\(\int (c+d x^4) (a+b x^8)^p \, dx\) [9]

Optimal result

Integrand size = 17, antiderivative size = 96 \[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=c x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+\frac {1}{5} d x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right ) \] Output:

c*x*(b*x^8+a)^p*hypergeom([1/8, -p],[9/8],-b*x^8/a)/((1+b*x^8/a)^p)+1/5*d* 
x^5*(b*x^8+a)^p*hypergeom([5/8, -p],[13/8],-b*x^8/a)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{5} x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (5 c \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+d x^4 \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )\right ) \] Input:

Integrate[(c + d*x^4)*(a + b*x^8)^p,x]
 

Output:

(x*(a + b*x^8)^p*(5*c*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)] + d*x^ 
4*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8)/a)]))/(5*(1 + (b*x^8)/a)^p)
 

Rubi [A] (verified)

Time = 0.23 (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 = {1763, 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.

\(\displaystyle \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx\)

\(\Big \downarrow \) 1763

\(\displaystyle \int \left (c \left (a+b x^8\right )^p+d x^4 \left (a+b x^8\right )^p\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle c x \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+\frac {1}{5} d x^5 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{8},-p,\frac {13}{8},-\frac {b x^8}{a}\right )\)

Input:

Int[(c + d*x^4)*(a + b*x^8)^p,x]
 

Output:

(c*x*(a + b*x^8)^p*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)])/(1 + (b* 
x^8)/a)^p + (d*x^5*(a + b*x^8)^p*Hypergeometric2F1[5/8, -p, 13/8, -((b*x^8 
)/a)])/(5*(1 + (b*x^8)/a)^p)
 

Defintions of rubi rules used

rule 1763

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> I 
nt[ExpandIntegrand[(d + e*x^n)*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a, c, d, 
 e, n}, x] && EqQ[n2, 2*n]
 

rule 2009

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

Maple [F]

\[\int \left (x^{4} d +c \right ) \left (b \,x^{8}+a \right )^{p}d x\]

Input:

int((d*x^4+c)*(b*x^8+a)^p,x)
 

Output:

int((d*x^4+c)*(b*x^8+a)^p,x)
 

Fricas [F]

\[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((d*x^4+c)*(b*x^8+a)^p,x, algorithm="fricas")
 

Output:

integral((d*x^4 + c)*(b*x^8 + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 144.46 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.78 \[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {a^{p} c x \Gamma \left (\frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{8}, - p \\ \frac {9}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {9}{8}\right )} + \frac {a^{p} d x^{5} \Gamma \left (\frac {5}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{8}, - p \\ \frac {13}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {13}{8}\right )} \] Input:

integrate((d*x**4+c)*(b*x**8+a)**p,x)
 

Output:

a**p*c*x*gamma(1/8)*hyper((1/8, -p), (9/8,), b*x**8*exp_polar(I*pi)/a)/(8* 
gamma(9/8)) + a**p*d*x**5*gamma(5/8)*hyper((5/8, -p), (13/8,), b*x**8*exp_ 
polar(I*pi)/a)/(8*gamma(13/8))
 

Maxima [F]

\[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((d*x^4+c)*(b*x^8+a)^p,x, algorithm="maxima")
 

Output:

integrate((d*x^4 + c)*(b*x^8 + a)^p, x)
 

Giac [F]

\[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((d*x^4+c)*(b*x^8+a)^p,x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)*(b*x^8 + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\int {\left (b\,x^8+a\right )}^p\,\left (d\,x^4+c\right ) \,d x \] Input:

int((a + b*x^8)^p*(c + d*x^4),x)
 

Output:

int((a + b*x^8)^p*(c + d*x^4), x)
 

Reduce [F]

\[ \int \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {8 \left (b \,x^{8}+a \right )^{p} c p x +5 \left (b \,x^{8}+a \right )^{p} c x +8 \left (b \,x^{8}+a \right )^{p} d p \,x^{5}+\left (b \,x^{8}+a \right )^{p} d \,x^{5}+4096 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a c \,p^{4}+5632 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a c \,p^{3}+2240 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a c \,p^{2}+200 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a c p +4096 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{4}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a d \,p^{4}+3584 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{4}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a d \,p^{3}+704 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{4}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a d \,p^{2}+40 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{4}}{64 b \,p^{2} x^{8}+48 b p \,x^{8}+5 b \,x^{8}+64 a \,p^{2}+48 a p +5 a}d x \right ) a d p}{64 p^{2}+48 p +5} \] Input:

int((d*x^4+c)*(b*x^8+a)^p,x)
 

Output:

(8*(a + b*x**8)**p*c*p*x + 5*(a + b*x**8)**p*c*x + 8*(a + b*x**8)**p*d*p*x 
**5 + (a + b*x**8)**p*d*x**5 + 4096*int((a + b*x**8)**p/(64*a*p**2 + 48*a* 
p + 5*a + 64*b*p**2*x**8 + 48*b*p*x**8 + 5*b*x**8),x)*a*c*p**4 + 5632*int( 
(a + b*x**8)**p/(64*a*p**2 + 48*a*p + 5*a + 64*b*p**2*x**8 + 48*b*p*x**8 + 
 5*b*x**8),x)*a*c*p**3 + 2240*int((a + b*x**8)**p/(64*a*p**2 + 48*a*p + 5* 
a + 64*b*p**2*x**8 + 48*b*p*x**8 + 5*b*x**8),x)*a*c*p**2 + 200*int((a + b* 
x**8)**p/(64*a*p**2 + 48*a*p + 5*a + 64*b*p**2*x**8 + 48*b*p*x**8 + 5*b*x* 
*8),x)*a*c*p + 4096*int(((a + b*x**8)**p*x**4)/(64*a*p**2 + 48*a*p + 5*a + 
 64*b*p**2*x**8 + 48*b*p*x**8 + 5*b*x**8),x)*a*d*p**4 + 3584*int(((a + b*x 
**8)**p*x**4)/(64*a*p**2 + 48*a*p + 5*a + 64*b*p**2*x**8 + 48*b*p*x**8 + 5 
*b*x**8),x)*a*d*p**3 + 704*int(((a + b*x**8)**p*x**4)/(64*a*p**2 + 48*a*p 
+ 5*a + 64*b*p**2*x**8 + 48*b*p*x**8 + 5*b*x**8),x)*a*d*p**2 + 40*int(((a 
+ b*x**8)**p*x**4)/(64*a*p**2 + 48*a*p + 5*a + 64*b*p**2*x**8 + 48*b*p*x** 
8 + 5*b*x**8),x)*a*d*p)/(64*p**2 + 48*p + 5)