\(\int (A+B x+C x^2) (4 c^3-27 c d^2 x^2-27 d^3 x^3)^{3/2} \, dx\) [36]

Optimal result

Integrand size = 38, antiderivative size = 398 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 c^3 \left (c^2 C+3 B c d+9 A d^2\right ) (c-3 d x) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{5 d^3 (2 c+3 d x)^3}+\frac {6 c^2 \left (c^2 C+2 B c d+3 A d^2\right ) (c-3 d x)^2 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{7 d^3 (2 c+3 d x)^3}-\frac {2 c \left (10 c^2 C+12 B c d+9 A d^2\right ) (c-3 d x)^3 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{27 d^3 (2 c+3 d x)^3}+\frac {2 \left (46 c^2 C+30 B c d+9 A d^2\right ) (c-3 d x)^4 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{297 d^3 (2 c+3 d x)^3}-\frac {2 (11 c C+3 B d) (c-3 d x)^5 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{351 d^3 (2 c+3 d x)^3}+\frac {2 C (c-3 d x)^6 \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2}}{405 d^3 (2 c+3 d x)^3} \] Output:

-2/5*c^3*(9*A*d^2+3*B*c*d+C*c^2)*(-3*d*x+c)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^ 
3)^(3/2)/d^3/(3*d*x+2*c)^3+6/7*c^2*(3*A*d^2+2*B*c*d+C*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/(3*d*x+2*c)^3-2/27*c*(9*A*d^2+12* 
B*c*d+10*C*c^2)*(-3*d*x+c)^3*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d^3/(3 
*d*x+2*c)^3+2/297*(9*A*d^2+30*B*c*d+46*C*c^2)*(-3*d*x+c)^4*(-27*d^3*x^3-27 
*c*d^2*x^2+4*c^3)^(3/2)/d^3/(3*d*x+2*c)^3-2/351*(3*B*d+11*C*c)*(-3*d*x+c)^ 
5*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d^3/(3*d*x+2*c)^3+2/405*C*(-3*d*x 
+c)^6*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d^3/(3*d*x+2*c)^3
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.40 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 (c-3 d x)^3 (2 c+3 d x) \left (6400 c^5 C+48 c^4 d (801 B+1000 C x)+1701 d^5 x^3 (195 A+11 x (15 B+13 C x))+5670 c d^4 x^2 (156 A+x (129 B+110 C x))+540 c^2 d^3 x (1599 A+7 x (186 B+157 C x))+72 c^3 d^2 (4602 A+5 x (801 B+700 C x))\right )}{405405 d^3 \sqrt {(c-3 d x) (2 c+3 d x)^2}} \] Input:

Integrate[(A + B*x + C*x^2)*(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)*(6400*c^5*C + 48*c^4*d*(801*B + 1000*C*x) 
+ 1701*d^5*x^3*(195*A + 11*x*(15*B + 13*C*x)) + 5670*c*d^4*x^2*(156*A + x* 
(129*B + 110*C*x)) + 540*c^2*d^3*x*(1599*A + 7*x*(186*B + 157*C*x)) + 72*c 
^3*d^2*(4602*A + 5*x*(801*B + 700*C*x))))/(405405*d^3*Sqrt[(c - 3*d*x)*(2* 
c + 3*d*x)^2])
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.98, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2526, 27, 2490, 2483, 27, 86, 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.

\(\displaystyle \int \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \left (A+B x+C x^2\right ) \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle -\frac {\int -27 d^2 (3 A d-(2 c C-3 B d) x) \left (4 c^3-27 d^2 x^2 c-27 d^3 x^3\right )^{3/2}dx}{81 d^3}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (3 A d-(2 c C-3 B d) x) \left (4 c^3-27 d^2 x^2 c-27 d^3 x^3\right )^{3/2}dx}{3 d}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

\(\Big \downarrow \) 2490

\(\displaystyle \frac {\int \left ((3 B d-2 c C) \left (\frac {c}{3 d}+x\right )-\frac {27 c d^2 (3 B d-2 c C)-243 A d^4}{81 d^3}\right ) \left (2 c^3+9 d \left (\frac {c}{3 d}+x\right ) c^2-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^{3/2}d\left (\frac {c}{3 d}+x\right )}{3 d}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

