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

Optimal result

Integrand size = 29, antiderivative size = 328 \[ \int \sqrt {a+b x^3+c x^6} \left (A+B x^3+C x^6\right ) \, dx=\frac {C x \left (a+b x^3+c x^6\right )^{3/2}}{10 c}+\frac {(10 A c-a C) x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{10 c \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}}+\frac {(20 B c-11 b C) x^4 \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},-\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{80 c \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \] Output:

1/10*C*x*(c*x^6+b*x^3+a)^(3/2)/c+1/10*(10*A*c-C*a)*x*(c*x^6+b*x^3+a)^(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/(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)+1/80*(20*B*c-11*C*b)*x^4*(c*x^6+b*x^3+a)^(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/(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)
 

Mathematica [A] (warning: unable to verify)

Time = 11.24 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.35 \[ \int \sqrt {a+b x^3+c x^6} \left (A+B x^3+C x^6\right ) \, dx=\frac {x \left (8 \left (a+b x^3+c x^6\right ) \left (-33 b^2 C+12 b c \left (5 B+2 C x^3\right )+4 c \left (70 A c+21 a C+40 B c x^3+28 c C x^6\right )\right )+24 a \left (-20 b B c+11 b^2 C+28 c (10 A c-a C)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+3 \left (-100 b^2 B c+320 a B c^2+55 b^3 C+4 b c (70 A c-51 a C)\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{8960 c^2 \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(x*(8*(a + b*x^3 + c*x^6)*(-33*b^2*C + 12*b*c*(5*B + 2*C*x^3) + 4*c*(70*A* 
c + 21*a*C + 40*B*c*x^3 + 28*c*C*x^6)) + 24*a*(-20*b*B*c + 11*b^2*C + 28*c 
*(10*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])]*App 
ellF1[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])] + 3*(-100*b^2*B*c + 320*a*B*c^2 + 55*b^3*C + 4*b*c 
*(70*A*c - 51*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[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])]))/(8960*c^2*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.54 (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.

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

\(\Big \downarrow \) 2321

\(\displaystyle \int \left (\frac {1}{20} \left (20 B-\frac {11 b C}{c}\right ) x^3+\frac {1}{10} \left (10 A-\frac {a C}{c}\right )\right ) \sqrt {c x^6+b x^3+a}dx+\frac {C x \left (a+b x^3+c x^6\right )^{3/2}}{10 c}\)

\(\Big \downarrow \) 1762

\(\displaystyle \int \left (\frac {(20 B c-11 b C) \sqrt {c x^6+b x^3+a} x^3}{20 c}+\frac {(10 A c-a C) \sqrt {c x^6+b x^3+a}}{10 c}\right )dx+\frac {C x \left (a+b x^3+c x^6\right )^{3/2}}{10 c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x (10 A c-a C) \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{10 c \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}}+\frac {x^4 \sqrt {a+b x^3+c x^6} (20 B c-11 b C) \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},-\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{80 c \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}}+\frac {C x \left (a+b x^3+c x^6\right )^{3/2}}{10 c}\)

Input:

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

Output:

(C*x*(a + b*x^3 + c*x^6)^(3/2))/(10*c) + ((10*A*c - a*C)*x*Sqrt[a + b*x^3 
+ c*x^6]*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])])/(10*c*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])]) + ((20*B*c - 1 
1*b*C)*x^4*Sqrt[a + b*x^3 + c*x^6]*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])])/(80*c*Sqr 
t[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])])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

integrate((c*x^6+b*x^3+a)^(1/2)*(C*x^6+B*x^3+A),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. \[ \int \sqrt {a+b x^3+c x^6} \left (A+B x^3+C x^6\right ) \, dx=\int \left (C\,x^6+B\,x^3+A\right )\,\sqrt {c\,x^6+b\,x^3+a} \,d x \] 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]

\[ \int \sqrt {a+b x^3+c x^6} \left (A+B x^3+C x^6\right ) \, dx=\frac {728 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a c x +54 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{2} x +368 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b c \,x^{4}+224 \sqrt {c \,x^{6}+b \,x^{3}+a}\, c^{2} x^{7}+1512 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a^{2} c -54 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a \,b^{2}+1188 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a b c -135 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b^{3}}{2240 c} \] Input:

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

Output:

(728*sqrt(a + b*x**3 + c*x**6)*a*c*x + 54*sqrt(a + b*x**3 + c*x**6)*b**2*x 
 + 368*sqrt(a + b*x**3 + c*x**6)*b*c*x**4 + 224*sqrt(a + b*x**3 + c*x**6)* 
c**2*x**7 + 1512*int(sqrt(a + b*x**3 + c*x**6)/(a + b*x**3 + c*x**6),x)*a* 
*2*c - 54*int(sqrt(a + b*x**3 + c*x**6)/(a + b*x**3 + c*x**6),x)*a*b**2 + 
1188*int((sqrt(a + b*x**3 + c*x**6)*x**3)/(a + b*x**3 + c*x**6),x)*a*b*c - 
 135*int((sqrt(a + b*x**3 + c*x**6)*x**3)/(a + b*x**3 + c*x**6),x)*b**3)/( 
2240*c)