\(\int \frac {1}{x^3 (a+b x^6)^2 \sqrt {c+d x^6}} \, dx\) [89]

Optimal result

Integrand size = 24, antiderivative size = 64 \[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=-\frac {\sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (-\frac {1}{3},2,\frac {1}{2},\frac {2}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 x^2 \sqrt {c+d x^6}} \] Output:

-1/2*(1+d*x^6/c)^(1/2)*AppellF1(-1/3,2,1/2,2/3,-b*x^6/a,-d*x^6/c)/a^2/x^2/ 
(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. \(2(64)=128\).

Time = 10.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.53 \[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {20 a \left (c+d x^6\right ) \left (3 a^2 d-4 b^2 c x^6-3 a b \left (c-d x^6\right )\right )-5 \left (8 b^2 c^2-15 a b c d+3 a^2 d^2\right ) x^6 \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )+2 b d (4 b c-3 a d) x^{12} \left (a+b x^6\right ) \sqrt {1+\frac {d x^6}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^6}{c},-\frac {b x^6}{a}\right )}{120 a^3 c (b c-a d) x^2 \left (a+b x^6\right ) \sqrt {c+d x^6}} \] Input:

Integrate[1/(x^3*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]
 

Output:

(20*a*(c + d*x^6)*(3*a^2*d - 4*b^2*c*x^6 - 3*a*b*(c - d*x^6)) - 5*(8*b^2*c 
^2 - 15*a*b*c*d + 3*a^2*d^2)*x^6*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[ 
2/3, 1/2, 1, 5/3, -((d*x^6)/c), -((b*x^6)/a)] + 2*b*d*(4*b*c - 3*a*d)*x^12 
*(a + b*x^6)*Sqrt[1 + (d*x^6)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^6)/c), 
-((b*x^6)/a)])/(120*a^3*c*(b*c - a*d)*x^2*(a + b*x^6)*Sqrt[c + d*x^6])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {965, 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.

\(\displaystyle \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx\)

\(\Big \downarrow \) 965

\(\displaystyle \frac {1}{2} \int \frac {1}{x^4 \left (b x^6+a\right )^2 \sqrt {d x^6+c}}dx^2\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {\sqrt {\frac {d x^6}{c}+1} \int \frac {1}{x^4 \left (b x^6+a\right )^2 \sqrt {\frac {d x^6}{c}+1}}dx^2}{2 \sqrt {c+d x^6}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\sqrt {\frac {d x^6}{c}+1} \operatorname {AppellF1}\left (-\frac {1}{3},2,\frac {1}{2},\frac {2}{3},-\frac {b x^6}{a},-\frac {d x^6}{c}\right )}{2 a^2 x^2 \sqrt {c+d x^6}}\)

Input:

Int[1/(x^3*(a + b*x^6)^2*Sqrt[c + d*x^6]),x]
 

Output:

-1/2*(Sqrt[1 + (d*x^6)/c]*AppellF1[-1/3, 2, 1/2, 2/3, -((b*x^6)/a), -((d*x 
^6)/c)])/(a^2*x^2*Sqrt[c + d*x^6])
 

Defintions of rubi rules used

rule 965

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 
 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free 
Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
 

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]

\[\int \frac {1}{x^{3} \left (b \,x^{6}+a \right )^{2} \sqrt {d \,x^{6}+c}}d x\]

Input:

int(1/x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Output:

int(1/x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\text {Timed out} \] Input:

integrate(1/x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^{3} \left (a + b x^{6}\right )^{2} \sqrt {c + d x^{6}}}\, dx \] Input:

integrate(1/x**3/(b*x**6+a)**2/(d*x**6+c)**(1/2),x)
 

Output:

Integral(1/(x**3*(a + b*x**6)**2*sqrt(c + d*x**6)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(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^3), x)
 

Giac [F]

\[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int { \frac {1}{{\left (b x^{6} + a\right )}^{2} \sqrt {d x^{6} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(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^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\int \frac {1}{x^3\,{\left (b\,x^6+a\right )}^2\,\sqrt {d\,x^6+c}} \,d x \] Input:

int(1/(x^3*(a + b*x^6)^2*(c + d*x^6)^(1/2)),x)
 

Output:

int(1/(x^3*(a + b*x^6)^2*(c + d*x^6)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^3 \left (a+b x^6\right )^2 \sqrt {c+d x^6}} \, dx=\frac {-\sqrt {d \,x^{6}+c}-5 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{9}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a b d \,x^{2}-5 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{9}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) b^{2} d \,x^{8}+\left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{3}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a^{2} d \,x^{2}-8 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{3}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a b c \,x^{2}+\left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{3}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) a b d \,x^{8}-8 \left (\int \frac {\sqrt {d \,x^{6}+c}\, x^{3}}{b^{2} d \,x^{18}+2 a b d \,x^{12}+b^{2} c \,x^{12}+a^{2} d \,x^{6}+2 a b c \,x^{6}+a^{2} c}d x \right ) b^{2} c \,x^{8}}{2 a c \,x^{2} \left (b \,x^{6}+a \right )} \] Input:

int(1/x^3/(b*x^6+a)^2/(d*x^6+c)^(1/2),x)
 

Output:

( - sqrt(c + d*x**6) - 5*int((sqrt(c + d*x**6)*x**9)/(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* 
*2 - 5*int((sqrt(c + d*x**6)*x**9)/(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**8 + int((sqrt(c 
+ d*x**6)*x**3)/(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**2 - 8*int((sqrt(c + d*x**6)*x**3)/( 
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**3)/(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* 
*8 - 8*int((sqrt(c + d*x**6)*x**3)/(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**8)/(2*a*c*x**2*( 
a + b*x**6))