\(\int \frac {1}{(a+b x^3+c x^6)^{3/2}} \, dx\) [230]

Optimal result

Integrand size = 16, antiderivative size = 138 \[ \int \frac {1}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {x \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},\frac {3}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{a \sqrt {a+b x^3+c x^6}} \] Output:

x*(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(1/3,3/2,3/2,4/3,-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/(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. \(359\) vs. \(2(138)=276\).

Time = 10.57 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.60 \[ \int \frac {1}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {x \left (-4 \left (b^2-2 a c+b c x^3\right )-2 \left (b^2-8 a c\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+b c x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{6 a \left (-b^2+4 a c\right ) \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1686, 936}

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.

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

\(\Big \downarrow \) 1686

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1686

Int[((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])))^FracPar 
t[p]))   Int[(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2*c*(x^n/(b - S 
qrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] 
&& NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

integrate(1/(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^12 + 2*b*c*x^9 + (b^2 + 2*a*c)*x^6 
 + 2*a*b*x^3 + a^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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