\(\int (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3)^{3/2} \, dx\) [94]

Optimal result

Integrand size = 32, antiderivative size = 216 \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=-\frac {1152 (b+c x) \sqrt {-64+(b+c x)^3}}{55 c}+\frac {2 (b+c x) \left (-64+(b+c x)^3\right )^{3/2}}{11 c}-\frac {36864\ 3^{3/4} \sqrt {2-\sqrt {3}} (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{55 c \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:

-1152/55*(c*x+b)*(-64+(c*x+b)^3)^(1/2)/c+2/11*(c*x+b)*(-64+(c*x+b)^3)^(3/2 
)/c-36864/55*3^(3/4)*(1/2*6^(1/2)-1/2*2^(1/2))*(-c*x-b+4)*((16+4*c*x+4*b+( 
c*x+b)^2)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)*EllipticF((4+4*3^(1/2)-b-c*x)/(4-4* 
3^(1/2)-b-c*x),2*I-I*3^(1/2))/c/(-(-c*x-b+4)/(4-4*3^(1/2)-b-c*x)^2)^(1/2)/ 
(-64+(c*x+b)^3)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 10.36 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.25 \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\frac {\frac {2 \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right ) \left (-896 b+5 b^4-896 c x+20 b^3 c x+30 b^2 c^2 x^2+20 b c^3 x^3+5 c^4 x^4\right )}{c}+\frac {110592 \sqrt {-4 i+4 \sqrt {3}-2 i b-2 i c x} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \left (2-2 i \sqrt {3}+b+c x\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )}{c \sqrt {2 i+2 \sqrt {3}+i b+i c x}}}{55 \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}} \] Input:

Integrate[(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(3/2),x]
 

Output:

((2*(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)*(-896*b + 5*b^4 - 896* 
c*x + 20*b^3*c*x + 30*b^2*c^2*x^2 + 20*b*c^3*x^3 + 5*c^4*x^4))/c + (110592 
*Sqrt[-4*I + 4*Sqrt[3] - (2*I)*b - (2*I)*c*x]*Sqrt[((-I)*(-4 + b + c*x))/( 
3*I + Sqrt[3])]*(2 - (2*I)*Sqrt[3] + b + c*x)*EllipticF[ArcSin[Sqrt[2*I + 
2*Sqrt[3] + I*b + I*c*x]/(2*3^(1/4))], (2*Sqrt[3])/(3*I + Sqrt[3])])/(c*Sq 
rt[2*I + 2*Sqrt[3] + I*b + I*c*x]))/(55*Sqrt[-64 + b^3 + 3*b^2*c*x + 3*b*c 
^2*x^2 + c^3*x^3])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.27, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2458, 748, 748, 760}

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.

\(\displaystyle \int \left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{3/2} \, dx\)

\(\Big \downarrow \) 2458

\(\displaystyle \int \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 748

\(\displaystyle \frac {2}{11} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}-\frac {576}{11} \int \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}d\left (\frac {b}{c}+x\right )\)

\(\Big \downarrow \) 748

\(\displaystyle \frac {2}{11} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}-\frac {576}{11} \left (\frac {2}{5} \left (\frac {b}{c}+x\right ) \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}-\frac {192}{5} \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {2}{11} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}-\frac {576}{11} \left (\frac {64\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{5 c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}+\frac {2}{5} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (\frac {b}{c}+x\right )\right )\)

Input:

Int[(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(3/2),x]
 

Output:

(2*(b/c + x)*(-64 + c^3*(b/c + x)^3)^(3/2))/11 - (576*((2*(b/c + x)*Sqrt[- 
64 + c^3*(b/c + x)^3])/5 + (64*3^(3/4)*Sqrt[2 - Sqrt[3]]*(4 - c*(b/c + x)) 
*Sqrt[(16 + 4*c*(b/c + x) + c^2*(b/c + x)^2)/(4*(1 - Sqrt[3]) - c*(b/c + x 
))^2]*EllipticF[ArcSin[(4*(1 + Sqrt[3]) - c*(b/c + x))/(4*(1 - Sqrt[3]) - 
c*(b/c + x))], -7 + 4*Sqrt[3]])/(5*c*Sqrt[-((4 - c*(b/c + x))/(4*(1 - Sqrt 
[3]) - c*(b/c + x))^2)]*Sqrt[-64 + c^3*(b/c + x)^3])))/11
 

