Integrand size = 17, antiderivative size = 521 \[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\frac {x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{8},-p,1,\frac {9}{8},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{c}-\frac {d x^2 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,1,\frac {5}{4},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{2 c^2}+\frac {d^2 x^3 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{8},-p,1,\frac {11}{8},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{3 c^3}-\frac {d^3 x^4 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{4 c^4}+\frac {d^4 x^5 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{8},-p,1,\frac {13}{8},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{5 c^5}-\frac {d^5 x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,1,\frac {7}{4},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{6 c^6}+\frac {d^6 x^7 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {7}{8},-p,1,\frac {15}{8},-\frac {b x^8}{a},\frac {d^8 x^8}{c^8}\right )}{7 c^7}-\frac {d^7 \left (a+b x^8\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {d^8 \left (a+b x^8\right )}{b c^8+a d^8}\right )}{8 \left (b c^8+a d^8\right ) (1+p)} \] Output:
x*(b*x^8+a)^p*AppellF1(1/8,1,-p,9/8,d^8*x^8/c^8,-b*x^8/a)/c/((1+b*x^8/a)^p )-1/2*d*x^2*(b*x^8+a)^p*AppellF1(1/4,1,-p,5/4,d^8*x^8/c^8,-b*x^8/a)/c^2/(( 1+b*x^8/a)^p)+1/3*d^2*x^3*(b*x^8+a)^p*AppellF1(3/8,1,-p,11/8,d^8*x^8/c^8,- b*x^8/a)/c^3/((1+b*x^8/a)^p)-1/4*d^3*x^4*(b*x^8+a)^p*AppellF1(1/2,1,-p,3/2 ,d^8*x^8/c^8,-b*x^8/a)/c^4/((1+b*x^8/a)^p)+1/5*d^4*x^5*(b*x^8+a)^p*AppellF 1(5/8,1,-p,13/8,d^8*x^8/c^8,-b*x^8/a)/c^5/((1+b*x^8/a)^p)-1/6*d^5*x^6*(b*x ^8+a)^p*AppellF1(3/4,1,-p,7/4,d^8*x^8/c^8,-b*x^8/a)/c^6/((1+b*x^8/a)^p)+1/ 7*d^6*x^7*(b*x^8+a)^p*AppellF1(7/8,1,-p,15/8,d^8*x^8/c^8,-b*x^8/a)/c^7/((1 +b*x^8/a)^p)-1/8*d^7*(b*x^8+a)^(p+1)*hypergeom([1, p+1],[2+p],d^8*(b*x^8+a )/(a*d^8+b*c^8))/(a*d^8+b*c^8)/(p+1)
\[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx \] Input:
Integrate[(a + b*x^8)^p/(c + d*x),x]
Output:
Integrate[(a + b*x^8)^p/(c + d*x), x]
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.
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {(c-d x) \left (a+b x^8\right )^p}{c^2-d^2 x^2}dx\) |
| \(\Big \downarrow \) 2003 |
| \(\displaystyle \int \frac {\left (a+b x^8\right )^p}{c+d x}dx\) |
Input:
Int[(a + b*x^8)^p/(c + d*x),x]
Output:
$Aborted
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbo l] :> Int[ExpandToSum[(c - d*x^n)^(-q), x]*((a + b*x^nn)^p/(c^2 - d^2*x^(2* n))^(-q)), x] /; FreeQ[{a, b, c, d, n, nn, p}, x] && !IntegerQ[p] && ILtQ[ q, 0] && IGtQ[Log[2, nn/n], 0]
\[\int \frac {\left (b \,x^{8}+a \right )^{p}}{d x +c}d x\]
Input:
int((b*x^8+a)^p/(d*x+c),x)
Output:
int((b*x^8+a)^p/(d*x+c),x)
\[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p}}{d x + c} \,d x } \] Input:
integrate((b*x^8+a)^p/(d*x+c),x, algorithm="fricas")
Output:
integral((b*x^8 + a)^p/(d*x + c), x)
Timed out. \[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\text {Timed out} \] Input:
integrate((b*x**8+a)**p/(d*x+c),x)
Output:
Timed out
\[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p}}{d x + c} \,d x } \] Input:
integrate((b*x^8+a)^p/(d*x+c),x, algorithm="maxima")
Output:
integrate((b*x^8 + a)^p/(d*x + c), x)
\[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p}}{d x + c} \,d x } \] Input:
integrate((b*x^8+a)^p/(d*x+c),x, algorithm="giac")
Output:
integrate((b*x^8 + a)^p/(d*x + c), x)
Timed out. \[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\int \frac {{\left (b\,x^8+a\right )}^p}{c+d\,x} \,d x \] Input:
int((a + b*x^8)^p/(c + d*x),x)
Output:
int((a + b*x^8)^p/(c + d*x), x)
\[ \int \frac {\left (a+b x^8\right )^p}{c+d x} \, dx=\frac {\left (b \,x^{8}+a \right )^{p}+8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{b d \,x^{9}+b c \,x^{8}+a d x +a c}d x \right ) a d p -8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{7}}{b d \,x^{9}+b c \,x^{8}+a d x +a c}d x \right ) b c p}{8 d p} \] Input:
int((b*x^8+a)^p/(d*x+c),x)
Output:
((a + b*x**8)**p + 8*int((a + b*x**8)**p/(a*c + a*d*x + b*c*x**8 + b*d*x** 9),x)*a*d*p - 8*int(((a + b*x**8)**p*x**7)/(a*c + a*d*x + b*c*x**8 + b*d*x **9),x)*b*c*p)/(8*d*p)