\(\int (a+b x^3+c x^6)^{3/2} \, dx\) [201]

Optimal result

Integrand size = 16, antiderivative size = 136 \[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {a x \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \] Output:

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)))/(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. \(408\) vs. \(2(136)=272\).

Time = 10.77 (sec) , antiderivative size = 408, normalized size of antiderivative = 3.00 \[ \int \left (a+b x^3+c x^6\right )^{3/2} \, dx=\frac {x \left (8 \left (27 a b^2+364 a^2 c+27 b^3 x^3+548 a b c x^3+211 b^2 c x^6+476 a c^2 x^6+296 b c^2 x^9+112 c^3 x^{12}\right )-216 a \left (b^2-28 a c\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )-27 b \left (5 b^2-44 a c\right ) x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right )}{8960 c \sqrt {a+b x^3+c x^6}} \] Input:

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

Output:

(x*(8*(27*a*b^2 + 364*a^2*c + 27*b^3*x^3 + 548*a*b*c*x^3 + 211*b^2*c*x^6 + 
 476*a*c^2*x^6 + 296*b*c^2*x^9 + 112*c^3*x^12) - 216*a*(b^2 - 28*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*b*(5*b^2 - 44*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*Sqrt[a + b*x^3 + c*x^6])
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1686, 936}

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.

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

\(\Big \downarrow \) 1686

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

(a*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])])/(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 936

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

rule 1686

Int[((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])))^FracPar 
t[p]))   Int[(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c])))^p*(1 + 2*c*(x^n/(b - S 
qrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] 
&& NeQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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