\(\int \frac {x^5 (a+b x^8)^p}{c+d x^4} \, dx\) [24]

Optimal result

Integrand size = 22, antiderivative size = 128 \[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\frac {x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,1,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{6 c}-\frac {d x^{10} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,1,\frac {9}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{10 c^2} \] Output:

1/6*x^6*(b*x^8+a)^p*AppellF1(3/4,1,-p,7/4,d^2*x^8/c^2,-b*x^8/a)/c/((1+b*x^ 
8/a)^p)-1/10*d*x^10*(b*x^8+a)^p*AppellF1(5/4,1,-p,9/4,d^2*x^8/c^2,-b*x^8/a 
)/c^2/((1+b*x^8/a)^p)
 

Mathematica [F]

\[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx \] Input:

Integrate[(x^5*(a + b*x^8)^p)/(c + d*x^4),x]
 

Output:

Integrate[(x^5*(a + b*x^8)^p)/(c + d*x^4), x]
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.39, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1815, 1675, 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.

\(\displaystyle \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx\)

\(\Big \downarrow \) 1815

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

\(\Big \downarrow \) 1675

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,1,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c}-\frac {x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,1,\frac {5}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{d}+\frac {x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},-p,\frac {5}{4},-\frac {b x^8}{a}\right )}{d}\right )\)

Input:

Int[(x^5*(a + b*x^8)^p)/(c + d*x^4),x]
 

Output:

(-((x^2*(a + b*x^8)^p*AppellF1[1/4, -p, 1, 5/4, -((b*x^8)/a), (d^2*x^8)/c^ 
2])/(d*(1 + (b*x^8)/a)^p)) + (x^6*(a + b*x^8)^p*AppellF1[3/4, -p, 1, 7/4, 
-((b*x^8)/a), (d^2*x^8)/c^2])/(3*c*(1 + (b*x^8)/a)^p) + (x^2*(a + b*x^8)^p 
*Hypergeometric2F1[1/4, -p, 5/4, -((b*x^8)/a)])/(d*(1 + (b*x^8)/a)^p))/2
 

Defintions of rubi rules used

rule 1675

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p 
_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^p, 
x], x] /; FreeQ[{a, c, d, e, f, m, p, q}, x] && (IGtQ[p, 0] || IGtQ[q, 0] | 
| IntegersQ[m, q])
 

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]

\[\int \frac {x^{5} \left (b \,x^{8}+a \right )^{p}}{x^{4} d +c}d x\]

Input:

int(x^5*(b*x^8+a)^p/(d*x^4+c),x)
 

Output:

int(x^5*(b*x^8+a)^p/(d*x^4+c),x)
 

Fricas [F]

\[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x^{5}}{d x^{4} + c} \,d x } \] Input:

integrate(x^5*(b*x^8+a)^p/(d*x^4+c),x, algorithm="fricas")
 

Output:

integral((b*x^8 + a)^p*x^5/(d*x^4 + c), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\text {Timed out} \] Input:

integrate(x**5*(b*x**8+a)**p/(d*x**4+c),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x^{5}}{d x^{4} + c} \,d x } \] Input:

integrate(x^5*(b*x^8+a)^p/(d*x^4+c),x, algorithm="maxima")
 

Output:

integrate((b*x^8 + a)^p*x^5/(d*x^4 + c), x)
 

Giac [F]

\[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x^{5}}{d x^{4} + c} \,d x } \] Input:

integrate(x^5*(b*x^8+a)^p/(d*x^4+c),x, algorithm="giac")
 

Output:

integrate((b*x^8 + a)^p*x^5/(d*x^4 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int \frac {x^5\,{\left (b\,x^8+a\right )}^p}{d\,x^4+c} \,d x \] Input:

int((x^5*(a + b*x^8)^p)/(c + d*x^4),x)
 

Output:

int((x^5*(a + b*x^8)^p)/(c + d*x^4), x)
 

Reduce [F]

\[ \int \frac {x^5 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\frac {\left (b \,x^{8}+a \right )^{p} x^{2}-32 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{9}}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) b c \,p^{2}-16 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{9}}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) b c p -2 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{9}}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) b c +32 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{5}}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) a d \,p^{2}+8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{5}}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) a d p -8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) a c p -2 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x}{4 b d p \,x^{12}+b d \,x^{12}+4 b c p \,x^{8}+b c \,x^{8}+4 a d p \,x^{4}+a d \,x^{4}+4 a c p +a c}d x \right ) a c}{2 d \left (4 p +1\right )} \] Input:

int(x^5*(b*x^8+a)^p/(d*x^4+c),x)
 

Output:

((a + b*x**8)**p*x**2 - 32*int(((a + b*x**8)**p*x**9)/(4*a*c*p + a*c + 4*a 
*d*p*x**4 + a*d*x**4 + 4*b*c*p*x**8 + b*c*x**8 + 4*b*d*p*x**12 + b*d*x**12 
),x)*b*c*p**2 - 16*int(((a + b*x**8)**p*x**9)/(4*a*c*p + a*c + 4*a*d*p*x** 
4 + a*d*x**4 + 4*b*c*p*x**8 + b*c*x**8 + 4*b*d*p*x**12 + b*d*x**12),x)*b*c 
*p - 2*int(((a + b*x**8)**p*x**9)/(4*a*c*p + a*c + 4*a*d*p*x**4 + a*d*x**4 
 + 4*b*c*p*x**8 + b*c*x**8 + 4*b*d*p*x**12 + b*d*x**12),x)*b*c + 32*int((( 
a + b*x**8)**p*x**5)/(4*a*c*p + a*c + 4*a*d*p*x**4 + a*d*x**4 + 4*b*c*p*x* 
*8 + b*c*x**8 + 4*b*d*p*x**12 + b*d*x**12),x)*a*d*p**2 + 8*int(((a + b*x** 
8)**p*x**5)/(4*a*c*p + a*c + 4*a*d*p*x**4 + a*d*x**4 + 4*b*c*p*x**8 + b*c* 
x**8 + 4*b*d*p*x**12 + b*d*x**12),x)*a*d*p - 8*int(((a + b*x**8)**p*x)/(4* 
a*c*p + a*c + 4*a*d*p*x**4 + a*d*x**4 + 4*b*c*p*x**8 + b*c*x**8 + 4*b*d*p* 
x**12 + b*d*x**12),x)*a*c*p - 2*int(((a + b*x**8)**p*x)/(4*a*c*p + a*c + 4 
*a*d*p*x**4 + a*d*x**4 + 4*b*c*p*x**8 + b*c*x**8 + 4*b*d*p*x**12 + b*d*x** 
12),x)*a*c)/(2*d*(4*p + 1))