Integrand size = 32, antiderivative size = 241 \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\frac {221184 (b+c x) \sqrt {-64+(b+c x)^3}}{187 c}-\frac {1920 (b+c x) \left (-64+(b+c x)^3\right )^{3/2}}{187 c}+\frac {2 (b+c x) \left (-64+(b+c x)^3\right )^{5/2}}{17 c}+\frac {7077888\ 3^{3/4} \sqrt {2-\sqrt {3}} (4-b-c x) \sqrt {\frac {16+4 (b+c x)+(b+c x)^2}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4+4 \sqrt {3}-b-c x}{4-4 \sqrt {3}-b-c x}\right ),-7+4 \sqrt {3}\right )}{187 c \sqrt {-\frac {4-b-c x}{\left (4-4 \sqrt {3}-b-c x\right )^2}} \sqrt {-64+(b+c x)^3}} \] Output:
221184/187*(c*x+b)*(-64+(c*x+b)^3)^(1/2)/c-1920/187*(c*x+b)*(-64+(c*x+b)^3 )^(3/2)/c+2/17*(c*x+b)*(-64+(c*x+b)^3)^(5/2)/c+7077888/187*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)
Result contains complex when optimal does not.
Time = 10.46 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.25 \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\frac {2 \left (\left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right ) \left (11 b^7+77 b^6 c x+231 b^5 c^2 x^2+3 b^2 c^2 x^2 \left (-4736+77 c^3 x^3\right )+b^3 c x \left (-9472+385 c^3 x^3\right )+b^4 \left (-2368+385 c^3 x^3\right )+c x \left (217088-2368 c^3 x^3+11 c^6 x^6\right )+b \left (217088-9472 c^3 x^3+77 c^6 x^6\right )\right )+10616832 i \sqrt {2} \sqrt {-\frac {i (-4+b+c x)}{3 i+\sqrt {3}}} \sqrt {16+b^2+4 c x+c^2 x^2+2 b (2+c x)} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 i+2 \sqrt {3}+i b+i c x}}{2 \sqrt [4]{3}}\right ),\frac {2 \sqrt {3}}{3 i+\sqrt {3}}\right )\right )}{187 c \sqrt {-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3}} \] Input:
Integrate[(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(5/2),x]
Output:
(2*((-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)*(11*b^7 + 77*b^6*c*x + 231*b^5*c^2*x^2 + 3*b^2*c^2*x^2*(-4736 + 77*c^3*x^3) + b^3*c*x*(-9472 + 3 85*c^3*x^3) + b^4*(-2368 + 385*c^3*x^3) + c*x*(217088 - 2368*c^3*x^3 + 11* c^6*x^6) + b*(217088 - 9472*c^3*x^3 + 77*c^6*x^6)) + (10616832*I)*Sqrt[2]* Sqrt[((-I)*(-4 + b + c*x))/(3*I + Sqrt[3])]*Sqrt[16 + b^2 + 4*c*x + c^2*x^ 2 + 2*b*(2 + 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])]))/(187*c*Sqrt[-64 + b^3 + 3*b^2*c *x + 3*b*c^2*x^2 + c^3*x^3])
Time = 0.33 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2458, 748, 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.
