\(\int \frac {x^5 (a+b x^8)^p}{(c+d x^4)^2} \, dx\) [36]

Optimal result

Integrand size = 22, antiderivative size = 194 \[ \int \frac {x^5 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\frac {x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,2,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{6 c^2}-\frac {d x^{10} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,2,\frac {9}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{5 c^3}+\frac {d^2 x^{14} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{4},-p,2,\frac {11}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{14 c^4} \] Output:

1/6*x^6*(b*x^8+a)^p*AppellF1(3/4,2,-p,7/4,d^2*x^8/c^2,-b*x^8/a)/c^2/((1+b* 
x^8/a)^p)-1/5*d*x^10*(b*x^8+a)^p*AppellF1(5/4,2,-p,9/4,d^2*x^8/c^2,-b*x^8/ 
a)/c^3/((1+b*x^8/a)^p)+1/14*d^2*x^14*(b*x^8+a)^p*AppellF1(7/4,2,-p,11/4,d^ 
2*x^8/c^2,-b*x^8/a)/c^4/((1+b*x^8/a)^p)
 

Mathematica [F]

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

Integrate[(x^5*(a + b*x^8)^p)/(c + d*x^4)^2,x]
 

Output:

Integrate[(x^5*(a + b*x^8)^p)/(c + d*x^4)^2, x]
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.66, 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.

\(\displaystyle \int \frac {x^5 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx\)

\(\Big \downarrow \) 1815

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

\(\Big \downarrow \) 1675

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,1,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c^2}+\frac {2 x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,2,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c^2}+\frac {x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,1,\frac {5}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{c d}-\frac {x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,2,\frac {5}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{c d}-\frac {d x^{10} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,2,\frac {9}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{5 c^3}\right )\)

Input:

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

\[\int \frac {x^{5} \left (b \,x^{8}+a \right )^{p}}{\left (x^{4} d +c \right )^{2}}d x\]

Input:

int(x^5*(b*x^8+a)^p/(d*x^4+c)^2,x)
 

Output:

int(x^5*(b*x^8+a)^p/(d*x^4+c)^2,x)
 

Fricas [F]

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

integrate(x^5*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="fricas")
 

Output:

integral((b*x^8 + a)^p*x^5/(d^2*x^8 + 2*c*d*x^4 + c^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x**5*(b*x**8+a)**p/(d*x**4+c)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^5*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="maxima")
 

Output:

integrate((b*x^8 + a)^p*x^5/(d*x^4 + c)^2, x)
 

Giac [F]

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

integrate(x^5*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="giac")
 

Output:

integrate((b*x^8 + a)^p*x^5/(d*x^4 + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int \frac {x^5\,{\left (b\,x^8+a\right )}^p}{{\left (d\,x^4+c\right )}^2} \,d x \] Input:

int((x^5*(a + b*x^8)^p)/(c + d*x^4)^2,x)
 

Output:

int((x^5*(a + b*x^8)^p)/(c + d*x^4)^2, x)
 

Reduce [F]

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

int(x^5*(b*x^8+a)^p/(d*x^4+c)^2,x)
 

Output:

int(((a + b*x**8)**p*x**5)/(c**2 + 2*c*d*x**4 + d**2*x**8),x)