\(\int (c+d x^2) (a+b x^4)^p \, dx\) [483]

Optimal result

Integrand size = 17, antiderivative size = 96 \[ \int \left (c+d x^2\right ) \left (a+b x^4\right )^p \, dx=c x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right )+\frac {1}{3} d x^3 \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^4}{a}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \left (c+d x^2\right ) \left (a+b x^4\right )^p \, dx=\frac {1}{3} x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (3 c \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^4}{a}\right )+d x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^4}{a}\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {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.

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

\(\Big \downarrow \) 1516

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

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 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 16.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \left (c+d x^2\right ) \left (a+b x^4\right )^p \, dx=\frac {a^{p} c x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{p} d x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (c+d x^2\right ) \left (a+b x^4\right )^p \, dx=\frac {4 \left (b \,x^{4}+a \right )^{p} c p x +3 \left (b \,x^{4}+a \right )^{p} c x +4 \left (b \,x^{4}+a \right )^{p} d p \,x^{3}+\left (b \,x^{4}+a \right )^{p} d \,x^{3}+256 \left (\int \frac {\left (b \,x^{4}+a \right )^{p}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c \,p^{4}+448 \left (\int \frac {\left (b \,x^{4}+a \right )^{p}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c \,p^{3}+240 \left (\int \frac {\left (b \,x^{4}+a \right )^{p}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c \,p^{2}+36 \left (\int \frac {\left (b \,x^{4}+a \right )^{p}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c p +256 \left (\int \frac {\left (b \,x^{4}+a \right )^{p} x^{2}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d \,p^{4}+320 \left (\int \frac {\left (b \,x^{4}+a \right )^{p} x^{2}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d \,p^{3}+112 \left (\int \frac {\left (b \,x^{4}+a \right )^{p} x^{2}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d \,p^{2}+12 \left (\int \frac {\left (b \,x^{4}+a \right )^{p} x^{2}}{16 b \,p^{2} x^{4}+16 b p \,x^{4}+3 b \,x^{4}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d p}{16 p^{2}+16 p +3} \] Input:

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

Output:

(4*(a + b*x**4)**p*c*p*x + 3*(a + b*x**4)**p*c*x + 4*(a + b*x**4)**p*d*p*x 
**3 + (a + b*x**4)**p*d*x**3 + 256*int((a + b*x**4)**p/(16*a*p**2 + 16*a*p 
 + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4),x)*a*c*p**4 + 448*int((a 
 + b*x**4)**p/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3 
*b*x**4),x)*a*c*p**3 + 240*int((a + b*x**4)**p/(16*a*p**2 + 16*a*p + 3*a + 
 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4),x)*a*c*p**2 + 36*int((a + b*x**4 
)**p/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4), 
x)*a*c*p + 256*int(((a + b*x**4)**p*x**2)/(16*a*p**2 + 16*a*p + 3*a + 16*b 
*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4),x)*a*d*p**4 + 320*int(((a + b*x**4)** 
p*x**2)/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x** 
4),x)*a*d*p**3 + 112*int(((a + b*x**4)**p*x**2)/(16*a*p**2 + 16*a*p + 3*a 
+ 16*b*p**2*x**4 + 16*b*p*x**4 + 3*b*x**4),x)*a*d*p**2 + 12*int(((a + b*x* 
*4)**p*x**2)/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**4 + 16*b*p*x**4 + 3* 
b*x**4),x)*a*d*p)/(16*p**2 + 16*p + 3)