\(\int x^3 \sqrt {a+b x^3+c x^6} \, dx\) [181]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int x^3 \sqrt {a+b x^3+c x^6} \, dx=\frac {x^4 \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {1}{2},-\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \] Output:

1/4*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)))/(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 [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(358\) vs. \(2(140)=280\).

Time = 10.61 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.56 \[ \int x^3 \sqrt {a+b x^3+c x^6} \, dx=\frac {x \left (8 \left (3 b+8 c x^3\right ) \left (a+b x^3+c x^6\right )-24 a b \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+3 \left (-5 b^2+16 a c\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{448 c \sqrt {a+b x^3+c x^6}} \] Input:

Integrate[x^3*Sqrt[a + b*x^3 + c*x^6],x]
 

Output:

(x*(8*(3*b + 8*c*x^3)*(a + b*x^3 + c*x^6) - 24*a*b*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])] + 3*(-5*b^2 + 16 
*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 + Sqr 
t[b^2 - 4*a*c])]))/(448*c*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1721, 1012}

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.

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

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

Int[x^3*Sqrt[a + b*x^3 + c*x^6],x]
 

Output:

(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])])/(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]) 
])
 

Defintions of rubi rules used

rule 1012

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

rule 1721

Int[((d_.)*(x_))^(m_.)*((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])))^FracPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c 
])))^p*(1 + 2*c*(x^n/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, 
 d, m, n, p}, x] && EqQ[n2, 2*n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \sqrt {a+b x^3+c x^6} \, dx=\frac {6 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b x +16 \sqrt {c \,x^{6}+b \,x^{3}+a}\, c \,x^{4}-6 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a b +48 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a c -15 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b^{2}}{112 c} \] Input:

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

Output:

(6*sqrt(a + b*x**3 + c*x**6)*b*x + 16*sqrt(a + b*x**3 + c*x**6)*c*x**4 - 6 
*int(sqrt(a + b*x**3 + c*x**6)/(a + b*x**3 + c*x**6),x)*a*b + 48*int((sqrt 
(a + b*x**3 + c*x**6)*x**3)/(a + b*x**3 + c*x**6),x)*a*c - 15*int((sqrt(a 
+ b*x**3 + c*x**6)*x**3)/(a + b*x**3 + c*x**6),x)*b**2)/(112*c)