\(\int x (c+d x^4) (a+b x^8)^p \, dx\) [3]

Optimal result

Integrand size = 18, antiderivative size = 101 \[ \int x \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{2} c x^2 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )+\frac {1}{6} d x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right ) \] Output:

1/2*c*x^2*(b*x^8+a)^p*hypergeom([1/4, -p],[5/4],-b*x^8/a)/((1+b*x^8/a)^p)+ 
1/6*d*x^6*(b*x^8+a)^p*hypergeom([3/4, -p],[7/4],-b*x^8/a)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int x \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {1}{6} x^2 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (3 c \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )+d x^4 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )\right ) \] Input:

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

Output:

(x^2*(a + b*x^8)^p*(3*c*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^8)/a)] + d* 
x^4*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^8)/a)]))/(6*(1 + (b*x^8)/a)^p)
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1815, 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.

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

\(\Big \downarrow \) 1815

\(\displaystyle \frac {1}{2} \int \left (d x^4+c\right ) \left (b x^8+a\right )^pdx^2\)

\(\Big \downarrow \) 1516

\(\displaystyle \frac {1}{2} \int \left (d x^4 \left (b x^8+a\right )^p+c \left (b x^8+a\right )^p\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (c x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )+\frac {1}{3} d x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^8}{a}\right )\right )\)

Input:

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

Output:

((c*x^2*(a + b*x^8)^p*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^8)/a)])/(1 + 
(b*x^8)/a)^p + (d*x^6*(a + b*x^8)^p*Hypergeometric2F1[3/4, -p, 7/4, -((b*x 
^8)/a)])/(3*(1 + (b*x^8)/a)^p))/2
 

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 1815

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_ 
.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/ 
k - 1)*(d + e*x^(n/k))^q*(a + c*x^(2*(n/k)))^p, x], x, x^k], x] /; k != 1] 
/; FreeQ[{a, c, d, e, p, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IntegerQ[m 
]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x^5 + c*x)*(b*x^8 + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 91.71 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.75 \[ \int x \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {a^{p} c x^{2} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, - p \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {5}{4}\right )} + \frac {a^{p} d x^{6} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

a**p*c*x**2*gamma(1/4)*hyper((1/4, -p), (5/4,), b*x**8*exp_polar(I*pi)/a)/ 
(8*gamma(5/4)) + a**p*d*x**6*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**8*ex 
p_polar(I*pi)/a)/(8*gamma(7/4))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x \left (c+d x^4\right ) \left (a+b x^8\right )^p \, dx=\frac {4 \left (b \,x^{8}+a \right )^{p} c p \,x^{2}+3 \left (b \,x^{8}+a \right )^{p} c \,x^{2}+4 \left (b \,x^{8}+a \right )^{p} d p \,x^{6}+\left (b \,x^{8}+a \right )^{p} d \,x^{6}+512 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{5}}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d \,p^{4}+640 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{5}}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d \,p^{3}+224 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{5}}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d \,p^{2}+24 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{5}}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a d p +512 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c \,p^{4}+896 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c \,p^{3}+480 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c \,p^{2}+72 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x}{16 b \,p^{2} x^{8}+16 b p \,x^{8}+3 b \,x^{8}+16 a \,p^{2}+16 a p +3 a}d x \right ) a c p}{32 p^{2}+32 p +6} \] Input:

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

Output:

(4*(a + b*x**8)**p*c*p*x**2 + 3*(a + b*x**8)**p*c*x**2 + 4*(a + b*x**8)**p 
*d*p*x**6 + (a + b*x**8)**p*d*x**6 + 512*int(((a + b*x**8)**p*x**5)/(16*a* 
p**2 + 16*a*p + 3*a + 16*b*p**2*x**8 + 16*b*p*x**8 + 3*b*x**8),x)*a*d*p**4 
 + 640*int(((a + b*x**8)**p*x**5)/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x* 
*8 + 16*b*p*x**8 + 3*b*x**8),x)*a*d*p**3 + 224*int(((a + b*x**8)**p*x**5)/ 
(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**8 + 16*b*p*x**8 + 3*b*x**8),x)*a* 
d*p**2 + 24*int(((a + b*x**8)**p*x**5)/(16*a*p**2 + 16*a*p + 3*a + 16*b*p* 
*2*x**8 + 16*b*p*x**8 + 3*b*x**8),x)*a*d*p + 512*int(((a + b*x**8)**p*x)/( 
16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**8 + 16*b*p*x**8 + 3*b*x**8),x)*a*c 
*p**4 + 896*int(((a + b*x**8)**p*x)/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2* 
x**8 + 16*b*p*x**8 + 3*b*x**8),x)*a*c*p**3 + 480*int(((a + b*x**8)**p*x)/( 
16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x**8 + 16*b*p*x**8 + 3*b*x**8),x)*a*c 
*p**2 + 72*int(((a + b*x**8)**p*x)/(16*a*p**2 + 16*a*p + 3*a + 16*b*p**2*x 
**8 + 16*b*p*x**8 + 3*b*x**8),x)*a*c*p)/(2*(16*p**2 + 16*p + 3))