\(\int \frac {1}{x^2 (a+b x^3+c x^6)^{3/2}} \, dx\) [231]

Optimal result

Integrand size = 20, antiderivative size = 141 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (-\frac {1}{3},\frac {3}{2},\frac {3}{2},\frac {2}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{a x \sqrt {a+b x^3+c x^6}} \] 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(-1/3,3/2,3/2,2/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^ 
3/(b+(-4*a*c+b^2)^(1/2)))/a/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. \(407\) vs. \(2(141)=282\).

Time = 10.91 (sec) , antiderivative size = 407, normalized size of antiderivative = 2.89 \[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )^{3/2}} \, dx=-\frac {5 b \left (-5 b^2+12 a c\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {5}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )-4 \left (60 a^2 c-25 b^2 x^3 \left (b+c x^3\right )+5 a \left (-3 b^2+18 b c x^3+16 c^2 x^6\right )+2 c \left (5 b^2-16 a c\right ) x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},\frac {1}{2},\frac {8}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{60 a^2 \left (b^2-4 a c\right ) x \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

-1/60*(5*b*(-5*b^2 + 12*a*c)*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])] - 4*(60*a^2*c - 25*b^2*x^3*(b + c* 
x^3) + 5*a*(-3*b^2 + 18*b*c*x^3 + 16*c^2*x^6) + 2*c*(5*b^2 - 16*a*c)*x^6*S 
qrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^2 - 4*a*c])]*Sqrt[(b + S 
qrt[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])]))/(a^2*(b^2 - 4*a*c)*x*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 141, 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.

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

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*x^6 + b*x^3 + a)/(c^2*x^14 + 2*b*c*x^11 + (b^2 + 2*a*c)*x^ 
8 + 2*a*b*x^5 + a^2*x^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x^2 \left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {-2 \sqrt {c \,x^{6}+b \,x^{3}+a}-8 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{4}}{c^{2} x^{12}+2 b c \,x^{9}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a c x -8 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{4}}{c^{2} x^{12}+2 b c \,x^{9}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b c \,x^{4}-8 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{4}}{c^{2} x^{12}+2 b c \,x^{9}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) c^{2} x^{7}-5 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x}{c^{2} x^{12}+2 b c \,x^{9}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) a b x -5 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x}{c^{2} x^{12}+2 b c \,x^{9}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b^{2} x^{4}-5 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x}{c^{2} x^{12}+2 b c \,x^{9}+2 a c \,x^{6}+b^{2} x^{6}+2 a b \,x^{3}+a^{2}}d x \right ) b c \,x^{7}}{2 a x \left (c \,x^{6}+b \,x^{3}+a \right )} \] Input:

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

Output:

( - 2*sqrt(a + b*x**3 + c*x**6) - 8*int((sqrt(a + b*x**3 + c*x**6)*x**4)/( 
a**2 + 2*a*b*x**3 + 2*a*c*x**6 + b**2*x**6 + 2*b*c*x**9 + c**2*x**12),x)*a 
*c*x - 8*int((sqrt(a + b*x**3 + c*x**6)*x**4)/(a**2 + 2*a*b*x**3 + 2*a*c*x 
**6 + b**2*x**6 + 2*b*c*x**9 + c**2*x**12),x)*b*c*x**4 - 8*int((sqrt(a + b 
*x**3 + c*x**6)*x**4)/(a**2 + 2*a*b*x**3 + 2*a*c*x**6 + b**2*x**6 + 2*b*c* 
x**9 + c**2*x**12),x)*c**2*x**7 - 5*int((sqrt(a + b*x**3 + c*x**6)*x)/(a** 
2 + 2*a*b*x**3 + 2*a*c*x**6 + b**2*x**6 + 2*b*c*x**9 + c**2*x**12),x)*a*b* 
x - 5*int((sqrt(a + b*x**3 + c*x**6)*x)/(a**2 + 2*a*b*x**3 + 2*a*c*x**6 + 
b**2*x**6 + 2*b*c*x**9 + c**2*x**12),x)*b**2*x**4 - 5*int((sqrt(a + b*x**3 
 + c*x**6)*x)/(a**2 + 2*a*b*x**3 + 2*a*c*x**6 + b**2*x**6 + 2*b*c*x**9 + c 
**2*x**12),x)*b*c*x**7)/(2*a*x*(a + b*x**3 + c*x**6))