\(\int (d x)^m (a+b x^2+c x^4)^p \, dx\) [1104]

Optimal result

Integrand size = 20, antiderivative size = 155 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^p \, dx=\frac {(d x)^{1+m} \left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (\frac {1+m}{2},-p,-p,\frac {3+m}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d (1+m)} \] Output:

(d*x)^(1+m)*(c*x^4+b*x^2+a)^p*AppellF1(1/2+1/2*m,-p,-p,3/2+1/2*m,-2*c*x^2/ 
(b-(-4*a*c+b^2)^(1/2)),-2*c*x^2/(b+(-4*a*c+b^2)^(1/2)))/d/(1+m)/((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] (warning: unable to verify)

Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15 \[ \int (d x)^m \left (a+b x^2+c x^4\right )^p \, dx=\frac {x (d x)^m \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^2+c x^4\right )^p \operatorname {AppellF1}\left (\frac {1+m}{2},-p,-p,\frac {3+m}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )}{1+m} \] Input:

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

Output:

(x*(d*x)^m*(a + b*x^2 + c*x^4)^p*AppellF1[(1 + m)/2, -p, -p, (3 + m)/2, (- 
2*c*x^2)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])])/((1 
 + m)*((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 = 0.47 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

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

\(\Big \downarrow \) 1461

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

\(\Big \downarrow \) 394

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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