\(\int (c+d x^2)^2 (a+b x^4)^p \, dx\) [482]

Optimal result

Integrand size = 19, antiderivative size = 142 \[ \int \left (c+d x^2\right )^2 \left (a+b x^4\right )^p \, dx=\frac {d^2 x \left (a+b x^4\right )^{1+p}}{b (5+4 p)}+\left (c^2-\frac {a d^2}{5 b+4 b p}\right ) x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right )+\frac {2}{3} c d x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^4}{a}\right ) \] Output:

d^2*x*(b*x^4+a)^(p+1)/b/(5+4*p)+(c^2-a*d^2/(4*b*p+5*b))*x*(b*x^4+a)^p*hype 
rgeom([1/4, -p],[5/4],-b*x^4/a)/((1+b*x^4/a)^p)+2/3*c*d*x^3*(b*x^4+a)^p*hy 
pergeom([3/4, -p],[7/4],-b*x^4/a)/((1+b*x^4/a)^p)
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int \left (c+d x^2\right )^2 \left (a+b x^4\right )^p \, dx=\frac {1}{15} x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (15 c^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right )+d x^2 \left (10 c \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^4}{a}\right )+3 d x^2 \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^4}{a}\right )\right )\right ) \] Input:

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

Output:

(x*(a + b*x^4)^p*(15*c^2*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^4)/a)] + d 
*x^2*(10*c*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^4)/a)] + 3*d*x^2*Hyperge 
ometric2F1[5/4, -p, 9/4, -((b*x^4)/a)])))/(15*(1 + (b*x^4)/a)^p)
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.11, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1519, 25, 1516, 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.

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

\(\Big \downarrow \) 1519

\(\displaystyle \frac {\int -\left (\left (-b (4 p+5) c^2-2 b d (4 p+5) x^2 c+a d^2\right ) \left (b x^4+a\right )^p\right )dx}{b (4 p+5)}+\frac {d^2 x \left (a+b x^4\right )^{p+1}}{b (4 p+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 x \left (a+b x^4\right )^{p+1}}{b (4 p+5)}-\frac {\int \left (-b (4 p+5) c^2-2 b d (4 p+5) x^2 c+a d^2\right ) \left (b x^4+a\right )^pdx}{b (4 p+5)}\)

\(\Big \downarrow \) 1516

\(\displaystyle \frac {d^2 x \left (a+b x^4\right )^{p+1}}{b (4 p+5)}-\frac {\int \left (a d^2 \left (1-\frac {b c^2 (4 p+5)}{a d^2}\right ) \left (b x^4+a\right )^p-2 b c d (4 p+5) x^2 \left (b x^4+a\right )^p\right )dx}{b (4 p+5)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 x \left (a+b x^4\right )^{p+1}}{b (4 p+5)}-\frac {x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (a d^2-b c^2 (4 p+5)\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right )-\frac {2}{3} b c d (4 p+5) x^3 \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^4}{a}\right )}{b (4 p+5)}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 1516

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[Expa 
ndIntegrand[(d + e*x^2)*(a + c*x^4)^p, x], x] /; FreeQ[{a, c, d, e}, x] && 
NeQ[c*d^2 + a*e^2, 0]
 

rule 1519

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Sim 
p[e^q*x^(2*q - 3)*((a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c 
*(4*p + 2*q + 1))   Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e* 
x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x 
], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 32.91 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.84 \[ \int \left (c+d x^2\right )^2 \left (a+b x^4\right )^p \, dx=\frac {a^{p} c^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{p} c d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{p} d^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:

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

Output:

a**p*c**2*x*gamma(1/4)*hyper((1/4, -p), (5/4,), b*x**4*exp_polar(I*pi)/a)/ 
(4*gamma(5/4)) + a**p*c*d*x**3*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**4* 
exp_polar(I*pi)/a)/(2*gamma(7/4)) + a**p*d**2*x**5*gamma(5/4)*hyper((5/4, 
-p), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*gamma(9/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(16*(a + b*x**4)**p*a*d**2*p**2*x + 12*(a + b*x**4)**p*a*d**2*p*x + 16*(a 
+ b*x**4)**p*b*c**2*p**2*x + 32*(a + b*x**4)**p*b*c**2*p*x + 15*(a + b*x** 
4)**p*b*c**2*x + 32*(a + b*x**4)**p*b*c*d*p**2*x**3 + 48*(a + b*x**4)**p*b 
*c*d*p*x**3 + 10*(a + b*x**4)**p*b*c*d*x**3 + 16*(a + b*x**4)**p*b*d**2*p* 
*2*x**5 + 16*(a + b*x**4)**p*b*d**2*p*x**5 + 3*(a + b*x**4)**p*b*d**2*x**5 
 - 1024*int((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b 
*p**3*x**4 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a**2*d**2*p**5 
- 3072*int((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b* 
p**3*x**4 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a**2*d**2*p**4 - 
 3200*int((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p 
**3*x**4 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a**2*d**2*p**3 - 
1344*int((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p* 
*3*x**4 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a**2*d**2*p**2 - 1 
80*int((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3 
*x**4 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a**2*d**2*p + 4096*i 
nt((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x** 
4 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a*b*c**2*p**6 + 17408*in 
t((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x**4 
 + 144*b*p**2*x**4 + 92*b*p*x**4 + 15*b*x**4),x)*a*b*c**2*p**5 + 28160*int 
((a + b*x**4)**p/(64*a*p**3 + 144*a*p**2 + 92*a*p + 15*a + 64*b*p**3*x*...