Integrand size = 21, antiderivative size = 59 \[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {x \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {1}{8},2,\frac {1}{2},\frac {9}{8},-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 \sqrt {c+d x^8}} \] Output:
x*(1+d*x^8/c)^(1/2)*AppellF1(1/8,2,1/2,9/8,-b*x^8/a,-d*x^8/c)/a^2/(d*x^8+c )^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(328\) vs. \(2(59)=118\).
Time = 10.22 (sec) , antiderivative size = 328, normalized size of antiderivative = 5.56 \[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=-\frac {x \left (b d x^8 \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {9}{8},\frac {1}{2},1,\frac {17}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+\frac {3 a \left (9 a c \left (8 a d-b \left (8 c+d x^8\right )\right ) \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{2},1,\frac {9}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+4 b x^8 \left (c+d x^8\right ) \left (2 b c \operatorname {AppellF1}\left (\frac {9}{8},\frac {1}{2},2,\frac {17}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+a d \operatorname {AppellF1}\left (\frac {9}{8},\frac {3}{2},1,\frac {17}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )\right )}{\left (a+b x^8\right ) \left (-9 a c \operatorname {AppellF1}\left (\frac {1}{8},\frac {1}{2},1,\frac {9}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+4 x^8 \left (2 b c \operatorname {AppellF1}\left (\frac {9}{8},\frac {1}{2},2,\frac {17}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+a d \operatorname {AppellF1}\left (\frac {9}{8},\frac {3}{2},1,\frac {17}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )\right )\right )}\right )}{24 a^2 (-b c+a d) \sqrt {c+d x^8}} \] Input:
Integrate[1/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
Output:
-1/24*(x*(b*d*x^8*Sqrt[1 + (d*x^8)/c]*AppellF1[9/8, 1/2, 1, 17/8, -((d*x^8 )/c), -((b*x^8)/a)] + (3*a*(9*a*c*(8*a*d - b*(8*c + d*x^8))*AppellF1[1/8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*b*x^8*(c + d*x^8)*(2*b*c*Appe llF1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/ 2, 1, 17/8, -((d*x^8)/c), -((b*x^8)/a)])))/((a + b*x^8)*(-9*a*c*AppellF1[1 /8, 1/2, 1, 9/8, -((d*x^8)/c), -((b*x^8)/a)] + 4*x^8*(2*b*c*AppellF1[9/8, 1/2, 2, 17/8, -((d*x^8)/c), -((b*x^8)/a)] + a*d*AppellF1[9/8, 3/2, 1, 17/8 , -((d*x^8)/c), -((b*x^8)/a)])))))/(a^2*(-(b*c) + a*d)*Sqrt[c + d*x^8])
Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {937, 936}
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.
\(\displaystyle \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {d x^8}{c}+1} \int \frac {1}{\left (b x^8+a\right )^2 \sqrt {\frac {d x^8}{c}+1}}dx}{\sqrt {c+d x^8}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt {\frac {d x^8}{c}+1} \operatorname {AppellF1}\left (\frac {1}{8},2,\frac {1}{2},\frac {9}{8},-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 \sqrt {c+d x^8}}\) |
Input:
Int[1/((a + b*x^8)^2*Sqrt[c + d*x^8]),x]
Output:
(x*Sqrt[1 + (d*x^8)/c]*AppellF1[1/8, 2, 1/2, 9/8, -((b*x^8)/a), -((d*x^8)/ c)])/(a^2*Sqrt[c + d*x^8])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) ], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q }, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {1}{\left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x\]
Input:
int(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
Output:
int(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
Timed out. \[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{\left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \] Input:
integrate(1/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
Output:
Integral(1/((a + b*x**8)**2*sqrt(c + d*x**8)), x)
\[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:
integrate(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)
\[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c}} \,d x } \] Input:
integrate(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^8 + a)^2*sqrt(d*x^8 + c)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \] Input:
int(1/((a + b*x^8)^2*(c + d*x^8)^(1/2)),x)
Output:
int(1/((a + b*x^8)^2*(c + d*x^8)^(1/2)), x)
\[ \int \frac {1}{\left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {\sqrt {d \,x^{8}+c}}{b^{2} d \,x^{24}+2 a b d \,x^{16}+b^{2} c \,x^{16}+a^{2} d \,x^{8}+2 a b c \,x^{8}+a^{2} c}d x \] Input:
int(1/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
Output:
int(sqrt(c + d*x**8)/(a**2*c + a**2*d*x**8 + 2*a*b*c*x**8 + 2*a*b*d*x**16 + b**2*c*x**16 + b**2*d*x**24),x)