\(\int (d+e x^2)^q (a+c x^4)^2 \, dx\) [474]

Optimal result

Integrand size = 19, antiderivative size = 321 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right )^2 \, dx=-\frac {3 c d \left (35 c d^2+2 a e^2 \left (63+32 q+4 q^2\right )\right ) x \left (d+e x^2\right )^{1+q}}{e^4 (3+2 q) (5+2 q) (7+2 q) (9+2 q)}+\frac {c \left (35 c d^2+2 a e^2 \left (63+32 q+4 q^2\right )\right ) x^3 \left (d+e x^2\right )^{1+q}}{e^3 (5+2 q) (7+2 q) (9+2 q)}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{1+q}}{e^2 (7+2 q) (9+2 q)}+\frac {c^2 x^7 \left (d+e x^2\right )^{1+q}}{e (9+2 q)}+\frac {\left (a^2 e^4 \left (63+32 q+4 q^2\right )+\frac {3 c d^2 \left (35 c d^2+2 a e^2 \left (63+32 q+4 q^2\right )\right )}{(3+2 q) (5+2 q)}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{e^4 (7+2 q) (9+2 q)} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.33 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right )^2 \, dx=\frac {1}{45} x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \left (45 a^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )+c x^4 \left (18 a \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-q,\frac {7}{2},-\frac {e x^2}{d}\right )+5 c x^4 \operatorname {Hypergeometric2F1}\left (\frac {9}{2},-q,\frac {11}{2},-\frac {e x^2}{d}\right )\right )\right ) \] Input:

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

Output:

(x*(d + e*x^2)^q*(45*a^2*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)] + c 
*x^4*(18*a*Hypergeometric2F1[5/2, -q, 7/2, -((e*x^2)/d)] + 5*c*x^4*Hyperge 
ometric2F1[9/2, -q, 11/2, -((e*x^2)/d)])))/(45*(1 + (e*x^2)/d)^q)
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1474, 2346, 1474, 299, 238, 237}

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.

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

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\int \left (e x^2+d\right )^q \left (-7 c^2 d x^6+2 a c e (2 q+9) x^4+a^2 e (2 q+9)\right )dx}{e (2 q+9)}+\frac {c^2 x^7 \left (d+e x^2\right )^{q+1}}{e (2 q+9)}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\int \left (e x^2+d\right )^q \left (c \left (35 c d^2+2 a e^2 \left (4 q^2+32 q+63\right )\right ) x^4+a^2 e^2 \left (4 q^2+32 q+63\right )\right )dx}{e (2 q+7)}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{q+1}}{e (2 q+7)}}{e (2 q+9)}+\frac {c^2 x^7 \left (d+e x^2\right )^{q+1}}{e (2 q+9)}\)

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\frac {\frac {\int \left (e x^2+d\right )^q \left (a^2 e^3 \left (8 q^3+84 q^2+286 q+315\right )-3 c d \left (35 c d^2+2 a e^2 \left (4 q^2+32 q+63\right )\right ) x^2\right )dx}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+5)}}{e (2 q+7)}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{q+1}}{e (2 q+7)}}{e (2 q+9)}+\frac {c^2 x^7 \left (d+e x^2\right )^{q+1}}{e (2 q+9)}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\frac {\frac {\left (a^2 e^4 \left (8 q^3+84 q^2+286 q+315\right )+\frac {3 c d^2 \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{2 q+3}\right ) \int \left (e x^2+d\right )^qdx}{e}-\frac {3 c d x \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+3)}}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+5)}}{e (2 q+7)}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{q+1}}{e (2 q+7)}}{e (2 q+9)}+\frac {c^2 x^7 \left (d+e x^2\right )^{q+1}}{e (2 q+9)}\)

\(\Big \downarrow \) 238

\(\displaystyle \frac {\frac {\frac {\frac {\left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \left (a^2 e^4 \left (8 q^3+84 q^2+286 q+315\right )+\frac {3 c d^2 \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{2 q+3}\right ) \int \left (\frac {e x^2}{d}+1\right )^qdx}{e}-\frac {3 c d x \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+3)}}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+5)}}{e (2 q+7)}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{q+1}}{e (2 q+7)}}{e (2 q+9)}+\frac {c^2 x^7 \left (d+e x^2\right )^{q+1}}{e (2 q+9)}\)

