Integrand size = 24, antiderivative size = 62 \[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (-\frac {1}{6},1,\frac {1}{2},\frac {5}{6},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a x \sqrt {c+d x^6}} \] Output:
-(1+d*x^6/c)^(1/2)*AppellF1(-1/6,1,1/2,5/6,-b*x^6/a,-d*x^6/c)/a/x/(d*x^6+c )^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(62)=124\).
Time = 10.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.27 \[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\frac {-55 a \left (c+d x^6\right )-11 (b c-2 a d) x^6 \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+10 b d x^{12} \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {11}{6},\frac {1}{2},1,\frac {17}{6},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{55 a^2 c x \sqrt {c+d x^6}} \] Input:
Integrate[1/(x^2*(a + b*x^6)*Sqrt[c + d*x^6]),x]
Output:
(-55*a*(c + d*x^6) - 11*(b*c - 2*a*d)*x^6*Sqrt[1 + (d*x^6)/c]*AppellF1[5/6 , 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)] + 10*b*d*x^12*Sqrt[1 + (d*x^6) /c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6)/c), -((b*x^6)/a)])/(55*a^2*c*x* Sqrt[c + d*x^6])
Time = 0.34 (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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {\sqrt {\frac {d x^6}{c}+1} \int \frac {1}{x^2 \left (b x^6+a\right ) \sqrt {\frac {d x^6}{c}+1}}dx}{\sqrt {c+d x^6}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle -\frac {\sqrt {\frac {d x^6}{c}+1} \operatorname {AppellF1}\left (-\frac {1}{6},1,\frac {1}{2},\frac {5}{6},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a x \sqrt {c+d x^6}}\) |
Input:
Int[1/(x^2*(a + b*x^6)*Sqrt[c + d*x^6]),x]
Output:
-((Sqrt[1 + (d*x^6)/c]*AppellF1[-1/6, 1, 1/2, 5/6, -((b*x^6)/a), -((d*x^6) /c)])/(a*x*Sqrt[c + d*x^6]))
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])
\[\int \frac {1}{x^{2} \left (b \,x^{6}+a \right ) \sqrt {d \,x^{6}+c}}d x\]
Input:
int(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x)
Output:
int(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x)
\[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(d*x^6 + c)/(b*d*x^14 + (b*c + a*d)*x^8 + a*c*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{6}\right ) \sqrt {c + d x^{6}}}\, dx \] Input:
integrate(1/x**2/(b*x**6+a)/(d*x**6+c)**(1/2),x)
Output:
Integral(1/(x**2*(a + b*x**6)*sqrt(c + d*x**6)), x)
\[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )} \sqrt {d x^{6} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^6 + a)*sqrt(d*x^6 + c)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^2\,\left (b\,x^6+a\right )\,\sqrt {d\,x^6+c}} \,d x \] Input:
int(1/(x^2*(a + b*x^6)*(c + d*x^6)^(1/2)),x)
Output:
int(1/(x^2*(a + b*x^6)*(c + d*x^6)^(1/2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \, dx=\int \frac {\sqrt {d \,x^{6}+c}}{b d \,x^{14}+a d \,x^{8}+b c \,x^{8}+a c \,x^{2}}d x \] Input:
int(1/x^2/(b*x^6+a)/(d*x^6+c)^(1/2),x)
Output:
int(sqrt(c + d*x**6)/(a*c*x**2 + a*d*x**8 + b*c*x**8 + b*d*x**14),x)