\(\int \frac {(d+e x^2)^q (A+B x^2+C x^4)}{a+c x^4} \, dx\) [129]

Optimal result

Integrand size = 31, antiderivative size = 234 \[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=-\frac {\left (\sqrt {-a} B \sqrt {c}-A c+a C\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-q,1,\frac {3}{2},-\frac {e x^2}{d},-\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )}{2 a c}+\frac {\left (\sqrt {-a} B \sqrt {c}+A c-a C\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-q,1,\frac {3}{2},-\frac {e x^2}{d},\frac {\sqrt {c} x^2}{\sqrt {-a}}\right )}{2 a c}+\frac {C x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{c} \] Output:

-1/2*((-a)^(1/2)*B*c^(1/2)-A*c+a*C)*x*(e*x^2+d)^q*AppellF1(1/2,1,-q,3/2,-c 
^(1/2)*x^2/(-a)^(1/2),-e*x^2/d)/a/c/((1+e*x^2/d)^q)+1/2*((-a)^(1/2)*B*c^(1 
/2)+A*c-a*C)*x*(e*x^2+d)^q*AppellF1(1/2,1,-q,3/2,c^(1/2)*x^2/(-a)^(1/2),-e 
*x^2/d)/a/c/((1+e*x^2/d)^q)+C*x*(e*x^2+d)^q*hypergeom([1/2, -q],[3/2],-e*x 
^2/d)/c/((1+e*x^2/d)^q)
 

Mathematica [F]

\[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx \] Input:

Integrate[((d + e*x^2)^q*(A + B*x^2 + C*x^4))/(a + c*x^4),x]
 

Output:

Integrate[((d + e*x^2)^q*(A + B*x^2 + C*x^4))/(a + c*x^4), x]
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2257, 2009}

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.

\(\displaystyle \int \frac {\left (A+B x^2+C x^4\right ) \left (d+e x^2\right )^q}{a+c x^4} \, dx\)

\(\Big \downarrow \) 2257

\(\displaystyle \int \left (\frac {\left (d+e x^2\right )^q \left (-a C+A c+B c x^2\right )}{c \left (a+c x^4\right )}+\frac {C \left (d+e x^2\right )^q}{c}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \left (\sqrt {-a} B \sqrt {c}+a C-A c\right ) \operatorname {AppellF1}\left (\frac {1}{2},1,-q,\frac {3}{2},-\frac {\sqrt {c} x^2}{\sqrt {-a}},-\frac {e x^2}{d}\right )}{2 a c}+\frac {x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \left (\sqrt {-a} B \sqrt {c}-a C+A c\right ) \operatorname {AppellF1}\left (\frac {1}{2},1,-q,\frac {3}{2},\frac {\sqrt {c} x^2}{\sqrt {-a}},-\frac {e x^2}{d}\right )}{2 a c}+\frac {C x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{c}\)

Input:

Int[((d + e*x^2)^q*(A + B*x^2 + C*x^4))/(a + c*x^4),x]
 

Output:

-1/2*((Sqrt[-a]*B*Sqrt[c] - A*c + a*C)*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q 
, 3/2, -((Sqrt[c]*x^2)/Sqrt[-a]), -((e*x^2)/d)])/(a*c*(1 + (e*x^2)/d)^q) + 
 ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*x*(d + e*x^2)^q*AppellF1[1/2, 1, -q, 3/ 
2, (Sqrt[c]*x^2)/Sqrt[-a], -((e*x^2)/d)])/(2*a*c*(1 + (e*x^2)/d)^q) + (C*x 
*(d + e*x^2)^q*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)])/(c*(1 + (e*x 
^2)/d)^q)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2257

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol 
] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a 
, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

Maple [F]

\[\int \frac {\left (e \,x^{2}+d \right )^{q} \left (C \,x^{4}+B \,x^{2}+A \right )}{c \,x^{4}+a}d x\]

Input:

int((e*x^2+d)^q*(C*x^4+B*x^2+A)/(c*x^4+a),x)
 

Output:

int((e*x^2+d)^q*(C*x^4+B*x^2+A)/(c*x^4+a),x)
 

Fricas [F]

\[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{q}}{c x^{4} + a} \,d x } \] Input:

integrate((e*x^2+d)^q*(C*x^4+B*x^2+A)/(c*x^4+a),x, algorithm="fricas")
 

Output:

integral((C*x^4 + B*x^2 + A)*(e*x^2 + d)^q/(c*x^4 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**q*(C*x**4+B*x**2+A)/(c*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{q}}{c x^{4} + a} \,d x } \] Input:

integrate((e*x^2+d)^q*(C*x^4+B*x^2+A)/(c*x^4+a),x, algorithm="maxima")
 

