\(\int (a+b x^3+c x^6)^{3/2} (A+B x^3+C x^6) \, dx\) [2]

Optimal result

Integrand size = 29, antiderivative size = 330 \[ \int \left (a+b x^3+c x^6\right )^{3/2} \left (A+B x^3+C x^6\right ) \, dx=\frac {C x \left (a+b x^3+c x^6\right )^{5/2}}{16 c}+\frac {a (16 A c-a C) x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{16 c \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}}+\frac {a (32 B c-17 b C) x^4 \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {3}{2},-\frac {3}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{128 c \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \] Output:

1/16*C*x*(c*x^6+b*x^3+a)^(5/2)/c+1/16*a*(16*A*c-C*a)*x*(c*x^6+b*x^3+a)^(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)))/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)+1/128*a*(32*B*c-17*C*b)*x^4*(c*x^6+b*x^3+a) 
^(1/2)*AppellF1(4/3,-3/2,-3/2,7/3,-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 = 11.97 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.82 \[ \int \left (a+b x^3+c x^6\right )^{3/2} \left (A+B x^3+C x^6\right ) \, dx=\frac {x \left (8 \left (a+b x^3+c x^6\right ) \left (5049 b^4 C-216 b^3 c \left (44 B+17 C x^3\right )+216 b^2 c \left (104 A c-151 a C+32 B c x^3+14 c C x^6\right )+32 b c^2 \left (1836 a B+4784 A c x^3+621 a C x^3+3248 B c x^6+2450 c C x^9\right )+16 c^2 \left (2457 a^2 C+56 c^2 x^6 \left (104 A+80 B x^3+65 C x^6\right )+4 a c \left (4732 A+2560 B x^3+1729 C x^6\right )\right )\right )-216 a \left (-352 b^3 B c+2176 a b B c^2+187 b^4 C+8 b^2 c (104 A c-151 a C)+1456 a c^2 (-16 A c+a C)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )-27 \left (-1760 b^4 B c+12928 a b^2 B c^2-20480 a^2 B c^3+935 b^5 C+8 b^3 c (520 A c-891 a C)+16 a b c^2 (-2288 A c+823 a C)\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{7454720 c^3 \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(x*(8*(a + b*x^3 + c*x^6)*(5049*b^4*C - 216*b^3*c*(44*B + 17*C*x^3) + 216* 
b^2*c*(104*A*c - 151*a*C + 32*B*c*x^3 + 14*c*C*x^6) + 32*b*c^2*(1836*a*B + 
 4784*A*c*x^3 + 621*a*C*x^3 + 3248*B*c*x^6 + 2450*c*C*x^9) + 16*c^2*(2457* 
a^2*C + 56*c^2*x^6*(104*A + 80*B*x^3 + 65*C*x^6) + 4*a*c*(4732*A + 2560*B* 
x^3 + 1729*C*x^6))) - 216*a*(-352*b^3*B*c + 2176*a*b*B*c^2 + 187*b^4*C + 8 
*b^2*c*(104*A*c - 151*a*C) + 1456*a*c^2*(-16*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 + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] - 27*(-176 
0*b^4*B*c + 12928*a*b^2*B*c^2 - 20480*a^2*B*c^3 + 935*b^5*C + 8*b^3*c*(520 
*A*c - 891*a*C) + 16*a*b*c^2*(-2288*A*c + 823*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])]))/(7454720* 
c^3*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 330, 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.

