Integrand size = 24, antiderivative size = 62 \[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=-\frac {\sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (-\frac {1}{8},2,\frac {1}{2},\frac {7}{8},-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 x \sqrt {c+d x^8}} \] Output:
-(1+d*x^8/c)^(1/2)*AppellF1(-1/8,2,1/2,7/8,-b*x^8/a,-d*x^8/c)/a^2/x/(d*x^8 +c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(62)=124\).
Time = 10.41 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\frac {35 a \left (c+d x^8\right ) \left (8 a^2 d-9 b^2 c x^8-8 a b \left (c-d x^8\right )\right )-5 \left (9 b^2 c^2-40 a b c d+24 a^2 d^2\right ) x^8 \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {7}{8},\frac {1}{2},1,\frac {15}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )+7 b d (9 b c-8 a d) x^{16} \left (a+b x^8\right ) \sqrt {1+\frac {d x^8}{c}} \operatorname {AppellF1}\left (\frac {15}{8},\frac {1}{2},1,\frac {23}{8},-\frac {d x^8}{c},-\frac {b x^8}{a}\right )}{280 a^3 c (b c-a d) x \left (a+b x^8\right ) \sqrt {c+d x^8}} \] Input:
Integrate[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]
Output:
(35*a*(c + d*x^8)*(8*a^2*d - 9*b^2*c*x^8 - 8*a*b*(c - d*x^8)) - 5*(9*b^2*c ^2 - 40*a*b*c*d + 24*a^2*d^2)*x^8*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1 [7/8, 1/2, 1, 15/8, -((d*x^8)/c), -((b*x^8)/a)] + 7*b*d*(9*b*c - 8*a*d)*x^ 16*(a + b*x^8)*Sqrt[1 + (d*x^8)/c]*AppellF1[15/8, 1/2, 1, 23/8, -((d*x^8)/ c), -((b*x^8)/a)])/(280*a^3*c*(b*c - a*d)*x*(a + b*x^8)*Sqrt[c + d*x^8])
Time = 0.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1013, 1012}
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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {\frac {d x^8}{c}+1} \int \frac {1}{x^2 \left (b x^8+a\right )^2 \sqrt {\frac {d x^8}{c}+1}}dx}{\sqrt {c+d x^8}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {\sqrt {\frac {d x^8}{c}+1} \operatorname {AppellF1}\left (-\frac {1}{8},2,\frac {1}{2},\frac {7}{8},-\frac {b x^8}{a},-\frac {d x^8}{c}\right )}{a^2 x \sqrt {c+d x^8}}\) |
Input:
Int[1/(x^2*(a + b*x^8)^2*Sqrt[c + d*x^8]),x]
Output:
-((Sqrt[1 + (d*x^8)/c]*AppellF1[-1/8, 2, 1/2, 7/8, -((b*x^8)/a), -((d*x^8) /c)])/(a^2*x*Sqrt[c + d*x^8]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
\[\int \frac {1}{x^{2} \left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x\]
Input:
int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
Output:
int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
\[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(d*x^8 + c)/(b^2*d*x^26 + (b^2*c + 2*a*b*d)*x^18 + (2*a*b*c + a^2*d)*x^10 + a^2*c*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{8}\right )^{2} \sqrt {c + d x^{8}}}\, dx \] Input:
integrate(1/x**2/(b*x**8+a)**2/(d*x**8+c)**(1/2),x)
Output:
Integral(1/(x**2*(a + b*x**8)**2*sqrt(c + d*x**8)), x)
\[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(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^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )}^{2} \sqrt {d x^{8} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(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^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^8+a\right )}^2\,\sqrt {d\,x^8+c}} \,d x \] Input:
int(1/(x^2*(a + b*x^8)^2*(c + d*x^8)^(1/2)),x)
Output:
int(1/(x^2*(a + b*x^8)^2*(c + d*x^8)^(1/2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^8\right )^2 \sqrt {c+d x^8}} \, dx=\int \frac {1}{x^{2} \left (b \,x^{8}+a \right )^{2} \sqrt {d \,x^{8}+c}}d x \] Input:
int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)
Output:
int(1/x^2/(b*x^8+a)^2/(d*x^8+c)^(1/2),x)