\(\int (A+B x+C x^2) (27 a^3+27 a^2 b x-4 b^3 x^3)^{5/2} \, dx\) [8]

Optimal result

Integrand size = 36, antiderivative size = 541 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=-\frac {118098 a^5 \left (A b^2+3 a (b B+3 a C)\right ) (3 a-b x) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{7 b^3 (3 a+2 b x)^5}+\frac {1458 a^4 \left (10 A b^2+39 a b B+144 a^2 C\right ) (3 a-b x)^2 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{b^3 (3 a+2 b x)^5}-\frac {1458 a^3 \left (40 A b^2+210 a b B+981 a^2 C\right ) (3 a-b x)^3 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{11 b^3 (3 a+2 b x)^5}+\frac {1620 a^2 \left (8 A b^2+60 a b B+369 a^2 C\right ) (3 a-b x)^4 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{13 b^3 (3 a+2 b x)^5}-\frac {48 a \left (2 A b^2+24 a b B+207 a^2 C\right ) (3 a-b x)^5 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{b^3 (3 a+2 b x)^5}+\frac {32 \left (2 A b^2+51 a b B+693 a^2 C\right ) (3 a-b x)^6 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{17 b^3 (3 a+2 b x)^5}-\frac {32 (2 b B+57 a C) (3 a-b x)^7 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{19 b^3 (3 a+2 b x)^5}+\frac {64 C (3 a-b x)^8 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{21 b^3 (3 a+2 b x)^5} \] Output:

-118098/7*a^5*(A*b^2+3*a*(B*b+3*C*a))*(-b*x+3*a)*(-4*b^3*x^3+27*a^2*b*x+27 
*a^3)^(5/2)/b^3/(2*b*x+3*a)^5+1458*a^4*(10*A*b^2+39*B*a*b+144*C*a^2)*(-b*x 
+3*a)^2*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)/b^3/(2*b*x+3*a)^5-1458/11*a^3 
*(40*A*b^2+210*B*a*b+981*C*a^2)*(-b*x+3*a)^3*(-4*b^3*x^3+27*a^2*b*x+27*a^3 
)^(5/2)/b^3/(2*b*x+3*a)^5+1620/13*a^2*(8*A*b^2+60*B*a*b+369*C*a^2)*(-b*x+3 
*a)^4*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)/b^3/(2*b*x+3*a)^5-48*a*(2*A*b^2 
+24*B*a*b+207*C*a^2)*(-b*x+3*a)^5*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)/b^3 
/(2*b*x+3*a)^5+32/17*(2*A*b^2+51*B*a*b+693*C*a^2)*(-b*x+3*a)^6*(-4*b^3*x^3 
+27*a^2*b*x+27*a^3)^(5/2)/b^3/(2*b*x+3*a)^5-32/19*(2*B*b+57*C*a)*(-b*x+3*a 
)^7*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)/b^3/(2*b*x+3*a)^5+64/21*C*(-b*x+3 
*a)^8*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)/b^3/(2*b*x+3*a)^5
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.40 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\frac {2 (-3 a+b x)^3 \sqrt {(3 a-b x) (3 a+2 b x)^2} \left (889234200 a^7 C+1458 a^6 b (366231 B+711550 C x)+48048 a b^6 x^4 (399 A+17 x (21 B+19 C x))+4576 b^7 x^5 (399 A+17 x (21 B+19 C x))+33264 a^2 b^5 x^3 (2679 A+13 x (187 B+171 C x))+1512 a^3 b^4 x^2 (159429 A+11 x (13821 B+13091 C x))+729 a^5 b^2 (526737 A+7 x (122077 B+152475 C x))+1134 a^4 b^3 x (360411 A+x (394689 B+407759 C x))\right )}{969969 b^3 (3 a+2 b x)} \] Input:

Integrate[(A + B*x + C*x^2)*(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(5/2),x]
 

Output:

(2*(-3*a + b*x)^3*Sqrt[(3*a - b*x)*(3*a + 2*b*x)^2]*(889234200*a^7*C + 145 
8*a^6*b*(366231*B + 711550*C*x) + 48048*a*b^6*x^4*(399*A + 17*x*(21*B + 19 
*C*x)) + 4576*b^7*x^5*(399*A + 17*x*(21*B + 19*C*x)) + 33264*a^2*b^5*x^3*( 
2679*A + 13*x*(187*B + 171*C*x)) + 1512*a^3*b^4*x^2*(159429*A + 11*x*(1382 
1*B + 13091*C*x)) + 729*a^5*b^2*(526737*A + 7*x*(122077*B + 152475*C*x)) + 
 1134*a^4*b^3*x*(360411*A + x*(394689*B + 407759*C*x))))/(969969*b^3*(3*a 
+ 2*b*x))
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2526, 27, 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.