Output:

integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^q/(c*x^4 + a), x)
 

Giac [F]

\[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{q}}{c x^{4} + a} \,d x } \] Input:

integrate((e*x^2+d)^q*(C*x^4+B*x^2+A)/(c*x^4+a),x, algorithm="giac")
 

Output:

integrate((C*x^4 + B*x^2 + A)*(e*x^2 + d)^q/(c*x^4 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int \frac {{\left (e\,x^2+d\right )}^q\,\left (C\,x^4+B\,x^2+A\right )}{c\,x^4+a} \,d x \] Input:

int(((d + e*x^2)^q*(A + B*x^2 + C*x^4))/(a + c*x^4),x)
 

Output:

int(((d + e*x^2)^q*(A + B*x^2 + C*x^4))/(a + c*x^4), x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^q \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\frac {\left (e \,x^{2}+d \right )^{q} x +4 \left (\int \frac {\left (e \,x^{2}+d \right )^{q}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) a d \,q^{2}+2 \left (\int \frac {\left (e \,x^{2}+d \right )^{q}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) a d q +4 \left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{4}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) b e \,q^{2}+4 \left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{4}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) b e q +\left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{4}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) b e +4 \left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{4}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) c d \,q^{2}+2 \left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{4}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) c d q +4 \left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{2}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) b d \,q^{2}+4 \left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{2}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) b d q +\left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{2}}{2 c e q \,x^{6}+c e \,x^{6}+2 c d q \,x^{4}+c d \,x^{4}+2 a e q \,x^{2}+a e \,x^{2}+2 a d q +a d}d x \right ) b d}{2 q +1} \] Input:

int((e*x^2+d)^q*(C*x^4+B*x^2+A)/(c*x^4+a),x)
 

Output:

((d + e*x**2)**q*x + 4*int((d + e*x**2)**q/(2*a*d*q + a*d + 2*a*e*q*x**2 + 
 a*e*x**2 + 2*c*d*q*x**4 + c*d*x**4 + 2*c*e*q*x**6 + c*e*x**6),x)*a*d*q**2 
 + 2*int((d + e*x**2)**q/(2*a*d*q + a*d + 2*a*e*q*x**2 + a*e*x**2 + 2*c*d* 
q*x**4 + c*d*x**4 + 2*c*e*q*x**6 + c*e*x**6),x)*a*d*q + 4*int(((d + e*x**2 
)**q*x**4)/(2*a*d*q + a*d + 2*a*e*q*x**2 + a*e*x**2 + 2*c*d*q*x**4 + c*d*x 
**4 + 2*c*e*q*x**6 + c*e*x**6),x)*b*e*q**2 + 4*int(((d + e*x**2)**q*x**4)/ 
(2*a*d*q + a*d + 2*a*e*q*x**2 + a*e*x**2 + 2*c*d*q*x**4 + c*d*x**4 + 2*c*e 
*q*x**6 + c*e*x**6),x)*b*e*q + int(((d + e*x**2)**q*x**4)/(2*a*d*q + a*d + 
 2*a*e*q*x**2 + a*e*x**2 + 2*c*d*q*x**4 + c*d*x**4 + 2*c*e*q*x**6 + c*e*x* 
*6),x)*b*e + 4*int(((d + e*x**2)**q*x**4)/(2*a*d*q + a*d + 2*a*e*q*x**2 + 
a*e*x**2 + 2*c*d*q*x**4 + c*d*x**4 + 2*c*e*q*x**6 + c*e*x**6),x)*c*d*q**2 
+ 2*int(((d + e*x**2)**q*x**4)/(2*a*d*q + a*d + 2*a*e*q*x**2 + a*e*x**2 + 
2*c*d*q*x**4 + c*d*x**4 + 2*c*e*q*x**6 + c*e*x**6),x)*c*d*q + 4*int(((d + 
e*x**2)**q*x**2)/(2*a*d*q + a*d + 2*a*e*q*x**2 + a*e*x**2 + 2*c*d*q*x**4 + 
 c*d*x**4 + 2*c*e*q*x**6 + c*e*x**6),x)*b*d*q**2 + 4*int(((d + e*x**2)**q* 
x**2)/(2*a*d*q + a*d + 2*a*e*q*x**2 + a*e*x**2 + 2*c*d*q*x**4 + c*d*x**4 + 
 2*c*e*q*x**6 + c*e*x**6),x)*b*d*q + int(((d + e*x**2)**q*x**2)/(2*a*d*q + 
 a*d + 2*a*e*q*x**2 + a*e*x**2 + 2*c*d*q*x**4 + c*d*x**4 + 2*c*e*q*x**6 + 
c*e*x**6),x)*b*d)/(2*q + 1)