\(\int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx\) [218]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=-\frac {\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2 \sqrt {a+b x^3+c x^6}} \] Output:

-1/2*(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(-2/3,1/2,1/2,1/3,-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^2/(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. \(342\) vs. \(2(140)=280\).

Time = 10.39 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\frac {-4 \left (a+b x^3+c x^6\right )-2 b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+c x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{8 a x^2 \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(-4*(a + b*x^3 + c*x^6) - 2*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[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a* 
c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 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[4/3, 1/2, 1/2, 7/3, (-2*c*x^3)/(b + 
Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])])/(8*a*x^2*Sqrt[a + 
 b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 140, 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.

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

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*x^6 + b*x^3 + a)/(c*x^9 + b*x^6 + a*x^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^3 \sqrt {a+b x^3+c x^6}} \, dx=\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{9}+b \,x^{6}+a \,x^{3}}d x \] Input:

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

Output:

int(sqrt(a + b*x**3 + c*x**6)/(a*x**3 + b*x**6 + c*x**9),x)