\(\int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx\) [217]

Optimal result

Integrand size = 20, antiderivative size = 138 \[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {a+b x^3+c x^6}} \] Output:

-(1+2*c*x^3/(b-(-4*a*c+b^2)^(1/2)))^(1/2)*(1+2*c*x^3/(b+(-4*a*c+b^2)^(1/2) 
))^(1/2)*AppellF1(-1/3,1/2,1/2,2/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^ 
3/(b+(-4*a*c+b^2)^(1/2)))/x/(c*x^6+b*x^3+a)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(343\) vs. \(2(138)=276\).

Time = 10.41 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.49 \[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\frac {-20 \left (a+b x^3+c x^6\right )+5 b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+8 c x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},\frac {1}{2},\frac {8}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{20 a x \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1721, 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.

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

\(\Big \downarrow \) 1721

\(\displaystyle \frac {\sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} \int \frac {1}{x^2 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}+1}}dx}{\sqrt {a+b x^3+c x^6}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{x \sqrt {a+b x^3+c x^6}}\)

Input:

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

Output:

-((Sqrt[1 + (2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^3)/(b + Sqr 
t[b^2 - 4*a*c])]*AppellF1[-1/3, 1/2, 1/2, 2/3, (-2*c*x^3)/(b - Sqrt[b^2 - 
4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(x*Sqrt[a + b*x^3 + c*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 1721

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x 
_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2* 
c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^2 - 4 
*a*c, 2])))^FracPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c 
])))^p*(1 + 2*c*(x^n/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, 
 d, m, n, p}, x] && EqQ[n2, 2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*x^6 + b*x^3 + a)/(c*x^8 + b*x^5 + a*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{8}+b \,x^{5}+a \,x^{2}}d x \] Input:

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

Output:

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