\(\int \frac {A+B x^3+C x^6}{(a+b x^3+c x^6)^{3/2}} \, dx\) [5]

Optimal result

Integrand size = 29, antiderivative size = 415 \[ \int \frac {A+B x^3+C x^6}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {2 x \left (A b^2-a b B-2 a A c+2 a^2 C+(A b c-2 a B c+a b C) x^3\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}+\frac {\left (A \left (b^2-8 a c\right )+2 a (b B-2 a C)\right ) x \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {(A b c-2 a B c+a b C) x^4 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{6 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}} \] Output:

2/3*x*(A*b^2-a*b*B-2*a*A*c+2*a^2*C+(A*b*c-2*B*a*c+C*a*b)*x^3)/a/(-4*a*c+b^ 
2)/(c*x^6+b*x^3+a)^(1/2)+1/3*(A*(-8*a*c+b^2)+2*a*(B*b-2*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)*Ap 
pellF1(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)))/a/(-4*a*c+b^2)/(c*x^6+b*x^3+a)^(1/2)-1/6*(A*b*c-2*B*a*c+C*a* 
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)))/a/(-4*a*c+b^2)/(c*x^6+b*x^3+a)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 11.09 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x^3+C x^6}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx=\frac {x \left (-4 \left (A \left (b^2-2 a c+b c x^3\right )+a \left (-b B+2 a C-2 B c x^3+b C x^3\right )\right )-2 \left (A \left (b^2-8 a c\right )+2 a (b B-2 a C)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+(A b c-2 a B c+a b C) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{6 a \left (-b^2+4 a c\right ) \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(x*(-4*(A*(b^2 - 2*a*c + b*c*x^3) + a*(-(b*B) + 2*a*C - 2*B*c*x^3 + b*C*x^ 
3)) - 2*(A*(b^2 - 8*a*c) + 2*a*(b*B - 2*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])] + (A*b*c - 2*a*B*c + 
a*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 + Sq 
rt[b^2 - 4*a*c])]))/(6*a*(-b^2 + 4*a*c)*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2317, 27, 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.

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

\(\Big \downarrow \) 2317

\(\displaystyle \frac {2 x \left (A \left (b^2-2 a c\right )+x^3 (a b C-2 a B c+A b c)-a (b B-2 a C)\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}-\frac {2 \int -\frac {b^2 c^2 \left (-2 (A b c-2 a B c+a b C) x^3+A \left (b^2-8 a c\right )+2 a (b B-2 a C)\right )}{2 \sqrt {c x^6+b x^3+a}}dx}{3 a b^2 c^2 \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {-2 (A b c-2 a B c+a b C) x^3+A \left (b^2-8 a c\right )+2 a (b B-2 a C)}{\sqrt {c x^6+b x^3+a}}dx}{3 a \left (b^2-4 a c\right )}+\frac {2 x \left (A \left (b^2-2 a c\right )+x^3 (a b C-2 a B c+A b c)-a (b B-2 a C)\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\)

\(\Big \downarrow \) 1762

\(\displaystyle \frac {\int \left (\frac {A \left (b^2-8 a c\right ) \left (\frac {2 a (b B-2 a C)}{A \left (b^2-8 a c\right )}+1\right )}{\sqrt {c x^6+b x^3+a}}-\frac {2 (A b c-2 a B c+a b C) x^3}{\sqrt {c x^6+b x^3+a}}\right )dx}{3 a \left (b^2-4 a c\right )}+\frac {2 x \left (A \left (b^2-2 a c\right )+x^3 (a b C-2 a B c+A b c)-a (b B-2 a C)\right )}{3 a \left (b^2-4 a c\right ) \sqrt {a+b x^3+c x^6}}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*x*(A*(b^2 - 2*a*c) - a*(b*B - 2*a*C) + (A*b*c - 2*a*B*c + a*b*C)*x^3))/ 
(3*a*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) + (((A*(b^2 - 8*a*c) + 2*a*(b* 
B - 2*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])]*AppellF1[1/3, 1/2, 1/2, 4/3, (-2*c*x^3)/(b - S 
qrt[b^2 - 4*a*c]), (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[a + b*x^3 + c 
*x^6] - ((A*b*c - 2*a*B*c + a*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])])/(2*Sqrt[a + b*x^3 + c*x^6]))/(3*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2317

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Wi 
th[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[(b*c)^(Floor[(q - 1)/ 
n] + 1)*Pq, a + b*x^n + c*x^(2*n), x], R = PolynomialRemainder[(b*c)^(Floor 
[(q - 1)/n] + 1)*Pq, a + b*x^n + c*x^(2*n), x], i}, Simp[1/(a*n*(p + 1)*(b^ 
2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1))   Int[(a + b*x^n + c*x^(2*n))^(p + 
 1)*ExpandToSum[a*n*(p + 1)*(b^2 - 4*a*c)*Q + Sum[((b^2*(n*(p + 1) + i + 1) 
 - 2*a*c*(2*n*(p + 1) + i + 1))*Coeff[R, x, i] - a*b*(i + 1)*Coeff[R, x, n 
+ i])*x^i + c*(n*(2*p + 3) + i + 1)*(b*Coeff[R, x, i] - 2*a*Coeff[R, x, n + 
 i])*x^(n + i), {i, 0, n - 1}], x], x], x] + Simp[(-x)*((a + b*x^n + c*x^(2 
*n))^(p + 1)/(a*n*(p + 1)*(b^2 - 4*a*c)*(b*c)^(Floor[(q - 1)/n] + 1)))*Sum[ 
((b^2 - 2*a*c)*Coeff[R, x, i] - a*b*Coeff[R, x, n + i])*x^i + c*(b*Coeff[R, 
 x, i] - 2*a*Coeff[R, x, n + i])*x^(n + i), {i, 0, n - 1}], x]] /; GeQ[q, 2 
*n]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4* 
a*c, 0] && IGtQ[n, 0] && LtQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((C*x^6 + B*x^3 + A)*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(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

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

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

Output:

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