\(\Big \downarrow \) 2483

\(\displaystyle \frac {\left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^{3/2} \int -216 \sqrt {3} c^6 \left (c+3 d \left (\frac {c}{3 d}+x\right )\right )^3 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2} \left (-\frac {2 C c^2}{d}+3 B c-9 A d+3 (2 c C-3 B d) \left (\frac {c}{3 d}+x\right )\right )d\left (\frac {c}{3 d}+x\right )}{1944 \sqrt {3} c^6 d \left (3 d \left (\frac {c}{3 d}+x\right )+c\right )^3 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2}}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^{3/2} \int \left (c+3 d \left (\frac {c}{3 d}+x\right )\right )^3 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2} \left (-\frac {2 C c^2}{d}+3 B c-9 A d+3 (2 c C-3 B d) \left (\frac {c}{3 d}+x\right )\right )d\left (\frac {c}{3 d}+x\right )}{9 d \left (3 d \left (\frac {c}{3 d}+x\right )+c\right )^3 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2}}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

\(\Big \downarrow \) 86

\(\displaystyle -\frac {\left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^{3/2} \int \left (\frac {(2 c C-3 B d) \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{11/2}}{c^8 d}+\frac {\left (-20 C c^2+30 B d c+9 A d^2\right ) \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{9/2}}{c^6 d}+\frac {9 \left (8 C c^2-12 B d c-9 A d^2\right ) \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{7/2}}{c^3 d}+\frac {27 \left (-4 C c^2+6 B d c+9 A d^2\right ) \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{5/2}}{d}+\frac {27 c^3 \left (2 C c^2-3 B d c-9 A d^2\right ) \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2}}{d}\right )d\left (\frac {c}{3 d}+x\right )}{9 d \left (3 d \left (\frac {c}{3 d}+x\right )+c\right )^3 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2}}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (2 c^3+9 c^2 d \left (\frac {c}{3 d}+x\right )-27 d^3 \left (\frac {c}{3 d}+x\right )^3\right )^{3/2} \left (\frac {18 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{7/2} \left (-9 A d^2-6 B c d+4 c^2 C\right )}{7 c^2 d^2}-\frac {18 c \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{5/2} \left (-9 A d^2-3 B c d+2 c^2 C\right )}{5 d^2}+\frac {2 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{11/2} \left (-9 A d^2-30 B c d+20 c^2 C\right )}{33 c^8 d^2}-\frac {2 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{9/2} \left (-9 A d^2-12 B c d+8 c^2 C\right )}{3 c^5 d^2}-\frac {2 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{13/2} (2 c C-3 B d)}{39 c^{10} d^2}\right )}{9 d \left (3 d \left (\frac {c}{3 d}+x\right )+c\right )^3 \left (2 c^3-3 c^2 d \left (\frac {c}{3 d}+x\right )\right )^{3/2}}-\frac {2 C \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{5/2}}{405 d^3}\)

Input:

Int[(A + B*x + C*x^2)*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(3/2),x]
 

Output:

(-2*C*(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3)^(5/2))/(405*d^3) - ((2*c^3 + 9*c 
^2*d*(c/(3*d) + x) - 27*d^3*(c/(3*d) + x)^3)^(3/2)*((-18*c*(2*c^2*C - 3*B* 
c*d - 9*A*d^2)*(2*c^3 - 3*c^2*d*(c/(3*d) + x))^(5/2))/(5*d^2) + (18*(4*c^2 
*C - 6*B*c*d - 9*A*d^2)*(2*c^3 - 3*c^2*d*(c/(3*d) + x))^(7/2))/(7*c^2*d^2) 
 - (2*(8*c^2*C - 12*B*c*d - 9*A*d^2)*(2*c^3 - 3*c^2*d*(c/(3*d) + x))^(9/2) 
)/(3*c^5*d^2) + (2*(20*c^2*C - 30*B*c*d - 9*A*d^2)*(2*c^3 - 3*c^2*d*(c/(3* 
d) + x))^(11/2))/(33*c^8*d^2) - (2*(2*c*C - 3*B*d)*(2*c^3 - 3*c^2*d*(c/(3* 
d) + x))^(13/2))/(39*c^10*d^2)))/(9*d*(c + 3*d*(c/(3*d) + x))^3*(2*c^3 - 3 
*c^2*d*(c/(3*d) + x))^(3/2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2483