\(\Big \downarrow \) 237

\(\displaystyle \frac {\frac {\frac {\frac {x \left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \left (a^2 e^4 \left (8 q^3+84 q^2+286 q+315\right )+\frac {3 c d^2 \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{2 q+3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{e}-\frac {3 c d x \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+3)}}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1} \left (2 a e^2 \left (4 q^2+32 q+63\right )+35 c d^2\right )}{e (2 q+5)}}{e (2 q+7)}-\frac {7 c^2 d x^5 \left (d+e x^2\right )^{q+1}}{e (2 q+7)}}{e (2 q+9)}+\frac {c^2 x^7 \left (d+e x^2\right )^{q+1}}{e (2 q+9)}\)

Input:

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

Output:

(c^2*x^7*(d + e*x^2)^(1 + q))/(e*(9 + 2*q)) + ((-7*c^2*d*x^5*(d + e*x^2)^( 
1 + q))/(e*(7 + 2*q)) + ((c*(35*c*d^2 + 2*a*e^2*(63 + 32*q + 4*q^2))*x^3*( 
d + e*x^2)^(1 + q))/(e*(5 + 2*q)) + ((-3*c*d*(35*c*d^2 + 2*a*e^2*(63 + 32* 
q + 4*q^2))*x*(d + e*x^2)^(1 + q))/(e*(3 + 2*q)) + ((a^2*e^4*(315 + 286*q 
+ 84*q^2 + 8*q^3) + (3*c*d^2*(35*c*d^2 + 2*a*e^2*(63 + 32*q + 4*q^2)))/(3 
+ 2*q))*x*(d + e*x^2)^q*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)])/(e* 
(1 + (e*x^2)/d)^q))/(e*(5 + 2*q)))/(e*(7 + 2*q)))/(e*(9 + 2*q))
 

Defintions of rubi rules used

rule 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !GtQ[a, 0]
 

rule 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 1474

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Si 
mp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))), x] + Simp[1/( 
e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 2*q + 1)*(a + c 
*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 2*q + 1)*x^(4*p), x], 
x], x] /; FreeQ[{a, c, d, e, q}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0] 
&&  !LtQ[q, -1]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 24.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.27 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right )^2 \, dx=a^{2} d^{q} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - q \\ \frac {3}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )} + \frac {2 a c d^{q} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - q \\ \frac {7}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{5} + \frac {c^{2} d^{q} x^{9} {{}_{2}F_{1}\left (\begin {matrix} \frac {9}{2}, - q \\ \frac {11}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{9} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((e*x^2+d)^q*(c*x^4+a)^2,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 - 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 + 32*(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 - 210*(d + e*x**2)**q*c**2*d**4*q*x + 140*(d + e*x**2)**q*c**2* 
d**3*e*q**2*x**3 + 70*(d + e*x**2)**q*c**2*d**3*e*q*x**3 - 56*(d + e*x**2) 
**q*c**2*d**2*e**2*q**3*x**5 - 112*(d + e*x**2)**q*c**2*d**2*e**2*q**2*x** 
5 - 42*(d + e*x**2)**q*c**2*d**2*e**2*q*x**5 + 16*(d + e*x**2)**q*c**2*d*e 
**3*q**4*x**7 + 72*(d + e*x**2)**q*c**2*d*e**3*q**3*x**7 + 92*(d + e*x**2) 
**q*c**2*d*e**3*q**2*x**7 + 30*(d + e*x**2)**q*c**2*d*e**3*q*x**7 + 16*(d 
+ e*x**2)**q*c**2*e**4*q**4*x**9 + 128*(d + e*x**2)**q*c**2*e**4*q**3*x**9 
 + 344*(d + e*x**2)**q*c**2*e**4*q**2*x**9 + 352*(d + e*x**2)**q*c**2*e**4 
*q*x**9 + 105*(d + e*x**2)**q*c**2*e**4*x**9 + 1024*int((d + e*x**2)**q/(3 
2*d*q**5 + 400*d*q**4 + 1840*d*q**3 + 3800*d*q**2 + 3378*d*q + 945*d + 32* 
e*q**5*x**2 + 400*e*q**4*x**2 + 1840*e*q**3*x**2 + 3800*e*q**2*x**2 + 3...