| \(\displaystyle \int \left (b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3-64\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 2458 |
| \(\displaystyle \int \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {2}{17} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}-\frac {960}{17} \int \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}d\left (\frac {b}{c}+x\right )\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {2}{17} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}-\frac {960}{17} \left (\frac {2}{11} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}-\frac {576}{11} \int \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}d\left (\frac {b}{c}+x\right )\right )\) |
| \(\Big \downarrow \) 748 |
| \(\displaystyle \frac {2}{17} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}-\frac {960}{17} \left (\frac {2}{11} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}-\frac {576}{11} \left (\frac {2}{5} \left (\frac {b}{c}+x\right ) \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}-\frac {192}{5} \int \frac {1}{\sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}d\left (\frac {b}{c}+x\right )\right )\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {2}{17} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{5/2}-\frac {960}{17} \left (\frac {2}{11} \left (\frac {b}{c}+x\right ) \left (c^3 \left (\frac {b}{c}+x\right )^3-64\right )^{3/2}-\frac {576}{11} \left (\frac {64\ 3^{3/4} \sqrt {2-\sqrt {3}} \left (4-c \left (\frac {b}{c}+x\right )\right ) \sqrt {\frac {c^2 \left (\frac {b}{c}+x\right )^2+4 c \left (\frac {b}{c}+x\right )+16}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {4 \left (1+\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}{4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )}\right ),-7+4 \sqrt {3}\right )}{5 c \sqrt {-\frac {4-c \left (\frac {b}{c}+x\right )}{\left (4 \left (1-\sqrt {3}\right )-c \left (\frac {b}{c}+x\right )\right )^2}} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64}}+\frac {2}{5} \sqrt {c^3 \left (\frac {b}{c}+x\right )^3-64} \left (\frac {b}{c}+x\right )\right )\right )\) |
Input:
Int[(-64 + b^3 + 3*b^2*c*x + 3*b*c^2*x^2 + c^3*x^3)^(5/2),x]
Output:
(2*(b/c + x)*(-64 + c^3*(b/c + x)^3)^(5/2))/17 - (960*((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[Arc Sin[(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))/17
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]])
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (208 ) = 416\).
Time = 0.70 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.82
| method | result | size |
| risch | \(\frac {2 \left (11 c^{7} x^{7}+77 b \,c^{6} x^{6}+231 b^{2} c^{5} x^{5}+385 b^{3} c^{4} x^{4}+385 b^{4} c^{3} x^{3}+231 b^{5} c^{2} x^{2}+77 b^{6} c x -2368 c^{4} x^{4}+11 b^{7}-9472 b \,c^{3} x^{3}-14208 b^{2} c^{2} x^{2}-9472 b^{3} c x -2368 b^{4}+217088 c x +217088 b \right ) \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{187 c}-\frac {42467328 \left (\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}\right ) \sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}\, \sqrt {\frac {x -\frac {-b -2+2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\, \sqrt {\frac {x -\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b -4}{c}}{\frac {-b -2-2 i \sqrt {3}}{c}+\frac {b -4}{c}}}, \sqrt {\frac {-\frac {b -4}{c}-\frac {-b -2-2 i \sqrt {3}}{c}}{-\frac {b -4}{c}-\frac {-b -2+2 i \sqrt {3}}{c}}}\right )}{187 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}\) | \(439\) |
| default | \(\text {Expression too large to display}\) | \(3749\) |
| elliptic | \(\text {Expression too large to display}\) | \(3749\) |
Input:
int((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/187/c*(11*c^7*x^7+77*b*c^6*x^6+231*b^2*c^5*x^5+385*b^3*c^4*x^4+385*b^4*c ^3*x^3+231*b^5*c^2*x^2+77*b^6*c*x-2368*c^4*x^4+11*b^7-9472*b*c^3*x^3-14208 *b^2*c^2*x^2-9472*b^3*c*x-2368*b^4+217088*c*x+217088*b)*(c^3*x^3+3*b*c^2*x ^2+3*b^2*c*x+b^3-64)^(1/2)-42467328/187*((-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))
Time = 0.10 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.72 \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\frac {2 \, {\left ({\left (11 \, c^{9} x^{7} + 77 \, b c^{8} x^{6} + 231 \, b^{2} c^{7} x^{5} + {\left (385 \, b^{3} - 2368\right )} c^{6} x^{4} + {\left (385 \, b^{4} - 9472 \, b\right )} c^{5} x^{3} + 3 \, {\left (77 \, b^{5} - 4736 \, b^{2}\right )} c^{4} x^{2} + {\left (77 \, b^{6} - 9472 \, b^{3} + 217088\right )} c^{3} x + {\left (11 \, b^{7} - 2368 \, b^{4} + 217088 \, b\right )} c^{2}\right )} \sqrt {c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64} - 21233664 \, \sqrt {c^{3}} {\rm weierstrassPInverse}\left (0, \frac {256}{c^{3}}, \frac {c x + b}{c}\right )\right )}}{187 \, c^{3}} \] Input:
integrate((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(5/2),x, algorithm="frica s")
Output:
2/187*((11*c^9*x^7 + 77*b*c^8*x^6 + 231*b^2*c^7*x^5 + (385*b^3 - 2368)*c^6 *x^4 + (385*b^4 - 9472*b)*c^5*x^3 + 3*(77*b^5 - 4736*b^2)*c^4*x^2 + (77*b^ 6 - 9472*b^3 + 217088)*c^3*x + (11*b^7 - 2368*b^4 + 217088*b)*c^2)*sqrt(c^ 3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64) - 21233664*sqrt(c^3)*weierstra ssPInverse(0, 256/c^3, (c*x + b)/c))/c^3
\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\int \left (b^{3} + 3 b^{2} c x + 3 b c^{2} x^{2} + c^{3} x^{3} - 64\right )^{\frac {5}{2}}\, dx \] Input:
integrate((c**3*x**3+3*b*c**2*x**2+3*b**2*c*x+b**3-64)**(5/2),x)
Output:
Integral((b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)**(5/2), x)
\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\int { {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(5/2),x, algorithm="maxim a")
Output:
integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(5/2), x)
\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\int { {\left (c^{3} x^{3} + 3 \, b c^{2} x^{2} + 3 \, b^{2} c x + b^{3} - 64\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(5/2),x, algorithm="giac" )
Output:
integrate((c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x + b^3 - 64)^(5/2), x)
Timed out. \[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\int {\left (b^3+3\,b^2\,c\,x+3\,b\,c^2\,x^2+c^3\,x^3-64\right )}^{5/2} \,d x \] Input:
int((b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(5/2),x)
Output:
int((b^3 + c^3*x^3 + 3*b*c^2*x^2 + 3*b^2*c*x - 64)^(5/2), x)
\[ \int \left (-64+b^3+3 b^2 c x+3 b c^2 x^2+c^3 x^3\right )^{5/2} \, dx=\frac {\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{9}}{17}+\frac {14 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{8} c x}{17}+\frac {42 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{7} c^{2} x^{2}}{17}+\frac {70 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{6} c^{3} x^{3}}{17}-\frac {4736 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{6}}{187}+\frac {70 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{5} c^{4} x^{4}}{17}-\frac {18944 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{5} c x}{187}+\frac {42 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{4} c^{5} x^{5}}{17}-\frac {28416 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{4} c^{2} x^{2}}{187}+\frac {14 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{3} c^{6} x^{6}}{17}-\frac {18944 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{3} c^{3} x^{3}}{187}+\frac {434176 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{3}}{187}+\frac {2 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{2} c^{7} x^{7}}{17}-\frac {4736 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{2} c^{4} x^{4}}{187}+\frac {434176 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, b^{2} c x}{187}-\frac {14155776 \sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}}{187}+\frac {21233664 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, x^{2}}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) c^{3}}{187}+\frac {42467328 \left (\int \frac {\sqrt {c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}\, x}{c^{3} x^{3}+3 b \,c^{2} x^{2}+3 b^{2} c x +b^{3}-64}d x \right ) b \,c^{2}}{187}}{b^{2} c} \] Input:
int((c^3*x^3+3*b*c^2*x^2+3*b^2*c*x+b^3-64)^(5/2),x)
Output:
(2*(11*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**9 + 77* sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**8*c*x + 231*sq rt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**7*c**2*x**2 + 38 5*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**6*c**3*x**3 - 2368*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**6 + 385 *sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**5*c**4*x**4 - 9472*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**5*c*x + 231*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**4*c**5*x** 5 - 14208*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 + 77*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64)*b**3 *c**6*x**6 - 9472*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) *b**3*c**3*x**3 + 217088*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x** 3 - 64)*b**3 + 11*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c**3*x**3 - 64) *b**2*c**7*x**7 - 2368*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 + 217088*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c** 3*x**3 - 64)*b**2*c*x - 7077888*sqrt(b**3 + 3*b**2*c*x + 3*b*c**2*x**2 + c **3*x**3 - 64) + 10616832*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 + 21233664*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...