\(\int \frac {A+B x^3+C x^6}{\sqrt {a+b x^3+c x^6}} \, dx\) [4]

Optimal result

Integrand size = 29, antiderivative size = 328 \[ \int \frac {A+B x^3+C x^6}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {C x \sqrt {a+b x^3+c x^6}}{4 c}+\frac {(4 A c-a C) x \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 c \sqrt {a+b x^3+c x^6}}+\frac {(8 B c-5 b C) x^4 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{32 c \sqrt {a+b x^3+c x^6}} \] Output:

1/4*C*x*(c*x^6+b*x^3+a)^(1/2)/c+1/4*(4*A*c-C*a)*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(1/3,1/2 
,1/2,4/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/ 
c/(c*x^6+b*x^3+a)^(1/2)+1/32*(8*B*c-5*C*b)*x^4*(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(4/3,1/2,1/ 
2,7/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/c/( 
c*x^6+b*x^3+a)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 10.82 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x^3+C x^6}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {x \left (8 C \left (a+b x^3+c x^6\right )+8 (4 A c-a C) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+(8 B c-5 b C) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{32 c \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(x*(8*C*(a + b*x^3 + c*x^6) + 8*(4*A*c - 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 + Sqr 
t[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + (8*B*c - 5*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])]))/(32*c*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2321, 1762, 2009}

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.

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

\(\Big \downarrow \) 2321

\(\displaystyle \int \frac {\frac {1}{8} \left (8 B-\frac {5 b C}{c}\right ) x^3+\frac {1}{4} \left (4 A-\frac {a C}{c}\right )}{\sqrt {c x^6+b x^3+a}}dx+\frac {C x \sqrt {a+b x^3+c x^6}}{4 c}\)

\(\Big \downarrow \) 1762

\(\displaystyle \int \left (\frac {(8 B c-5 b C) x^3}{8 c \sqrt {c x^6+b x^3+a}}+\frac {4 A c-a C}{4 c \sqrt {c x^6+b x^3+a}}\right )dx+\frac {C x \sqrt {a+b x^3+c x^6}}{4 c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x (4 A c-a C) \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 c \sqrt {a+b x^3+c x^6}}+\frac {x^4 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1} (8 B c-5 b C) \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{32 c \sqrt {a+b x^3+c x^6}}+\frac {C x \sqrt {a+b x^3+c x^6}}{4 c}\)

Input:

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

Output:

(C*x*Sqrt[a + b*x^3 + c*x^6])/(4*c) + ((4*A*c - a*C)*x*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])]*Appel 
lF1[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])])/(4*c*Sqrt[a + b*x^3 + c*x^6]) + ((8*B*c - 5*b*C)*x^ 
4*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[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])])/(32*c*Sqrt[a + b*x^3 + c*x^6])
 

Defintions of rubi rules used

rule 1762

Int[((d_) + (e_.)*(x_)^(n_))*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p 
_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^n)*(a + b*x^n + c*x^(2*n))^p, 
 x], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 
 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2321

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> W 
ith[{q = Expon[Pq, x]}, With[{Pqq = Coeff[Pq, x, q]}, Int[ExpandToSum[Pq - 
Pqq*x^q - Pqq*((a*(q - 2*n + 1)*x^(q - 2*n) + b*(q + n*(p - 1) + 1)*x^(q - 
n))/(c*(q + 2*n*p + 1))), x]*(a + b*x^n + c*x^(2*n))^p, x] + Simp[Pqq*x^(q 
- 2*n + 1)*((a + b*x^n + c*x^(2*n))^(p + 1)/(c*(q + 2*n*p + 1))), x]] /; Ge 
Q[q, 2*n] && NeQ[q + 2*n*p + 1, 0] && (IntegerQ[2*p] || (EqQ[n, 1] && Integ 
erQ[4*p]) || IntegerQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, p}, x] && EqQ 
[n2, 2*n] && PolyQ[Pq, x^n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((C*x^6 + B*x^3 + A)/sqrt(c*x^6 + b*x^3 + a), x)
 

Sympy [F]

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

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

Output:

Integral((A + B*x**3 + C*x**6)/sqrt(a + b*x**3 + c*x**6), x)
 

Maxima [F]

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

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

Output:

integrate((C*x^6 + B*x^3 + A)/sqrt(c*x^6 + b*x^3 + a), x)
 

Giac [F]

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

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

Output:

integrate((C*x^6 + B*x^3 + A)/sqrt(c*x^6 + b*x^3 + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^3+C x^6}{\sqrt {a+b x^3+c x^6}} \, dx=\frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x}{4}+\frac {3 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a}{4}+\frac {3 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b}{8} \] Input:

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

Output:

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