\(\int x^3 (a+b x^3+c x^6)^{3/2} \, dx\) [199]

Optimal result

Integrand size = 20, antiderivative size = 141 \[ \int x^3 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {a x^4 \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {4}{3},-\frac {3}{2},-\frac {3}{2},\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{4 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \] Output:

1/4*a*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)))/(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. \(453\) vs. \(2(141)=282\).

Time = 10.98 (sec) , antiderivative size = 453, normalized size of antiderivative = 3.21 \[ \int x^3 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {x \left (8 \left (-297 b^4 x^3-81 b^3 c x^6+3464 b^2 c^2 x^9+5488 b c^3 x^{12}+2240 c^4 x^{15}+4 a^2 c \left (459 b+1280 c x^3\right )+a \left (-297 b^3+2052 b^2 c x^3+10204 b c^2 x^6+7360 c^3 x^9\right )\right )+216 a b \left (11 b^2-68 a c\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+27 \left (55 b^4-404 a b^2 c+640 a^2 c^2\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{232960 c^2 \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(x*(8*(-297*b^4*x^3 - 81*b^3*c*x^6 + 3464*b^2*c^2*x^9 + 5488*b*c^3*x^12 + 
2240*c^4*x^15 + 4*a^2*c*(459*b + 1280*c*x^3) + a*(-297*b^3 + 2052*b^2*c*x^ 
3 + 10204*b*c^2*x^6 + 7360*c^3*x^9)) + 216*a*b*(11*b^2 - 68*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*(55*b^4 - 404*a*b^2*c + 640*a^2*c^2)*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])]))/(232960*c^2*Sqrt[a + 
b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 141, 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.

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

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

(a*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])])/(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*x^9 + b*x^6 + a*x^3)*sqrt(c*x^6 + b*x^3 + a), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^3 \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {3672 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a b c x +10240 \sqrt {c \,x^{6}+b \,x^{3}+a}\, a \,c^{2} x^{4}-594 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{3} x +432 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b^{2} c \,x^{4}+6496 \sqrt {c \,x^{6}+b \,x^{3}+a}\, b \,c^{2} x^{7}+4480 \sqrt {c \,x^{6}+b \,x^{3}+a}\, c^{3} x^{10}-3672 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a^{2} b c +594 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a \,b^{3}+17280 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a^{2} c^{2}-10908 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) a \,b^{2} c +1485 \left (\int \frac {\sqrt {c \,x^{6}+b \,x^{3}+a}\, x^{3}}{c \,x^{6}+b \,x^{3}+a}d x \right ) b^{4}}{58240 c^{2}} \] Input:

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

Output:

(3672*sqrt(a + b*x**3 + c*x**6)*a*b*c*x + 10240*sqrt(a + b*x**3 + c*x**6)* 
a*c**2*x**4 - 594*sqrt(a + b*x**3 + c*x**6)*b**3*x + 432*sqrt(a + b*x**3 + 
 c*x**6)*b**2*c*x**4 + 6496*sqrt(a + b*x**3 + c*x**6)*b*c**2*x**7 + 4480*s 
qrt(a + b*x**3 + c*x**6)*c**3*x**10 - 3672*int(sqrt(a + b*x**3 + c*x**6)/( 
a + b*x**3 + c*x**6),x)*a**2*b*c + 594*int(sqrt(a + b*x**3 + c*x**6)/(a + 
b*x**3 + c*x**6),x)*a*b**3 + 17280*int((sqrt(a + b*x**3 + c*x**6)*x**3)/(a 
 + b*x**3 + c*x**6),x)*a**2*c**2 - 10908*int((sqrt(a + b*x**3 + c*x**6)*x* 
*3)/(a + b*x**3 + c*x**6),x)*a*b**2*c + 1485*int((sqrt(a + b*x**3 + c*x**6 
)*x**3)/(a + b*x**3 + c*x**6),x)*b**4)/(58240*c**2)