\(\int (d+e x^2)^q (a+c x^4) (A+B x^2+C x^4) \, dx\) [127]

Optimal result

Integrand size = 29, antiderivative size = 425 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right ) \left (A+B x^2+C x^4\right ) \, dx=-\frac {\left (a e^2 \left (63+32 q+4 q^2\right ) (3 C d-B e (5+2 q))+3 c d \left (35 C d^2-e (9+2 q) (5 B d-A e (7+2 q))\right )\right ) x \left (d+e x^2\right )^{1+q}}{e^4 (3+2 q) (5+2 q) (7+2 q) (9+2 q)}+\frac {\left (a C e^2 \left (63+32 q+4 q^2\right )+c \left (35 C d^2-e (9+2 q) (5 B d-A e (7+2 q))\right )\right ) x^3 \left (d+e x^2\right )^{1+q}}{e^3 (5+2 q) (7+2 q) (9+2 q)}-\frac {c (7 C d-B e (9+2 q)) x^5 \left (d+e x^2\right )^{1+q}}{e^2 (7+2 q) (9+2 q)}+\frac {c C x^7 \left (d+e x^2\right )^{1+q}}{e (9+2 q)}+\frac {\left (a e^2 \left (63+32 q+4 q^2\right ) \left (3 C d^2-e (5+2 q) (B d-A e (3+2 q))\right )+3 c d^2 \left (35 C d^2-e (9+2 q) (5 B d-A e (7+2 q))\right )\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{e^4 (3+2 q) (5+2 q) (7+2 q) (9+2 q)} \] Output:

-(a*e^2*(4*q^2+32*q+63)*(3*C*d-B*e*(5+2*q))+3*c*d*(35*C*d^2-e*(9+2*q)*(5*B 
*d-A*e*(7+2*q))))*x*(e*x^2+d)^(1+q)/e^4/(3+2*q)/(5+2*q)/(7+2*q)/(9+2*q)+(a 
*C*e^2*(4*q^2+32*q+63)+c*(35*C*d^2-e*(9+2*q)*(5*B*d-A*e*(7+2*q))))*x^3*(e* 
x^2+d)^(1+q)/e^3/(5+2*q)/(7+2*q)/(9+2*q)-c*(7*C*d-B*e*(9+2*q))*x^5*(e*x^2+ 
d)^(1+q)/e^2/(7+2*q)/(9+2*q)+c*C*x^7*(e*x^2+d)^(1+q)/e/(9+2*q)+(a*e^2*(4*q 
^2+32*q+63)*(3*C*d^2-e*(5+2*q)*(B*d-A*e*(3+2*q)))+3*c*d^2*(35*C*d^2-e*(9+2 
*q)*(5*B*d-A*e*(7+2*q))))*x*(e*x^2+d)^q*hypergeom([1/2, -q],[3/2],-e*x^2/d 
)/e^4/(3+2*q)/(5+2*q)/(7+2*q)/(9+2*q)/((1+e*x^2/d)^q)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.38 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right ) \left (A+B x^2+C x^4\right ) \, dx=\frac {1}{315} x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \left (315 a A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )+105 a B x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-q,\frac {5}{2},-\frac {e x^2}{d}\right )+63 (A c+a C) x^4 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-q,\frac {7}{2},-\frac {e x^2}{d}\right )+5 c x^6 \left (9 B \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-q,\frac {9}{2},-\frac {e x^2}{d}\right )+7 C x^2 \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-q,\frac {11}{2},-\frac {e x^2}{d}\right )\right )\right ) \] Input:

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

Output:

(x*(d + e*x^2)^q*(315*a*A*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)] + 
105*a*B*x^2*Hypergeometric2F1[3/2, -q, 5/2, -((e*x^2)/d)] + 63*(A*c + a*C) 
*x^4*Hypergeometric2F1[5/2, -q, 7/2, -((e*x^2)/d)] + 5*c*x^6*(9*B*Hypergeo 
metric2F1[7/2, -q, 9/2, -((e*x^2)/d)] + 7*C*x^2*Hypergeometric2F1[9/2, -q, 
 11/2, -((e*x^2)/d)])))/(315*(1 + (e*x^2)/d)^q)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2256, 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.

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

\(\Big \downarrow \) 2256

\(\displaystyle \int \left (x^4 (a C+A c) \left (d+e x^2\right )^q+a A \left (d+e x^2\right )^q+a B x^2 \left (d+e x^2\right )^q+B c x^6 \left (d+e x^2\right )^q+c C x^8 \left (d+e x^2\right )^q\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} x^5 (a C+A c) \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-q,\frac {7}{2},-\frac {e x^2}{d}\right )+a A x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )+\frac {1}{3} a B x^3 \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-q,\frac {5}{2},-\frac {e x^2}{d}\right )+\frac {1}{7} B c x^7 \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-q,\frac {9}{2},-\frac {e x^2}{d}\right )+\frac {1}{9} c C x^9 \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-q,\frac {11}{2},-\frac {e x^2}{d}\right )\)

Input:

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

Output:

(a*A*x*(d + e*x^2)^q*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)])/(1 + ( 
e*x^2)/d)^q + (a*B*x^3*(d + e*x^2)^q*Hypergeometric2F1[3/2, -q, 5/2, -((e* 
x^2)/d)])/(3*(1 + (e*x^2)/d)^q) + ((A*c + a*C)*x^5*(d + e*x^2)^q*Hypergeom 
etric2F1[5/2, -q, 7/2, -((e*x^2)/d)])/(5*(1 + (e*x^2)/d)^q) + (B*c*x^7*(d 
+ e*x^2)^q*Hypergeometric2F1[7/2, -q, 9/2, -((e*x^2)/d)])/(7*(1 + (e*x^2)/ 
d)^q) + (c*C*x^9*(d + e*x^2)^q*Hypergeometric2F1[9/2, -q, 11/2, -((e*x^2)/ 
d)])/(9*(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 2256

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((C*c*x^8 + B*c*x^6 + (C*a + A*c)*x^4 + B*a*x^2 + A*a)*(e*x^2 + d) 
^q, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 50.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.42 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right ) \left (A+B x^2+C x^4\right ) \, dx=A a d^{q} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - q \\ \frac {3}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )} + \frac {A c d^{q} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - q \\ \frac {7}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{5} + \frac {B a d^{q} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - q \\ \frac {5}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{3} + \frac {B c d^{q} x^{7} {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, - q \\ \frac {9}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{7} + \frac {C a d^{q} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - q \\ \frac {7}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{5} + \frac {C c d^{q} x^{9} {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{2}, - q \\ \frac {11}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{9} \] Input:

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

Output:

A*a*d**q*x*hyper((1/2, -q), (3/2,), e*x**2*exp_polar(I*pi)/d) + A*c*d**q*x 
**5*hyper((5/2, -q), (7/2,), e*x**2*exp_polar(I*pi)/d)/5 + B*a*d**q*x**3*h 
yper((3/2, -q), (5/2,), e*x**2*exp_polar(I*pi)/d)/3 + B*c*d**q*x**7*hyper( 
(7/2, -q), (9/2,), e*x**2*exp_polar(I*pi)/d)/7 + C*a*d**q*x**5*hyper((5/2, 
 -q), (7/2,), e*x**2*exp_polar(I*pi)/d)/5 + C*c*d**q*x**9*hyper((9/2, -q), 
 (11/2,), e*x**2*exp_polar(I*pi)/d)/9
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(16*(d + e*x**2)**q*a**2*e**4*q**4*x + 192*(d + e*x**2)**q*a**2*e**4*q**3* 
x + 824*(d + e*x**2)**q*a**2*e**4*q**2*x + 1488*(d + e*x**2)**q*a**2*e**4* 
q*x + 945*(d + e*x**2)**q*a**2*e**4*x + 16*(d + e*x**2)**q*a*b*d*e**3*q**4 
*x + 168*(d + e*x**2)**q*a*b*d*e**3*q**3*x + 572*(d + e*x**2)**q*a*b*d*e** 
3*q**2*x + 630*(d + e*x**2)**q*a*b*d*e**3*q*x + 16*(d + e*x**2)**q*a*b*e** 
4*q**4*x**3 + 176*(d + e*x**2)**q*a*b*e**4*q**3*x**3 + 656*(d + e*x**2)**q 
*a*b*e**4*q**2*x**3 + 916*(d + e*x**2)**q*a*b*e**4*q*x**3 + 315*(d + e*x** 
2)**q*a*b*e**4*x**3 - 48*(d + e*x**2)**q*a*c*d**2*e**2*q**3*x - 384*(d + e 
*x**2)**q*a*c*d**2*e**2*q**2*x - 756*(d + e*x**2)**q*a*c*d**2*e**2*q*x + 3 
2*(d + e*x**2)**q*a*c*d*e**3*q**4*x**3 + 272*(d + e*x**2)**q*a*c*d*e**3*q* 
*3*x**3 + 632*(d + e*x**2)**q*a*c*d*e**3*q**2*x**3 + 252*(d + e*x**2)**q*a 
*c*d*e**3*q*x**3 + 32*(d + e*x**2)**q*a*c*e**4*q**4*x**5 + 320*(d + e*x**2 
)**q*a*c*e**4*q**3*x**5 + 1040*(d + e*x**2)**q*a*c*e**4*q**2*x**5 + 1200*( 
d + e*x**2)**q*a*c*e**4*q*x**5 + 378*(d + e*x**2)**q*a*c*e**4*x**5 + 60*(d 
 + e*x**2)**q*b*c*d**3*e*q**2*x + 270*(d + e*x**2)**q*b*c*d**3*e*q*x - 40* 
(d + e*x**2)**q*b*c*d**2*e**2*q**3*x**3 - 200*(d + e*x**2)**q*b*c*d**2*e** 
2*q**2*x**3 - 90*(d + e*x**2)**q*b*c*d**2*e**2*q*x**3 + 16*(d + e*x**2)**q 
*b*c*d*e**3*q**4*x**5 + 104*(d + e*x**2)**q*b*c*d*e**3*q**3*x**5 + 156*(d 
+ e*x**2)**q*b*c*d*e**3*q**2*x**5 + 54*(d + e*x**2)**q*b*c*d*e**3*q*x**5 + 
 16*(d + e*x**2)**q*b*c*e**4*q**4*x**7 + 144*(d + e*x**2)**q*b*c*e**4*q...