Integrand size = 27, antiderivative size = 214 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {18 c^3 (c-3 d x) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{5 d (2 c+3 d x)^3}+\frac {18 c^2 (c-3 d x)^2 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{7 d (2 c+3 d x)^3}-\frac {2 c (c-3 d x)^3 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{3 d (2 c+3 d x)^3}+\frac {2 (c-3 d x)^4 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{33 d (2 c+3 d x)^3} \] Output:
-18/5*c^3*(-3*d*x+c)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d/(3*d*x+2*c)^ 3+18/7*c^2*(-3*d*x+c)^2*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d/(3*d*x+2* c)^3-2/3*c*(-3*d*x+c)^3*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d/(3*d*x+2* c)^3+2/33*(-3*d*x+c)^4*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d/(3*d*x+2*c )^3
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.35 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 (c-3 d x)^3 (2 c+3 d x) \left (944 c^3+2460 c^2 d x+2520 c d^2 x^2+945 d^3 x^3\right )}{1155 d \sqrt {(c-3 d x) (2 c+3 d x)^2}} \] Input:
Integrate[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(3/2),x]
Output:
(-2*(c - 3*d*x)^3*(2*c + 3*d*x)*(944*c^3 + 2460*c^2*d*x + 2520*c*d^2*x^2 + 945*d^3*x^3))/(1155*d*Sqrt[(c - 3*d*x)*(2*c + 3*d*x)^2])
Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2480, 27, 53, 2009}
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.
\(\displaystyle \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 2480 |
\(\displaystyle \frac {\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \int 61004779879896 c^6 d^9 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}dx}{61004779879896 c^6 d^9 (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \int (2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}dx}{(2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \int \left (-\frac {\left (c^3 d^6-3 c^2 d^7 x\right )^{9/2}}{c^6 d^{18}}+\frac {9 \left (c^3 d^6-3 c^2 d^7 x\right )^{7/2}}{c^3 d^{12}}-\frac {27 \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}{d^6}+27 c^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}\right )dx}{(2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \left (-\frac {18 c \left (c^3 d^6-3 c^2 d^7 x\right )^{5/2}}{5 d^7}+\frac {18 \left (c^3 d^6-3 c^2 d^7 x\right )^{7/2}}{7 c^2 d^{13}}+\frac {2 \left (c^3 d^6-3 c^2 d^7 x\right )^{11/2}}{33 c^8 d^{25}}-\frac {2 \left (c^3 d^6-3 c^2 d^7 x\right )^{9/2}}{3 c^5 d^{19}}\right )}{(2 c+3 d x)^3 \left (c^3 d^6-3 c^2 d^7 x\right )^{3/2}}\) |
Input:
Int[(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(3/2),x]
Output:
((4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(3/2)*((-18*c*(c^3*d^6 - 3*c^2*d^7*x) ^(5/2))/(5*d^7) + (18*(c^3*d^6 - 3*c^2*d^7*x)^(7/2))/(7*c^2*d^13) - (2*(c^ 3*d^6 - 3*c^2*d^7*x)^(9/2))/(3*c^5*d^19) + (2*(c^3*d^6 - 3*c^2*d^7*x)^(11/ 2))/(33*c^8*d^25)))/((2*c + 3*d*x)^3*(c^3*d^6 - 3*c^2*d^7*x)^(3/2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] , c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Simp[Px^p/((c^3 - 4*b*c*d + 9* a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3*b*d)*x)^(2*p)) Int [(c^3 - 4*b*c*d + 9*a*d^2 + d*(c^2 - 3*b*d)*x)^p*(b*c - 9*a*d + 2*(c^2 - 3* b*d)*x)^(2*p), x], x] /; EqQ[b^2*c^2 - 4*a*c^3 - 4*b^3*d + 18*a*b*c*d - 27* a^2*d^2, 0] && NeQ[c^2 - 3*b*d, 0]] /; FreeQ[p, x] && PolyQ[Px, x, 3] && ! IntegerQ[p]
Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.36
method | result | size |
gosper | \(-\frac {2 \left (-3 d x +c \right ) \left (945 d^{3} x^{3}+2520 c \,d^{2} x^{2}+2460 c^{2} x d +944 c^{3}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {3}{2}}}{1155 d \left (3 d x +2 c \right )^{3}}\) | \(77\) |
default | \(-\frac {2 \left (-3 d x +c \right ) \left (945 d^{3} x^{3}+2520 c \,d^{2} x^{2}+2460 c^{2} x d +944 c^{3}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {3}{2}}}{1155 d \left (3 d x +2 c \right )^{3}}\) | \(77\) |
orering | \(-\frac {2 \left (-3 d x +c \right ) \left (945 d^{3} x^{3}+2520 c \,d^{2} x^{2}+2460 c^{2} x d +944 c^{3}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {3}{2}}}{1155 d \left (3 d x +2 c \right )^{3}}\) | \(77\) |
trager | \(-\frac {2 \left (8505 d^{5} x^{5}+17010 c \,d^{4} x^{4}+7965 c^{2} d^{3} x^{3}-3744 c^{3} d^{2} x^{2}-3204 c^{4} x d +944 c^{5}\right ) \sqrt {-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}}}{1155 \left (3 d x +2 c \right ) d}\) | \(93\) |
Input:
int((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/1155*(-3*d*x+c)*(945*d^3*x^3+2520*c*d^2*x^2+2460*c^2*d*x+944*c^3)*(-27* d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d/(3*d*x+2*c)^3
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.43 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (8505 \, d^{5} x^{5} + 17010 \, c d^{4} x^{4} + 7965 \, c^{2} d^{3} x^{3} - 3744 \, c^{3} d^{2} x^{2} - 3204 \, c^{4} d x + 944 \, c^{5}\right )} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}}}{1155 \, {\left (3 \, d^{2} x + 2 \, c d\right )}} \] Input:
integrate((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x, algorithm="fricas")
Output:
-2/1155*(8505*d^5*x^5 + 17010*c*d^4*x^4 + 7965*c^2*d^3*x^3 - 3744*c^3*d^2* x^2 - 3204*c^4*d*x + 944*c^5)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)/(3* d^2*x + 2*c*d)
\[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=\int \left (4 c^{3} - 27 c d^{2} x^{2} - 27 d^{3} x^{3}\right )^{\frac {3}{2}}\, dx \] Input:
integrate((-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**(3/2),x)
Output:
Integral((4*c**3 - 27*c*d**2*x**2 - 27*d**3*x**3)**(3/2), x)
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.30 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (8505 \, d^{5} x^{5} + 17010 \, c d^{4} x^{4} + 7965 \, c^{2} d^{3} x^{3} - 3744 \, c^{3} d^{2} x^{2} - 3204 \, c^{4} d x + 944 \, c^{5}\right )} \sqrt {-3 \, d x + c}}{1155 \, d} \] Input:
integrate((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x, algorithm="maxima")
Output:
-2/1155*(8505*d^5*x^5 + 17010*c*d^4*x^4 + 7965*c^2*d^3*x^3 - 3744*c^3*d^2* x^2 - 3204*c^4*d*x + 944*c^5)*sqrt(-3*d*x + c)/d
Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (206) = 412\).
Time = 0.12 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.01 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=\frac {2 \, {\left (27720 \, \sqrt {-3 \, d x + c} c^{5} \mathrm {sgn}\left (-3 \, d x - 2 \, c\right ) + 4620 \, {\left ({\left (-3 \, d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {-3 \, d x + c} c\right )} c^{4} \mathrm {sgn}\left (-3 \, d x - 2 \, c\right ) - 2310 \, {\left (3 \, {\left (3 \, d x - c\right )}^{2} \sqrt {-3 \, d x + c} - 10 \, {\left (-3 \, d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {-3 \, d