Integrand size = 20, antiderivative size = 194 \[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\frac {x^2 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,2,\frac {5}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{2 c^2}-\frac {d x^6 \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,2,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c^3}+\frac {d^2 x^{10} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,2,\frac {9}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{10 c^4} \] Output:
1/2*x^2*(b*x^8+a)^p*AppellF1(1/4,2,-p,5/4,d^2*x^8/c^2,-b*x^8/a)/c^2/((1+b* x^8/a)^p)-1/3*d*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^3/((1+b*x^8/a)^p)+1/10*d^2*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^4/((1+b*x^8/a)^p)
\[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx \] Input:
Integrate[(x*(a + b*x^8)^p)/(c + d*x^4)^2,x]
Output:
Integrate[(x*(a + b*x^8)^p)/(c + d*x^4)^2, x]
Time = 0.42 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1815, 1569, 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.
\(\displaystyle \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 1815 |
\(\displaystyle \frac {1}{2} \int \frac {\left (b x^8+a\right )^p}{\left (d x^4+c\right )^2}dx^2\) |
\(\Big \downarrow \) 1569 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {2 c d x^4 \left (b x^8+a\right )^p}{\left (c^2-d^2 x^8\right )^2}+\frac {c^2 \left (b x^8+a\right )^p}{\left (c^2-d^2 x^8\right )^2}+\frac {d^2 x^8 \left (b x^8+a\right )^p}{\left (d^2 x^8-c^2\right )^2}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{4},-p,2,\frac {5}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{c^2}+\frac {d^2 x^{10} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {5}{4},-p,2,\frac {9}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{5 c^4}-\frac {2 d x^6 \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{4},-p,2,\frac {7}{4},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c^3}\right )\) |
Input:
Int[(x*(a + b*x^8)^p)/(c + d*x^4)^2,x]
Output:
((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^2*(1 + (b*x^8)/a)^p) - (2*d*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^3*(1 + (b*x^8)/a)^p) + (d^2*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^4* (1 + (b*x^8)/a)^p))/2
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int [ExpandIntegrand[(a + c*x^4)^p, (d/(d^2 - e^2*x^4) - e*(x^2/(d^2 - e^2*x^4) ))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && ! IntegerQ[p] && ILtQ[q, 0]
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 ]
\[\int \frac {x \left (b \,x^{8}+a \right )^{p}}{\left (x^{4} d +c \right )^{2}}d x\]
Input:
int(x*(b*x^8+a)^p/(d*x^4+c)^2,x)
Output:
int(x*(b*x^8+a)^p/(d*x^4+c)^2,x)
\[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x}{{\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:
integrate(x*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="fricas")
Output:
integral((b*x^8 + a)^p*x/(d^2*x^8 + 2*c*d*x^4 + c^2), x)
Timed out. \[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x*(b*x**8+a)**p/(d*x**4+c)**2,x)
Output:
Timed out
\[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x}{{\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:
integrate(x*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="maxima")
Output:
integrate((b*x^8 + a)^p*x/(d*x^4 + c)^2, x)
\[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x}{{\left (d x^{4} + c\right )}^{2}} \,d x } \] Input:
integrate(x*(b*x^8+a)^p/(d*x^4+c)^2,x, algorithm="giac")
Output:
integrate((b*x^8 + a)^p*x/(d*x^4 + c)^2, x)
Timed out. \[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int \frac {x\,{\left (b\,x^8+a\right )}^p}{{\left (d\,x^4+c\right )}^2} \,d x \] Input:
int((x*(a + b*x^8)^p)/(c + d*x^4)^2,x)
Output:
int((x*(a + b*x^8)^p)/(c + d*x^4)^2, x)
\[ \int \frac {x \left (a+b x^8\right )^p}{\left (c+d x^4\right )^2} \, dx=\int \frac {\left (b \,x^{8}+a \right )^{p} x}{d^{2} x^{8}+2 c d \,x^{4}+c^{2}}d x \] Input:
int(x*(b*x^8+a)^p/(d*x^4+c)^2,x)
Output:
int(((a + b*x**8)**p*x)/(c**2 + 2*c*d*x**4 + d**2*x**8),x)