\(\int (c+d x)^2 (a+b x^8)^p \, dx\) [316]

Optimal result

Integrand size = 17, antiderivative size = 148 \[ \int (c+d x)^2 \left (a+b x^8\right )^p \, dx=c^2 x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+c d x^2 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )+\frac {1}{3} d^2 x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},-p,\frac {11}{8},-\frac {b x^8}{a}\right ) \] Output:

c^2*x*(b*x^8+a)^p*hypergeom([1/8, -p],[9/8],-b*x^8/a)/((1+b*x^8/a)^p)+c*d* 
x^2*(b*x^8+a)^p*hypergeom([1/4, -p],[5/4],-b*x^8/a)/((1+b*x^8/a)^p)+1/3*d^ 
2*x^3*(b*x^8+a)^p*hypergeom([3/8, -p],[11/8],-b*x^8/a)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.68 \[ \int (c+d x)^2 \left (a+b x^8\right )^p \, dx=\frac {1}{3} x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (3 c^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+d x \left (3 c \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )+d x \operatorname {Hypergeometric2F1}\left (\frac {3}{8},-p,\frac {11}{8},-\frac {b x^8}{a}\right )\right )\right ) \] Input:

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

Output:

(x*(a + b*x^8)^p*(3*c^2*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)] + d* 
x*(3*c*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^8)/a)] + d*x*Hypergeometric2 
F1[3/8, -p, 11/8, -((b*x^8)/a)])))/(3*(1 + (b*x^8)/a)^p)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 148, 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 = {2424, 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.

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

\(\Big \downarrow \) 2424

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

\(\Big \downarrow \) 2009

\(\displaystyle c^2 x \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )+c d x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )+\frac {1}{3} d^2 x^3 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{8},-p,\frac {11}{8},-\frac {b x^8}{a}\right )\)

Input:

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

Output:

(c^2*x*(a + b*x^8)^p*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)])/(1 + ( 
b*x^8)/a)^p + (c*d*x^2*(a + b*x^8)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b* 
x^8)/a)])/(1 + (b*x^8)/a)^p + (d^2*x^3*(a + b*x^8)^p*Hypergeometric2F1[3/8 
, -p, 11/8, -((b*x^8)/a)])/(3*(1 + (b*x^8)/a)^p)
 

Defintions of rubi rules used

rule 2009

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

rule 2424

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, 
 x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2 
*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, 
 x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 135.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.80 \[ \int (c+d x)^2 \left (a+b x^8\right )^p \, dx=\frac {a^{p} c^{2} x \Gamma \left (\frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{8}, - p \\ \frac {9}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {9}{8}\right )} + \frac {a^{p} c d x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{p} d^{2} x^{3} \Gamma \left (\frac {3}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{8}, - p \\ \frac {11}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {11}{8}\right )} \] Input:

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

Output:

a**p*c**2*x*gamma(1/8)*hyper((1/8, -p), (9/8,), b*x**8*exp_polar(I*pi)/a)/ 
(8*gamma(9/8)) + a**p*c*d*x**2*gamma(1/4)*hyper((1/4, -p), (5/4,), b*x**8* 
exp_polar(I*pi)/a)/(4*gamma(5/4)) + a**p*d**2*x**3*gamma(3/8)*hyper((3/8, 
-p), (11/8,), b*x**8*exp_polar(I*pi)/a)/(8*gamma(11/8))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(32*(a + b*x**8)**p*c**2*p**2*x + 20*(a + b*x**8)**p*c**2*p*x + 3*(a + b*x 
**8)**p*c**2*x + 64*(a + b*x**8)**p*c*d*p**2*x**2 + 32*(a + b*x**8)**p*c*d 
*p*x**2 + 3*(a + b*x**8)**p*c*d*x**2 + 32*(a + b*x**8)**p*d**2*p**2*x**3 + 
 12*(a + b*x**8)**p*d**2*p*x**3 + (a + b*x**8)**p*d**2*x**3 + 65536*int((a 
 + b*x**8)**p/(256*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 
192*b*p**2*x**8 + 44*b*p*x**8 + 3*b*x**8),x)*a*c**2*p**6 + 90112*int((a + 
b*x**8)**p/(256*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192 
*b*p**2*x**8 + 44*b*p*x**8 + 3*b*x**8),x)*a*c**2*p**5 + 48128*int((a + b*x 
**8)**p/(256*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192*b* 
p**2*x**8 + 44*b*p*x**8 + 3*b*x**8),x)*a*c**2*p**4 + 12416*int((a + b*x**8 
)**p/(256*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192*b*p** 
2*x**8 + 44*b*p*x**8 + 3*b*x**8),x)*a*c**2*p**3 + 1536*int((a + b*x**8)**p 
/(256*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192*b*p**2*x* 
*8 + 44*b*p*x**8 + 3*b*x**8),x)*a*c**2*p**2 + 72*int((a + b*x**8)**p/(256* 
a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192*b*p**2*x**8 + 4 
4*b*p*x**8 + 3*b*x**8),x)*a*c**2*p + 65536*int(((a + b*x**8)**p*x**2)/(256 
*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192*b*p**2*x**8 + 
44*b*p*x**8 + 3*b*x**8),x)*a*d**2*p**6 + 73728*int(((a + b*x**8)**p*x**2)/ 
(256*a*p**3 + 192*a*p**2 + 44*a*p + 3*a + 256*b*p**3*x**8 + 192*b*p**2*x** 
8 + 44*b*p*x**8 + 3*b*x**8),x)*a*d**2*p**5 + 31744*int(((a + b*x**8)**p...