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_S 
ymbol] :> Simp[(a + b*x + d*x^3)^p/((3*a - b*x)^p*(3*a + 2*b*x)^(2*p))   In 
t[(e + f*x)^m*(3*a - b*x)^p*(3*a + 2*b*x)^(2*p), x], x] /; FreeQ[{a, b, d, 
e, f, m, p}, x] && EqQ[4*b^3 + 27*a^2*d, 0] &&  !IntegerQ[p]
 

rule 2490

Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 
, x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su 
bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 
*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c 
, 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
 

rule 2526

Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] 
}, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp 
[1/(n*Coeff[Qn, x, n])   Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x 
, m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm 
, x] && PolyQ[Qn, x] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.48

method	result	size
gosper	\(-\frac {2 \left (-3 d x +c \right ) \left (243243 C \,d^{5} x^{5}+280665 B \,d^{5} x^{4}+623700 C c \,d^{4} x^{4}+331695 A \,d^{5} x^{3}+731430 B c \,d^{4} x^{3}+593460 C \,c^{2} d^{3} x^{3}+884520 A c \,d^{4} x^{2}+703080 B \,c^{2} d^{3} x^{2}+252000 C \,c^{3} d^{2} x^{2}+863460 A \,c^{2} d^{3} x +288360 B \,c^{3} d^{2} x +48000 C \,c^{4} d x +331344 A \,c^{3} d^{2}+38448 B \,c^{4} d +6400 C \,c^{5}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {3}{2}}}{405405 d^{3} \left (3 d x +2 c \right )^{3}}\)	\(191\)
default	\(-\frac {2 \left (-3 d x +c \right ) \left (243243 C \,d^{5} x^{5}+280665 B \,d^{5} x^{4}+623700 C c \,d^{4} x^{4}+331695 A \,d^{5} x^{3}+731430 B c \,d^{4} x^{3}+593460 C \,c^{2} d^{3} x^{3}+884520 A c \,d^{4} x^{2}+703080 B \,c^{2} d^{3} x^{2}+252000 C \,c^{3} d^{2} x^{2}+863460 A \,c^{2} d^{3} x +288360 B \,c^{3} d^{2} x +48000 C \,c^{4} d x +331344 A \,c^{3} d^{2}+38448 B \,c^{4} d +6400 C \,c^{5}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {3}{2}}}{405405 d^{3} \left (3 d x +2 c \right )^{3}}\)	\(191\)
orering	\(-\frac {2 \left (-3 d x +c \right ) \left (243243 C \,d^{5} x^{5}+280665 B \,d^{5} x^{4}+623700 C c \,d^{4} x^{4}+331695 A \,d^{5} x^{3}+731430 B c \,d^{4} x^{3}+593460 C \,c^{2} d^{3} x^{3}+884520 A c \,d^{4} x^{2}+703080 B \,c^{2} d^{3} x^{2}+252000 C \,c^{3} d^{2} x^{2}+863460 A \,c^{2} d^{3} x +288360 B \,c^{3} d^{2} x +48000 C \,c^{4} d x +331344 A \,c^{3} d^{2}+38448 B \,c^{4} d +6400 C \,c^{5}\right ) \left (-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}\right )^{\frac {3}{2}}}{405405 d^{3} \left (3 d x +2 c \right )^{3}}\)	\(191\)
trager	\(-\frac {2 \left (2189187 C \,d^{7} x^{7}+2525985 B \,d^{7} x^{6}+4153842 C c \,d^{6} x^{6}+2985255 A \,d^{7} x^{5}+4898880 B c \,d^{6} x^{5}+1842183 C \,c^{2} d^{5} x^{5}+5970510 A c \,d^{6} x^{4}+2219805 B \,c^{2} d^{5} x^{4}-669060 C \,c^{3} d^{4} x^{4}+2795715 A \,c^{2} d^{5} x^{3}-891810 B \,c^{3} d^{4} x^{3}-486540 C \,c^{4} d^{3} x^{3}-1314144 A \,d^{4} c^{3} x^{2}-681048 B \,c^{4} d^{3} x^{2}+21600 C \,c^{5} d^{2} x^{2}-1124604 A \,c^{4} d^{3} x +57672 B \,c^{5} d^{2} x +9600 C \,c^{6} d x +331344 A \,c^{5} d^{2}+38448 B \,c^{6} d +6400 C \,c^{7}\right ) \sqrt {-27 d^{3} x^{3}-27 c \,d^{2} x^{2}+4 c^{3}}}{405405 \left (3 d x +2 c \right ) d^{3}}\)	\(257\)

