\(\int \frac {1}{x^2 (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\) [88]

Optimal result

Integrand size = 24, antiderivative size = 62 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (-\frac {1}{6},2,\frac {1}{2},\frac {5}{6},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}} \] Output:

-(1+d*x^6/c)^(1/2)*AppellF1(-1/6,2,1/2,5/6,-b*x^6/a,-d*x^6/c)/a^2/x/(d*x^6 
+c)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(62)=124\).

Time = 10.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {55 a \left (c+d x^6\right ) \left (6 a^2 d-7 b^2 c x^6-6 a b \left (c-d x^6\right )\right )-11 \left (7 b^2 c^2-24 a b c d+12 a^2 d^2\right ) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {5}{6},\frac {1}{2},1,\frac {11}{6},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+10 b d (7 b c-6 a d) x^{12} \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {11}{6},\frac {1}{2},1,\frac {17}{6},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{330 a^3 c (b c-a d) x \left (a+b x^6\right ) \sqrt {c+d x^6}} \] Input:

Integrate[1/(x^2*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]
 

Output:

(55*a*(c + d*x^6)*(6*a^2*d - 7*b^2*c*x^6 - 6*a*b*(c - d*x^6)) - 11*(7*b^2* 
c^2 - 24*a*b*c*d + 12*a^2*d^2)*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF 
1[5/6, 1/2, 1, 11/6, -((d*x^6)/c), -((b*x^6)/a)] + 10*b*d*(7*b*c - 6*a*d)* 
x^12*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[11/6, 1/2, 1, 17/6, -((d*x^6 
)/c), -((b*x^6)/a)])/(330*a^3*c*(b*c - a*d)*x*(a + b*x^6)*Sqrt[c + d*x^6])
 

Rubi [A] (verified)

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.

\(\displaystyle \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {\sqrt {\frac {d x^6}{c}+1} \int \frac {1}{x^2 \left (b x^6+a\right )^2 \sqrt {\frac {d x^6}{c}+1}}dx}{\sqrt {c+d x^6}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\sqrt {\frac {d x^6}{c}+1} \operatorname {AppellF1}\left (-\frac {1}{6},2,\frac {1}{2},\frac {5}{6},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{a^2 x \sqrt {c+d x^6}}\)

Input:

Int[1/(x^2*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]
 

Output:

-((Sqrt[1 + (d*x^6)/c]*AppellF1[-1/6, 2, 1/2, 5/6, -((b*x^6)/a), -((d*x^6) 
/c)])/(a^2*x*Sqrt[c + d*x^6]))
 

Defintions of rubi rules used

rule 1012

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])
 

rule 1013

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])
 

Maple [F]

\[\int \frac {1}{x^{2} \left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}d x\]

Input:

int(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Output:

int(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{2}} \,d x } \] Input:

integrate(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")
 

Output:

integral(sqrt(d*x^6 + c)/(b^2*d*x^20 + (b^2*c + 2*a*b*d)*x^14 + (2*a*b*c + 
 a^2*d)*x^8 + a^2*c*x^2), x)
 

Sympy [F]

\[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \] Input:

integrate(1/x**2/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
 

Output:

Integral(1/(x**2*(a + b*x**6)**2*sqrt(c + d*x**6)), x)
 

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{2}} \,d x } \] Input:

integrate(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^2), x)
 

Giac [F]

\[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{2}} \,d x } \] Input:

integrate(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/((b*x^6 + a)^2*sqrt(d*x^6 + c)*x^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \] Input:

int(1/(x^2*(a + b*x^6)^2*(c + d*x^6)^(1/2)),x)
 

Output:

int(1/(x^2*(a + b*x^6)^2*(c + d*x^6)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^2 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {-\sqrt {d \,x^{6}+c}-4 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{10}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a b d x -4 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{10}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) b^{2} d \,x^{7}+2 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{4}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a^{2} d x -7 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{4}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a b c x +2 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{4}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a b d \,x^{7}-7 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{4}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) b^{2} c \,x^{7}}{a c x \left (b \,x^{6}+a \right )} \] Input:

int(1/x^2/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Output:

( - sqrt(c + d*x**6) - 4*int((sqrt(c + d*x**6)*x**10)/(a**2*c + a**2*d*x** 
6 + 2*a*b*c*x**6 + 2*a*b*d*x**12 + b**2*c*x**12 + b**2*d*x**18),x)*a*b*d*x 
 - 4*int((sqrt(c + d*x**6)*x**10)/(a**2*c + a**2*d*x**6 + 2*a*b*c*x**6 + 2 
*a*b*d*x**12 + b**2*c*x**12 + b**2*d*x**18),x)*b**2*d*x**7 + 2*int((sqrt(c 
 + d*x**6)*x**4)/(a**2*c + a**2*d*x**6 + 2*a*b*c*x**6 + 2*a*b*d*x**12 + b* 
*2*c*x**12 + b**2*d*x**18),x)*a**2*d*x - 7*int((sqrt(c + d*x**6)*x**4)/(a* 
*2*c + a**2*d*x**6 + 2*a*b*c*x**6 + 2*a*b*d*x**12 + b**2*c*x**12 + b**2*d* 
x**18),x)*a*b*c*x + 2*int((sqrt(c + d*x**6)*x**4)/(a**2*c + a**2*d*x**6 + 
2*a*b*c*x**6 + 2*a*b*d*x**12 + b**2*c*x**12 + b**2*d*x**18),x)*a*b*d*x**7 
- 7*int((sqrt(c + d*x**6)*x**4)/(a**2*c + a**2*d*x**6 + 2*a*b*c*x**6 + 2*a 
*b*d*x**12 + b**2*c*x**12 + b**2*d*x**18),x)*b**2*c*x**7)/(a*c*x*(a + b*x* 
*6))