((1 + 2* 
c*(x^2/(b + q)))^FracPart[p]*(1 + 2*c*(x^2/(b - q)))^FracPart[p]))   Int[(1 
 + 2*c*(x^2/(b + q)))^p*(1 + 2*c*(x^2/(b - q)))^p, x], x]] /; FreeQ[{a, b, 
c, p}, x] && NeQ[b^2 - 4*a*c, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

((a + b*x**4 + c*x**2)**p*x + 16*int((a + b*x**4 + c*x**2)**p/(4*a*p + a + 
 4*b*p*x**4 + b*x**4 + 4*c*p*x**2 + c*x**2),x)*a*p**2 + 4*int((a + b*x**4 
+ c*x**2)**p/(4*a*p + a + 4*b*p*x**4 + b*x**4 + 4*c*p*x**2 + c*x**2),x)*a* 
p + 8*int(((a + b*x**4 + c*x**2)**p*x**2)/(4*a*p + a + 4*b*p*x**4 + b*x**4 
 + 4*c*p*x**2 + c*x**2),x)*c*p**2 + 2*int(((a + b*x**4 + c*x**2)**p*x**2)/ 
(4*a*p + a + 4*b*p*x**4 + b*x**4 + 4*c*p*x**2 + c*x**2),x)*c*p)/(4*p + 1)