\(\int (a+b x^8)^p (c+d x^8)^2 \, dx\) [296]

Optimal result

Integrand size = 19, antiderivative size = 171 \[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=-\frac {d (9 a d-2 b c (17+8 p)) x \left (a+b x^8\right )^{1+p}}{b^2 (9+8 p) (17+8 p)}+\frac {d^2 x^9 \left (a+b x^8\right )^{1+p}}{b (17+8 p)}+\frac {\left (b^2 c^2 (9+8 p) (17+8 p)+a d (9 a d-2 b c (17+8 p))\right ) x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )}{b^2 (9+8 p) (17+8 p)} \] Output:

-d*(9*a*d-2*b*c*(17+8*p))*x*(b*x^8+a)^(p+1)/b^2/(9+8*p)/(17+8*p)+d^2*x^9*( 
b*x^8+a)^(p+1)/b/(17+8*p)+(b^2*c^2*(9+8*p)*(17+8*p)+a*d*(9*a*d-2*b*c*(17+8 
*p)))*x*(b*x^8+a)^p*hypergeom([1/8, -p],[9/8],-b*x^8/a)/b^2/(9+8*p)/(17+8* 
p)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 5.64 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.62 \[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=\frac {1}{153} x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (153 c^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+d x^8 \left (34 c \operatorname {Hypergeometric2F1}\left (\frac {9}{8},-p,\frac {17}{8},-\frac {b x^8}{a}\right )+9 d x^8 \operatorname {Hypergeometric2F1}\left (\frac {17}{8},-p,\frac {25}{8},-\frac {b x^8}{a}\right )\right )\right ) \] Input:

Integrate[(a + b*x^8)^p*(c + d*x^8)^2,x]
 

Output:

(x*(a + b*x^8)^p*(153*c^2*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)] + 
d*x^8*(34*c*Hypergeometric2F1[9/8, -p, 17/8, -((b*x^8)/a)] + 9*d*x^8*Hyper 
geometric2F1[17/8, -p, 25/8, -((b*x^8)/a)])))/(153*(1 + (b*x^8)/a)^p)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {933, 25, 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.

\(\displaystyle \int \left (c+d x^8\right )^2 \left (a+b x^8\right )^p \, dx\)

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int -\left (b x^8+a\right )^p \left (d (9 a d-b c (8 p+25)) x^8+c (a d-b c (8 p+17))\right )dx}{b (8 p+17)}+\frac {d x \left (c+d x^8\right ) \left (a+b x^8\right )^{p+1}}{b (8 p+17)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d x \left (c+d x^8\right ) \left (a+b x^8\right )^{p+1}}{b (8 p+17)}-\frac {\int \left (b x^8+a\right )^p \left (d (9 a d-b c (8 p+25)) x^8+c (a d-b c (8 p+17))\right )dx}{b (8 p+17)}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {d x \left (c+d x^8\right ) \left (a+b x^8\right )^{p+1}}{b (8 p+17)}-\frac {\frac {d x \left (a+b x^8\right )^{p+1} (9 a d-b c (8 p+25))}{b (8 p+9)}-\frac {\left (9 a^2 d^2-2 a b c d (8 p+17)+b^2 c^2 \left (64 p^2+208 p+153\right )\right ) \int \left (b x^8+a\right )^pdx}{b (8 p+9)}}{b (8 p+17)}\)

\(\Big \downarrow \) 779

\(\displaystyle \frac {d x \left (c+d x^8\right ) \left (a+b x^8\right )^{p+1}}{b (8 p+17)}-\frac {\frac {d x \left (a+b x^8\right )^{p+1} (9 a d-b c (8 p+25))}{b (8 p+9)}-\frac {\left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \left (9 a^2 d^2-2 a b c d (8 p+17)+b^2 c^2 \left (64 p^2+208 p+153\right )\right ) \int \left (\frac {b x^8}{a}+1\right )^pdx}{b (8 p+9)}}{b (8 p+17)}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {d x \left (c+d x^8\right ) \left (a+b x^8\right )^{p+1}}{b (8 p+17)}-\frac {\frac {d x \left (a+b x^8\right )^{p+1} (9 a d-b c (8 p+25))}{b (8 p+9)}-\frac {x \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \left (9 a^2 d^2-2 a b c d (8 p+17)+b^2 c^2 \left (64 p^2+208 p+153\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )}{b (8 p+9)}}{b (8 p+17)}\)

Input:

Int[(a + b*x^8)^p*(c + d*x^8)^2,x]
 

Output:

(d*x*(a + b*x^8)^(1 + p)*(c + d*x^8))/(b*(17 + 8*p)) - ((d*(9*a*d - b*c*(2 
5 + 8*p))*x*(a + b*x^8)^(1 + p))/(b*(9 + 8*p)) - ((9*a^2*d^2 - 2*a*b*c*d*( 
17 + 8*p) + b^2*c^2*(153 + 208*p + 64*p^2))*x*(a + b*x^8)^p*Hypergeometric 
2F1[1/8, -p, 9/8, -((b*x^8)/a)])/(b*(9 + 8*p)*(1 + (b*x^8)/a)^p))/(b*(17 + 
 8*p))
 