Input:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x,method=_RETURNV 
ERBOSE)
 

Output:

-2/405405*(-3*d*x+c)*(243243*C*d^5*x^5+280665*B*d^5*x^4+623700*C*c*d^4*x^4 
+331695*A*d^5*x^3+731430*B*c*d^4*x^3+593460*C*c^2*d^3*x^3+884520*A*c*d^4*x 
^2+703080*B*c^2*d^3*x^2+252000*C*c^3*d^2*x^2+863460*A*c^2*d^3*x+288360*B*c 
^3*d^2*x+48000*C*c^4*d*x+331344*A*c^3*d^2+38448*B*c^4*d+6400*C*c^5)*(-27*d 
^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2)/d^3/(3*d*x+2*c)^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.62 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (2189187 \, C d^{7} x^{7} + 6400 \, C c^{7} + 38448 \, B c^{6} d + 331344 \, A c^{5} d^{2} + 56133 \, {\left (74 \, C c d^{6} + 45 \, B d^{7}\right )} x^{6} + 5103 \, {\left (361 \, C c^{2} d^{5} + 960 \, B c d^{6} + 585 \, A d^{7}\right )} x^{5} - 2835 \, {\left (236 \, C c^{3} d^{4} - 783 \, B c^{2} d^{5} - 2106 \, A c d^{6}\right )} x^{4} - 135 \, {\left (3604 \, C c^{4} d^{3} + 6606 \, B c^{3} d^{4} - 20709 \, A c^{2} d^{5}\right )} x^{3} + 216 \, {\left (100 \, C c^{5} d^{2} - 3153 \, B c^{4} d^{3} - 6084 \, A c^{3} d^{4}\right )} x^{2} + 12 \, {\left (800 \, C c^{6} d + 4806 \, B c^{5} d^{2} - 93717 \, A c^{4} d^{3}\right )} x\right )} \sqrt {-27 \, d^{3} x^{3} - 27 \, c d^{2} x^{2} + 4 \, c^{3}}}{405405 \, {\left (3 \, d^{4} x + 2 \, c d^{3}\right )}} \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x, algorith 
m="fricas")
 

Output:

-2/405405*(2189187*C*d^7*x^7 + 6400*C*c^7 + 38448*B*c^6*d + 331344*A*c^5*d 
^2 + 56133*(74*C*c*d^6 + 45*B*d^7)*x^6 + 5103*(361*C*c^2*d^5 + 960*B*c*d^6 
 + 585*A*d^7)*x^5 - 2835*(236*C*c^3*d^4 - 783*B*c^2*d^5 - 2106*A*c*d^6)*x^ 
4 - 135*(3604*C*c^4*d^3 + 6606*B*c^3*d^4 - 20709*A*c^2*d^5)*x^3 + 216*(100 
*C*c^5*d^2 - 3153*B*c^4*d^3 - 6084*A*c^3*d^4)*x^2 + 12*(800*C*c^6*d + 4806 
*B*c^5*d^2 - 93717*A*c^4*d^3)*x)*sqrt(-27*d^3*x^3 - 27*c*d^2*x^2 + 4*c^3)/ 
(3*d^4*x + 2*c*d^3)
 

Sympy [F]

\[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=\int \left (- \left (- c + 3 d x\right ) \left (2 c + 3 d x\right )^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )\, dx \] Input:

integrate((C*x**2+B*x+A)*(-27*d**3*x**3-27*c*d**2*x**2+4*c**3)**(3/2),x)
 

Output:

Integral((-(-c + 3*d*x)*(2*c + 3*d*x)**2)**(3/2)*(A + B*x + C*x**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.58 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (8505 \, d^{5} x^{5} + 17010 \, c d^{4} x^{4} + 7965 \, c^{2} d^{3} x^{3} - 3744 \, c^{3} d^{2} x^{2} - 3204 \, c^{4} d x + 944 \, c^{5}\right )} \sqrt {-3 \, d x + c} A}{1155 \, d} - \frac {2 \, {\left (93555 \, d^{6} x^{6} + 181440 \, c d^{5} x^{5} + 82215 \, c^{2} d^{4} x^{4} - 33030 \, c^{3} d^{3} x^{3} - 25224 \, c^{4} d^{2} x^{2} + 2136 \, c^{5} d x + 1424 \, c^{6}\right )} \sqrt {-3 \, d x + c} B}{15015 \, d^{2}} - \frac {2 \, {\left (2189187 \, d^{7} x^{7} + 4153842 \, c d^{6} x^{6} + 1842183 \, c^{2} d^{5} x^{5} - 669060 \, c^{3} d^{4} x^{4} - 486540 \, c^{4} d^{3} x^{3} + 21600 \, c^{5} d^{2} x^{2} + 9600 \, c^{6} d x + 6400 \, c^{7}\right )} \sqrt {-3 \, d x + c} C}{405405 \, d^{3}} \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x, algorith 
m="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)*A/d - 2/15015*(93555*d^6*x^ 
6 + 181440*c*d^5*x^5 + 82215*c^2*d^4*x^4 - 33030*c^3*d^3*x^3 - 25224*c^4*d 
^2*x^2 + 2136*c^5*d*x + 1424*c^6)*sqrt(-3*d*x + c)*B/d^2 - 2/405405*(21891 
87*d^7*x^7 + 4153842*c*d^6*x^6 + 1842183*c^2*d^5*x^5 - 669060*c^3*d^4*x^4 
- 486540*c^4*d^3*x^3 + 21600*c^5*d^2*x^2 + 9600*c^6*d*x + 6400*c^7)*sqrt(- 
3*d*x + c)*C/d^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1733 vs. \(2 (386) = 772\).

Time = 0.14 (sec) , antiderivative size = 1733, normalized size of antiderivative = 4.35 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x, algorith 
m="giac")
 

Output:

2/1216215*(3243240*sqrt(-3*d*x + c)*A*c^5*sgn(-3*d*x - 2*c) + 540540*((-3* 
d*x + c)^(3/2) - 3*sqrt(-3*d*x + c)*c)*A*c^4*sgn(-3*d*x - 2*c) - 360360*(( 
-3*d*x + c)^(3/2) - 3*sqrt(-3*d*x + c)*c)*B*c^5*sgn(-3*d*x - 2*c)/d - 2702 
70*(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)*A*c^3*sgn(-3*d*x - 2*c) + 24024*(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*c^5*sgn(-3* 
d*x - 2*c)/d^2 - 36036*(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)*B*c^4*sgn(-3*d*x - 2*c)/d + 11583*(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)*A*c^2*sgn(-3*d*x - 2*c) - 
 5148*(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*c^4*sgn(-3*d* 
x - 2*c)/d^2 - 38610*(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)* 
B*c^3*sgn(-3*d*x - 2*c)/d + 5148*(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)*A*c*sgn(-3*d*x - 
2*c) - 1430*(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*c^3*sgn(-3*d*x - 2*c)/d^2 + 429*(...
 

Mupad [B] (verification not implemented)

Time = 12.61 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.86 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=-\frac {\sqrt {4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3}\,\left (\frac {1888\,A\,c^5}{3465\,d^2}-\frac {832\,A\,c^3\,x^2}{385}+\frac {54\,A\,d^3\,x^5}{11}+\frac {354\,A\,c^2\,d\,x^3}{77}-\frac {712\,A\,c^4\,x}{385\,d}+\frac {108\,A\,c\,d^2\,x^4}{11}\right )}{x+\frac {2\,c}{3\,d}}-\frac {\sqrt {4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3}\,\left (\frac {2848\,B\,c^6}{45045\,d^3}-\frac {1468\,B\,c^3\,x^3}{1001}+\frac {54\,B\,d^3\,x^6}{13}+\frac {522\,B\,c^2\,d\,x^4}{143}+\frac {1152\,B\,c\,d^2\,x^5}{143}+\frac {1424\,B\,c^5\,x}{15015\,d^2}-\frac {16816\,B\,c^4\,x^2}{15015\,d}\right )}{x+\frac {2\,c}{3\,d}}-\frac {2\,C\,\sqrt {4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3}\,\left (-4576\,c^6+11664\,c^5\,d\,x-6696\,c^4\,d^2\,x^2-233226\,c^3\,d^3\,x^3+15309\,c^2\,d^4\,x^4+898128\,c\,d^5\,x^5+729729\,d^6\,x^6\right )}{405405\,d^3}-\frac {384\,C\,c^7\,\sqrt {4\,c^3-27\,c\,d^2\,x^2-27\,d^3\,x^3}}{5005\,d^3\,\left (2\,c+3\,d\,x\right )} \] Input:

int((A + B*x + C*x^2)*(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*A*c^5)/(3465*d^2) - (8 
32*A*c^3*x^2)/385 + (54*A*d^3*x^5)/11 + (354*A*c^2*d*x^3)/77 - (712*A*c^4* 
x)/(385*d) + (108*A*c*d^2*x^4)/11))/(x + (2*c)/(3*d)) - ((4*c^3 - 27*d^3*x 
^3 - 27*c*d^2*x^2)^(1/2)*((2848*B*c^6)/(45045*d^3) - (1468*B*c^3*x^3)/1001 
 + (54*B*d^3*x^6)/13 + (522*B*c^2*d*x^4)/143 + (1152*B*c*d^2*x^5)/143 + (1 
424*B*c^5*x)/(15015*d^2) - (16816*B*c^4*x^2)/(15015*d)))/(x + (2*c)/(3*d)) 
 - (2*C*(4*c^3 - 27*d^3*x^3 - 27*c*d^2*x^2)^(1/2)*(729729*d^6*x^6 - 4576*c 
^6 + 898128*c*d^5*x^5 - 6696*c^4*d^2*x^2 - 233226*c^3*d^3*x^3 + 15309*c^2* 
d^4*x^4 + 11664*c^5*d*x))/(405405*d^3) - (384*C*c^7*(4*c^3 - 27*d^3*x^3 - 
27*c*d^2*x^2)^(1/2))/(5005*d^3*(2*c + 3*d*x))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.56 \[ \int \left (A+B x+C x^2\right ) \left (4 c^3-27 c d^2 x^2-27 d^3 x^3\right )^{3/2} \, dx=\frac {2 \sqrt {-3 d x +c}\, \left (-2189187 c \,d^{7} x^{7}-2525985 b \,d^{7} x^{6}-4153842 c^{2} d^{6} x^{6}-2985255 a \,d^{7} x^{5}-4898880 b c \,d^{6} x^{5}-1842183 c^{3} d^{5} x^{5}-5970510 a c \,d^{6} x^{4}-2219805 b \,c^{2} d^{5} x^{4}+669060 c^{4} d^{4} x^{4}-2795715 a \,c^{2} d^{5} x^{3}+891810 b \,c^{3} d^{4} x^{3}+486540 c^{5} d^{3} x^{3}+1314144 a \,c^{3} d^{4} x^{2}+681048 b \,c^{4} d^{3} x^{2}-21600 c^{6} d^{2} x^{2}+1124604 a \,c^{4} d^{3} x -57672 b \,c^{5} d^{2} x -9600 c^{7} d x -331344 a \,c^{5} d^{2}-38448 b \,c^{6} d -6400 c^{8}\right )}{405405 d^{3}} \] Input:

int((C*x^2+B*x+A)*(-27*d^3*x^3-27*c*d^2*x^2+4*c^3)^(3/2),x)
 

Output:

(2*sqrt(c - 3*d*x)*( - 331344*a*c**5*d**2 + 1124604*a*c**4*d**3*x + 131414 
4*a*c**3*d**4*x**2 - 2795715*a*c**2*d**5*x**3 - 5970510*a*c*d**6*x**4 - 29 
85255*a*d**7*x**5 - 38448*b*c**6*d - 57672*b*c**5*d**2*x + 681048*b*c**4*d 
**3*x**2 + 891810*b*c**3*d**4*x**3 - 2219805*b*c**2*d**5*x**4 - 4898880*b* 
c*d**6*x**5 - 2525985*b*d**7*x**6 - 6400*c**8 - 9600*c**7*d*x - 21600*c**6 
*d**2*x**2 + 486540*c**5*d**3*x**3 + 669060*c**4*d**4*x**4 - 1842183*c**3* 
d**5*x**5 - 4153842*c**2*d**6*x**6 - 2189187*c*d**7*x**7))/(405405*d**3)