\(\int (d+e x^2)^q (a+c x^4) \, dx\) [475]

Optimal result

Integrand size = 17, antiderivative size = 143 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right ) \, dx=-\frac {3 c d x \left (d+e x^2\right )^{1+q}}{e^2 (3+2 q) (5+2 q)}+\frac {c x^3 \left (d+e x^2\right )^{1+q}}{e (5+2 q)}+\frac {\left (3 c d^2+a e^2 \left (15+16 q+4 q^2\right )\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{e^2 (3+2 q) (5+2 q)} \] Output:

-3*c*d*x*(e*x^2+d)^(1+q)/e^2/(3+2*q)/(5+2*q)+c*x^3*(e*x^2+d)^(1+q)/e/(5+2* 
q)+(3*c*d^2+a*e^2*(4*q^2+16*q+15))*x*(e*x^2+d)^q*hypergeom([1/2, -q],[3/2] 
,-e*x^2/d)/e^2/(3+2*q)/(5+2*q)/((1+e*x^2/d)^q)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.52 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right ) \, dx=\frac {1}{5} x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \left (5 a \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )+c x^4 \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-q,\frac {7}{2},-\frac {e x^2}{d}\right )\right ) \] Input:

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

Output:

(x*(d + e*x^2)^q*(5*a*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)] + c*x^ 
4*Hypergeometric2F1[5/2, -q, 7/2, -((e*x^2)/d)]))/(5*(1 + (e*x^2)/d)^q)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {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.

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

\(\Big \downarrow \) 1474

\(\displaystyle \frac {\int \left (a e (2 q+5)-3 c d x^2\right ) \left (e x^2+d\right )^qdx}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1}}{e (2 q+5)}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\left (a e (2 q+5)+\frac {3 c d^2}{2 e q+3 e}\right ) \int \left (e x^2+d\right )^qdx-\frac {3 c d x \left (d+e x^2\right )^{q+1}}{e (2 q+3)}}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1}}{e (2 q+5)}\)

\(\Big \downarrow \) 238

\(\displaystyle \frac {\left (d+e x^2\right )^q \left (\frac {e x^2}{d}+1\right )^{-q} \left (a e (2 q+5)+\frac {3 c d^2}{2 e q+3 e}\right ) \int \left (\frac {e x^2}{d}+1\right )^qdx-\frac {3 c d x \left (d+e x^2\right )^{q+1}}{e (2 q+3)}}{e (2 q+5)}+\frac {c x^3 \left (d+e x^2\right )^{q+1}}{e (2 q+5)}\)

\(\Big \downarrow \) 237

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

Input:

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

Output:

(c*x^3*(d + e*x^2)^(1 + q))/(e*(5 + 2*q)) + ((-3*c*d*x*(d + e*x^2)^(1 + q) 
)/(e*(3 + 2*q)) + ((a*e*(5 + 2*q) + (3*c*d^2)/(3*e + 2*e*q))*x*(d + e*x^2) 
^q*Hypergeometric2F1[1/2, -q, 3/2, -((e*x^2)/d)])/(1 + (e*x^2)/d)^q)/(e*(5 
 + 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]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 6.85 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.37 \[ \int \left (d+e x^2\right )^q \left (a+c x^4\right ) \, dx=a d^{q} x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - q \\ \frac {3}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )} + \frac {c d^{q} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{2}, - q \\ \frac {7}{2} \end {matrix}\middle | {\frac {e x^{2} e^{i \pi }}{d}} \right )}}{5} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(4*(d + e*x**2)**q*a*e**2*q**2*x + 16*(d + e*x**2)**q*a*e**2*q*x + 15*(d + 
 e*x**2)**q*a*e**2*x - 6*(d + e*x**2)**q*c*d**2*q*x + 4*(d + e*x**2)**q*c* 
d*e*q**2*x**3 + 2*(d + e*x**2)**q*c*d*e*q*x**3 + 4*(d + e*x**2)**q*c*e**2* 
q**2*x**5 + 8*(d + e*x**2)**q*c*e**2*q*x**5 + 3*(d + e*x**2)**q*c*e**2*x** 
5 + 64*int((d + e*x**2)**q/(8*d*q**3 + 36*d*q**2 + 46*d*q + 15*d + 8*e*q** 
3*x**2 + 36*e*q**2*x**2 + 46*e*q*x**2 + 15*e*x**2),x)*a*d*e**2*q**6 + 544* 
int((d + e*x**2)**q/(8*d*q**3 + 36*d*q**2 + 46*d*q + 15*d + 8*e*q**3*x**2 
+ 36*e*q**2*x**2 + 46*e*q*x**2 + 15*e*x**2),x)*a*d*e**2*q**5 + 1760*int((d 
 + e*x**2)**q/(8*d*q**3 + 36*d*q**2 + 46*d*q + 15*d + 8*e*q**3*x**2 + 36*e 
*q**2*x**2 + 46*e*q*x**2 + 15*e*x**2),x)*a*d*e**2*q**4 + 2672*int((d + e*x 
**2)**q/(8*d*q**3 + 36*d*q**2 + 46*d*q + 15*d + 8*e*q**3*x**2 + 36*e*q**2* 
x**2 + 46*e*q*x**2 + 15*e*x**2),x)*a*d*e**2*q**3 + 1860*int((d + e*x**2)** 
q/(8*d*q**3 + 36*d*q**2 + 46*d*q + 15*d + 8*e*q**3*x**2 + 36*e*q**2*x**2 + 
 46*e*q*x**2 + 15*e*x**2),x)*a*d*e**2*q**2 + 450*int((d + e*x**2)**q/(8*d* 
q**3 + 36*d*q**2 + 46*d*q + 15*d + 8*e*q**3*x**2 + 36*e*q**2*x**2 + 46*e*q 
*x**2 + 15*e*x**2),x)*a*d*e**2*q + 48*int((d + e*x**2)**q/(8*d*q**3 + 36*d 
*q**2 + 46*d*q + 15*d + 8*e*q**3*x**2 + 36*e*q**2*x**2 + 46*e*q*x**2 + 15* 
e*x**2),x)*c*d**3*q**4 + 216*int((d + e*x**2)**q/(8*d*q**3 + 36*d*q**2 + 4 
6*d*q + 15*d + 8*e*q**3*x**2 + 36*e*q**2*x**2 + 46*e*q*x**2 + 15*e*x**2),x 
)*c*d**3*q**3 + 276*int((d + e*x**2)**q/(8*d*q**3 + 36*d*q**2 + 46*d*q ...