x + c} c^{2}\right )} c^{3} \mathrm {sgn}\left (-3 \, d x - 2 \, c\right ) + 99 \, {\left (5 \, {\left (3 \, d x - c\right )}^{3} \sqrt {-3 \, d x + c} + 21 \, {\left (3 \, d x - c\right )}^{2} \sqrt {-3 \, d x + c} c - 35 \, {\left (-3 \, d x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, \sqrt {-3 \, d x + c} c^{3}\right )} c^{2} \mathrm {sgn}\left (-3 \, d x - 2 \, c\right ) + 44 \, {\left (35 \, {\left (3 \, d x - c\right )}^{4} \sqrt {-3 \, d x + c} + 180 \, {\left (3 \, d x - c\right )}^{3} \sqrt {-3 \, d x + c} c + 378 \, {\left (3 \, d x - c\right )}^{2} \sqrt {-3 \, d x + c} c^{2} - 420 \, {\left (-3 \, d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {-3 \, d x + c} c^{4}\right )} c \mathrm {sgn}\left (-3 \, d x - 2 \, c\right ) + 5 \, {\left (63 \, {\left (3 \, d x - c\right )}^{5} \sqrt {-3 \, d x + c} + 385 \, {\left (3 \, d x - c\right )}^{4} \sqrt {-3 \, d x + c} c + 990 \, {\left (3 \, d x - c\right )}^{3} \sqrt {-3 \, d x + c} c^{2} + 1386 \, {\left (3 \, d x - c\right )}^{2} \sqrt {-3 \, d x + c} c^{3} - 1155 \, {\left (-3 \, d x + c\right )}^{\frac {3}{2}} c^{4} + 693 \, \sqrt {-3 \, d x + c} c^{5}\right )} \mathrm {sgn}\left (-3 \, d x - 2 \, c\right )\right )}}{10395 \, d} \] Input:
integrate((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x, algorithm="giac")
Output:
2/10395*(27720*sqrt(-3*d*x + c)*c^5*sgn(-3*d*x - 2*c) + 4620*((-3*d*x + c) ^(3/2) - 3*sqrt(-3*d*x + c)*c)*c^4*sgn(-3*d*x - 2*c) - 2310*(3*(3*d*x - c) ^2*sqrt(-3*d*x + c) - 10*(-3*d*x + c)^(3/2)*c + 15*sqrt(-3*d*x + c)*c^2)*c ^3*sgn(-3*d*x - 2*c) + 99*(5*(3*d*x - c)^3*sqrt(-3*d*x + c) + 21*(3*d*x - c)^2*sqrt(-3*d*x + c)*c - 35*(-3*d*x + c)^(3/2)*c^2 + 35*sqrt(-3*d*x + c)* c^3)*c^2*sgn(-3*d*x - 2*c) + 44*(35*(3*d*x - c)^4*sqrt(-3*d*x + c) + 180*( 3*d*x - c)^3*sqrt(-3*d*x + c)*c + 378*(3*d*x - c)^2*sqrt(-3*d*x + c)*c^2 - 420*(-3*d*x + c)^(3/2)*c^3 + 315*sqrt(-3*d*x + c)*c^4)*c*sgn(-3*d*x - 2*c ) + 5*(63*(3*d*x - c)^5*sqrt(-3*d*x + c) + 385*(3*d*x - c)^4*sqrt(-3*d*x + c)*c + 990*(3*d*x - c)^3*sqrt(-3*d*x + c)*c^2 + 1386*(3*d*x - c)^2*sqrt(- 3*d*x + c)*c^3 - 1155*(-3*d*x + c)^(3/2)*c^4 + 693*sqrt(-3*d*x + c)*c^5)*s gn(-3*d*x - 2*c))/d
Time = 12.32 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.42 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {\sqrt {4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3}\,\left (\frac {1888\,c^5}{3465\,d^2}-\frac {832\,c^3\,x^2}{385}+\frac {54\,d^3\,x^5}{11}+\frac {354\,c^2\,d\,x^3}{77}-\frac {712\,c^4\,x}{385\,d}+\frac {108\,c\,d^2\,x^4}{11}\right )}{x+\frac {2\,c}{3\,d}} \] Input:
int((4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(3/2),x)
Output:
-((4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(1/2)*((1888*c^5)/(3465*d^2) - (832* c^3*x^2)/385 + (54*d^3*x^5)/11 + (354*c^2*d*x^3)/77 - (712*c^4*x)/(385*d) + (108*c*d^2*x^4)/11))/(x + (2*c)/(3*d))
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.30 \[ \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=\frac {2 \sqrt {-3 d x +c}\, \left (-8505 d^{5} x^{5}-17010 c \,d^{4} x^{4}-7965 c^{2} d^{3} x^{3}+3744 c^{3} d^{2} x^{2}+3204 c^{4} d x -944 c^{5}\right )}{1155 d} \] Input:
int((-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x)
Output:
(2*sqrt(c - 3*d*x)*( - 944*c**5 + 3204*c**4*d*x + 3744*c**3*d**2*x**2 - 79 65*c**2*d**3*x**3 - 17010*c*d**4*x**4 - 8505*d**5*x**5))/(1155*d)