Defintions of rubi rules used

rule 748

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p 
+ 1)), x] + Simp[a*n*(p/(n*p + 1))   Int[(a + b*x^n)^(p - 1), x], x] /; Fre 
eQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || LtQ[Denominat 
or[p + 1/n], Denominator[p]])
 

rule 760

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- 
s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) 
*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x 
] && NegQ[a]
 

rule 2458

Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp 
on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x 
- S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp 
on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P 
n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.69

method	result	size
risch	\(\frac {2 \left (5 c^{4} x^{4}+20 b \,c^{3} x^{3}+30 b^{2} c^{2} x^{2}+20 b^{3} c x +5 b^{4}-896 c x -896 b \right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{55 c}+\frac {221184 \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{55 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}\)	\(366\)
default	\(\text {Expression too large to display}\)	\(1212\)
elliptic	\(\text {Expression too large to display}\)	\(1212\)

Input:

int((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/55/c*(5*c^4*x^4+20*b*c^3*x^3+30*b^2*c^2*x^2+20*b^3*c*x+5*b^4-896*c*x-896 
*b)*(c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(1/2)+221184/55*((-b-2-2*I*3^(1 
/2))/c+(b-4)/c)*((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c))^(1/2)*((x-(-b 
-2+2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^(1/2)*((x-(-b-2-2*I*3^ 
(1/2))/c)/(-(b-4)/c-(-b-2-2*I*3^(1/2))/c))^(1/2)/(c^3*x^3+3*b*c^2*x^2+3*b^ 
2*c*x+b^3-64)^(1/2)*EllipticF(((x+(b-4)/c)/((-b-2-2*I*3^(1/2))/c+(b-4)/c)) 
^(1/2),((-(b-4)/c-(-b-2-2*I*3^(1/2))/c)/(-(b-4)/c-(-b-2+2*I*3^(1/2))/c))^( 
1/2))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.53 \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\frac {2 \, {\left ({\left (5 \, c^{6} x^{4} + 20 \, b c^{5} x^{3} + 30 \, b^{2} c^{4} x^{2} + 4 \, {\left (5 \, b^{3} - 224\right )} c^{3} x + {\left (5 \, b^{4} - 896 \, b\right )} c^{2}\right )} \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64} + 110592 \, \sqrt {c^{3}} {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right )\right )}}{55 \, c^{3}} \] Input:

integrate((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x, algorithm="frica 
s")
 

Output:

2/55*((5*c^6*x^4 + 20*b*c^5*x^3 + 30*b^2*c^4*x^2 + 4*(5*b^3 - 224)*c^3*x + 
 (5*b^4 - 896*b)*c^2)*sqrt(c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64) + 
 110592*sqrt(c^3)*weierstrassPInverse(0, 256/c^3, (c*x + b)/c))/c^3
 

Sympy [F]

\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int \left (b^{3} + 3 b^{2} c x + 3 b c^{2} x^{2} + c^{3} x^{3} - 64\right )^{\frac {3}{2}}\, dx \] Input:

integrate((c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**(3/2),x)
 

Output:

Integral((b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**(3/2), x)
 

Maxima [F]

\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int { {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x, algorithm="maxim 
a")
 

Output:

integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(3/2), x)
 

Giac [F]

\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int { {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {3}{2}} \,d x } \] Input:

integrate((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x, algorithm="giac" 
)
 

Output:

integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\int {\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{3/2} \,d x \] Input:

int((b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(3/2),x)
 

Output:

int((b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(3/2), x)
 

Reduce [F]

\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{3/2} \, dx=\frac {\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{6}}{11}+\frac {8 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{5} c x}{11}+\frac {12 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{4} c^{2} x^{2}}{11}+\frac {8 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{3} c^{3} x^{3}}{11}-\frac {1792 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{3}}{55}+\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{2} c^{4} x^{4}}{11}-\frac {1792 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{2} c x}{55}+\frac {73728 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{55}-\frac {110592 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, x^{2}}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) c^{3}}{55}-\frac {221184 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, x}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) b \,c^{2}}{55}}{b^{2} c} \] Input:

int((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(2*(5*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**6 + 20*s 
qrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**5*c*x + 30*sqrt 
(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**4*c**2*x**2 + 20*s 
qrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**3*c**3*x**3 - 8 
96*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**3 + 5*sqrt( 
b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**2*c**4*x**4 - 896*s 
qrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**2*c*x + 36864*s 
qrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) - 55296*int((sqrt( 
b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*x**2)/(b**3 + 3*b**2*c 
*x + 3*b*c**2*x**2 + c**3*x**3 - 64),x)*c**3 - 110592*int((sqrt(b**3 + 3*b 
**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*x)/(b**3 + 3*b**2*c*x + 3*b*c**2 
*x**2 + c**3*x**3 - 64),x)*b*c**2))/(55*b**2*c)