Defintions of rubi rules used

rule 25

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

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]
 

rule 933

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[d*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*(n*(p + q) + 1))), 
x] + Simp[1/(b*(n*(p + q) + 1))   Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Sim 
p[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q 
- 1) + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d 
, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[ 
a, b, c, d, n, p, q, x]
 

Maple [F]

\[\int \left (b \,x^{8}+a \right )^{p} \left (d \,x^{8}+c \right )^{2}d x\]

Input:

int((b*x^8+a)^p*(d*x^8+c)^2,x)
 

Output:

int((b*x^8+a)^p*(d*x^8+c)^2,x)
 

Fricas [F]

\[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=\int { {\left (d x^{8} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((b*x^8+a)^p*(d*x^8+c)^2,x, algorithm="fricas")
 

Output:

integral((d^2*x^16 + 2*c*d*x^8 + c^2)*(b*x^8 + a)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=\text {Timed out} \] Input:

integrate((b*x**8+a)**p*(d*x**8+c)**2,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=\int { {\left (d x^{8} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((b*x^8+a)^p*(d*x^8+c)^2,x, algorithm="maxima")
 

Output:

integrate((d*x^8 + c)^2*(b*x^8 + a)^p, x)
 

Giac [F]

\[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=\int { {\left (d x^{8} + c\right )}^{2} {\left (b x^{8} + a\right )}^{p} \,d x } \] Input:

integrate((b*x^8+a)^p*(d*x^8+c)^2,x, algorithm="giac")
 

Output:

integrate((d*x^8 + c)^2*(b*x^8 + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx=\int {\left (b\,x^8+a\right )}^p\,{\left (d\,x^8+c\right )}^2 \,d x \] Input:

int((a + b*x^8)^p*(c + d*x^8)^2,x)
 

Output:

int((a + b*x^8)^p*(c + d*x^8)^2, x)
 

Reduce [F]

\[ \int \left (a+b x^8\right )^p \left (c+d x^8\right )^2 \, dx =\text {Too large to display} \] Input:

int((b*x^8+a)^p*(d*x^8+c)^2,x)
 

Output:

( - 72*(a + b*x**8)**p*a**2*d**2*p*x + 128*(a + b*x**8)**p*a*b*c*d*p**2*x 
+ 272*(a + b*x**8)**p*a*b*c*d*p*x + 64*(a + b*x**8)**p*a*b*d**2*p**2*x**9 
+ 8*(a + b*x**8)**p*a*b*d**2*p*x**9 + 64*(a + b*x**8)**p*b**2*c**2*p**2*x 
+ 208*(a + b*x**8)**p*b**2*c**2*p*x + 153*(a + b*x**8)**p*b**2*c**2*x + 12 
8*(a + b*x**8)**p*b**2*c*d*p**2*x**9 + 288*(a + b*x**8)**p*b**2*c*d*p*x**9 
 + 34*(a + b*x**8)**p*b**2*c*d*x**9 + 64*(a + b*x**8)**p*b**2*d**2*p**2*x* 
*17 + 80*(a + b*x**8)**p*b**2*d**2*p*x**17 + 9*(a + b*x**8)**p*b**2*d**2*x 
**17 + 36864*int((a + b*x**8)**p/(512*a*p**3 + 1728*a*p**2 + 1432*a*p + 15 
3*a + 512*b*p**3*x**8 + 1728*b*p**2*x**8 + 1432*b*p*x**8 + 153*b*x**8),x)* 
a**3*d**2*p**4 + 124416*int((a + b*x**8)**p/(512*a*p**3 + 1728*a*p**2 + 14 
32*a*p + 153*a + 512*b*p**3*x**8 + 1728*b*p**2*x**8 + 1432*b*p*x**8 + 153* 
b*x**8),x)*a**3*d**2*p**3 + 103104*int((a + b*x**8)**p/(512*a*p**3 + 1728* 
a*p**2 + 1432*a*p + 153*a + 512*b*p**3*x**8 + 1728*b*p**2*x**8 + 1432*b*p* 
x**8 + 153*b*x**8),x)*a**3*d**2*p**2 + 11016*int((a + b*x**8)**p/(512*a*p* 
*3 + 1728*a*p**2 + 1432*a*p + 153*a + 512*b*p**3*x**8 + 1728*b*p**2*x**8 + 
 1432*b*p*x**8 + 153*b*x**8),x)*a**3*d**2*p - 65536*int((a + b*x**8)**p/(5 
12*a*p**3 + 1728*a*p**2 + 1432*a*p + 153*a + 512*b*p**3*x**8 + 1728*b*p**2 
*x**8 + 1432*b*p*x**8 + 153*b*x**8),x)*a**2*b*c*d*p**5 - 360448*int((a + b 
*x**8)**p/(512*a*p**3 + 1728*a*p**2 + 1432*a*p + 153*a + 512*b*p**3*x**8 + 
 1728*b*p**2*x**8 + 1432*b*p*x**8 + 153*b*x**8),x)*a**2*b*c*d*p**4 - 65...