\(\int (a+b x^8)^p (c+d x^8) \, dx\) [297]

Optimal result

Integrand size = 17, antiderivative size = 93 \[ \int \left (a+b x^8\right )^p \left (c+d x^8\right ) \, dx=\frac {d x \left (a+b x^8\right )^{1+p}}{b (9+8 p)}-\frac {(a d-b c (9+8 p)) x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )}{b (9+8 p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97 \[ \int \left (a+b x^8\right )^p \left (c+d x^8\right ) \, dx=\frac {x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (d \left (a+b x^8\right ) \left (1+\frac {b x^8}{a}\right )^p+(-a d+b c (9+8 p)) \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )\right )}{b (9+8 p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {913, 779, 778}

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.

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

\(\Big \downarrow \) 913

\(\displaystyle \left (c-\frac {a d}{8 b p+9 b}\right ) \int \left (b x^8+a\right )^pdx+\frac {d x \left (a+b x^8\right )^{p+1}}{b (8 p+9)}\)

\(\Big \downarrow \) 779

\(\displaystyle \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \left (c-\frac {a d}{8 b p+9 b}\right ) \int \left (\frac {b x^8}{a}+1\right )^pdx+\frac {d x \left (a+b x^8\right )^{p+1}}{b (8 p+9)}\)

\(\Big \downarrow \) 778

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

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

rule 913

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b x^8\right )^p \left (c+d x^8\right ) \, dx=\frac {8 \left (b \,x^{8}+a \right )^{p} a d p x +8 \left (b \,x^{8}+a \right )^{p} b c p x +9 \left (b \,x^{8}+a \right )^{p} b c x +8 \left (b \,x^{8}+a \right )^{p} b d p \,x^{9}+\left (b \,x^{8}+a \right )^{p} b d \,x^{9}-512 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a^{2} d \,p^{3}-640 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a^{2} d \,p^{2}-72 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a^{2} d p +4096 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a b c \,p^{4}+9728 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a b c \,p^{3}+6336 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a b c \,p^{2}+648 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}+80 b p \,x^{8}+9 b \,x^{8}+64 a \,p^{2}+80 a p +9 a}d x \right ) a b c p}{b \left (64 p^{2}+80 p +9\right )} \] Input:

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

Output:

(8*(a + b*x**8)**p*a*d*p*x + 8*(a + b*x**8)**p*b*c*p*x + 9*(a + b*x**8)**p 
*b*c*x + 8*(a + b*x**8)**p*b*d*p*x**9 + (a + b*x**8)**p*b*d*x**9 - 512*int 
((a + b*x**8)**p/(64*a*p**2 + 80*a*p + 9*a + 64*b*p**2*x**8 + 80*b*p*x**8 
+ 9*b*x**8),x)*a**2*d*p**3 - 640*int((a + b*x**8)**p/(64*a*p**2 + 80*a*p + 
 9*a + 64*b*p**2*x**8 + 80*b*p*x**8 + 9*b*x**8),x)*a**2*d*p**2 - 72*int((a 
 + b*x**8)**p/(64*a*p**2 + 80*a*p + 9*a + 64*b*p**2*x**8 + 80*b*p*x**8 + 9 
*b*x**8),x)*a**2*d*p + 4096*int((a + b*x**8)**p/(64*a*p**2 + 80*a*p + 9*a 
+ 64*b*p**2*x**8 + 80*b*p*x**8 + 9*b*x**8),x)*a*b*c*p**4 + 9728*int((a + b 
*x**8)**p/(64*a*p**2 + 80*a*p + 9*a + 64*b*p**2*x**8 + 80*b*p*x**8 + 9*b*x 
**8),x)*a*b*c*p**3 + 6336*int((a + b*x**8)**p/(64*a*p**2 + 80*a*p + 9*a + 
64*b*p**2*x**8 + 80*b*p*x**8 + 9*b*x**8),x)*a*b*c*p**2 + 648*int((a + b*x* 
*8)**p/(64*a*p**2 + 80*a*p + 9*a + 64*b*p**2*x**8 + 80*b*p*x**8 + 9*b*x**8 
),x)*a*b*c*p)/(b*(64*p**2 + 80*p + 9))