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

\(\Big \downarrow \) 2321

\(\displaystyle \int \left (\frac {1}{32} \left (32 B-\frac {17 b C}{c}\right ) x^3+\frac {1}{16} \left (16 A-\frac {a C}{c}\right )\right ) \left (c x^6+b x^3+a\right )^{3/2}dx+\frac {C x \left (a+b x^3+c x^6\right )^{5/2}}{16 c}\)

\(\Big \downarrow \) 1762

\(\displaystyle \int \left (\frac {(32 B c-17 b C) \left (c x^6+b x^3+a\right )^{3/2} x^3}{32 c}+\frac {(16 A c-a C) \left (c x^6+b x^3+a\right )^{3/2}}{16 c}\right )dx+\frac {C x \left (a+b x^3+c x^6\right )^{5/2}}{16 c}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a x (16 A c-a C) \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{16 c \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}}+\frac {a x^4 \sqrt {a+b x^3+c x^6} (32 B c-17 b C) \operatorname {AppellF1}\left (\frac {4}{3},-\frac {3}{2},-\frac {3}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{128 c \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}}+\frac {C x \left (a+b x^3+c x^6\right )^{5/2}}{16 c}\)

Input:

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

Output:

(C*x*(a + b*x^3 + c*x^6)^(5/2))/(16*c) + (a*(16*A*c - a*C)*x*Sqrt[a + b*x^ 
3 + c*x^6]*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])])/(16*c*Sqrt[1 + (2*c*x^3)/(b - Sqr 
t[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])]) + (a*(32*B*c 
 - 17*b*C)*x^4*Sqrt[a + b*x^3 + c*x^6]*AppellF1[4/3, -3/2, -3/2, 7/3, (-2* 
c*x^3)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/(128* 
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])])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((C*c*x^12 + (C*b + B*c)*x^9 + (C*a + B*b + A*c)*x^6 + (B*a + A*b) 
*x^3 + A*a)*sqrt(c*x^6 + b*x^3 + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b x^3+c x^6\right )^{3/2} \left (A+B x^3+C x^6\right ) \, dx=\frac {136864 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a^{2} c^{2} x +19440 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a \,b^{2} c x +134720 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a b \,c^{2} x^{4}+81536 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a \,c^{3} x^{7}-1782 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{4} x +1296 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{3} c \,x^{4}+42784 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{2} c^{2} x^{7}+60032 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b \,c^{3} x^{10}+23296 \sqrt {c \,x^{6}+b \,x^{3}+a}\, c^{4} x^{13}+235872 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a^{3} c^{2}-19440 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a^{2} b^{2} c +1782 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a \,b^{4}+237168 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a^{2} b \,c^{2}-53784 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a \,b^{3} c +4455 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b^{5}}{372736 c^{2}} \] Input:

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

Output:

(136864*sqrt(a + b*x**3 + c*x**6)*a**2*c**2*x + 19440*sqrt(a + b*x**3 + c* 
x**6)*a*b**2*c*x + 134720*sqrt(a + b*x**3 + c*x**6)*a*b*c**2*x**4 + 81536* 
sqrt(a + b*x**3 + c*x**6)*a*c**3*x**7 - 1782*sqrt(a + b*x**3 + c*x**6)*b** 
4*x + 1296*sqrt(a + b*x**3 + c*x**6)*b**3*c*x**4 + 42784*sqrt(a + b*x**3 + 
 c*x**6)*b**2*c**2*x**7 + 60032*sqrt(a + b*x**3 + c*x**6)*b*c**3*x**10 + 2 
3296*sqrt(a + b*x**3 + c*x**6)*c**4*x**13 + 235872*int(sqrt(a + b*x**3 + c 
*x**6)/(a + b*x**3 + c*x**6),x)*a**3*c**2 - 19440*int(sqrt(a + b*x**3 + c* 
x**6)/(a + b*x**3 + c*x**6),x)*a**2*b**2*c + 1782*int(sqrt(a + b*x**3 + c* 
x**6)/(a + b*x**3 + c*x**6),x)*a*b**4 + 237168*int((sqrt(a + b*x**3 + c*x* 
*6)*x**3)/(a + b*x**3 + c*x**6),x)*a**2*b*c**2 - 53784*int((sqrt(a + b*x** 
3 + c*x**6)*x**3)/(a + b*x**3 + c*x**6),x)*a*b**3*c + 4455*int((sqrt(a + b 
*x**3 + c*x**6)*x**3)/(a + b*x**3 + c*x**6),x)*b**5)/(372736*c**2)