\(\displaystyle \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \left (A+B x+C x^2\right ) \, dx\)

\(\Big \downarrow \) 2526

\(\displaystyle -\frac {\int -3 b \left (9 C a^2+4 A b^2+4 b^2 B x\right ) \left (27 a^3+27 b x a^2-4 b^3 x^3\right )^{5/2}dx}{12 b^3}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{7/2}}{42 b^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (9 C a^2+4 A b^2+4 b^2 B x\right ) \left (27 a^3+27 b x a^2-4 b^3 x^3\right )^{5/2}dx}{4 b^2}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{7/2}}{42 b^3}\)

\(\Big \downarrow \) 2483

\(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \int 31381059609 \sqrt {3} a^{10} (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (9 C a^2+4 A b^2+4 b^2 B x\right )dx}{125524238436 \sqrt {3} a^{10} b^2 (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{7/2}}{42 b^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \int (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2} \left (9 C a^2+4 A b^2+4 b^2 B x\right )dx}{4 b^2 (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{7/2}}{42 b^3}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \int \left (\frac {128 b B \left (3 a^3-a^2 b x\right )^{17/2}}{a^{12}}-\frac {32 \left (9 C a^2+102 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{15/2}}{a^{10}}+\frac {720 \left (9 C a^2+48 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{13/2}}{a^7}-\frac {6480 \left (9 C a^2+30 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{11/2}}{a^4}+\frac {29160 \left (9 C a^2+21 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{9/2}}{a}-13122 a^2 \left (45 C a^2+78 b B a+20 A b^2\right ) \left (3 a^3-a^2 b x\right )^{7/2}+59049 a^5 \left (9 C a^2+12 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{5/2}\right )dx}{4 b^2 (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{7/2}}{42 b^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \left (-\frac {58320 \left (3 a^3-a^2 b x\right )^{11/2} \left (9 a^2 C+21 a b B+4 A b^2\right )}{11 a^3 b}+\frac {2916 \left (3 a^3-a^2 b x\right )^{9/2} \left (45 a^2 C+78 a b B+20 A b^2\right )}{b}-\frac {118098 a^3 \left (3 a^3-a^2 b x\right )^{7/2} \left (9 a^2 C+12 a b B+4 A b^2\right )}{7 b}-\frac {256 B \left (3 a^3-a^2 b x\right )^{19/2}}{19 a^{14}}+\frac {64 \left (3 a^3-a^2 b x\right )^{17/2} \left (9 a^2 C+102 a b B+4 A b^2\right )}{17 a^{12} b}-\frac {96 \left (3 a^3-a^2 b x\right )^{15/2} \left (9 a^2 C+48 a b B+4 A b^2\right )}{a^9 b}+\frac {12960 \left (3 a^3-a^2 b x\right )^{13/2} \left (9 a^2 C+30 a b B+4 A b^2\right )}{13 a^6 b}\right )}{4 b^2 (3 a+2 b x)^5 \left (3 a^3-a^2 b x\right )^{5/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{7/2}}{42 b^3}\)

Input:

Int[(A + B*x + C*x^2)*(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(5/2),x]
 

Output:

-1/42*(C*(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(7/2))/b^3 + ((27*a^3 + 27*a^2* 
b*x - 4*b^3*x^3)^(5/2)*((-118098*a^3*(4*A*b^2 + 12*a*b*B + 9*a^2*C)*(3*a^3 
 - a^2*b*x)^(7/2))/(7*b) + (2916*(20*A*b^2 + 78*a*b*B + 45*a^2*C)*(3*a^3 - 
 a^2*b*x)^(9/2))/b - (58320*(4*A*b^2 + 21*a*b*B + 9*a^2*C)*(3*a^3 - a^2*b* 
x)^(11/2))/(11*a^3*b) + (12960*(4*A*b^2 + 30*a*b*B + 9*a^2*C)*(3*a^3 - a^2 
*b*x)^(13/2))/(13*a^6*b) - (96*(4*A*b^2 + 48*a*b*B + 9*a^2*C)*(3*a^3 - a^2 
*b*x)^(15/2))/(a^9*b) + (64*(4*A*b^2 + 102*a*b*B + 9*a^2*C)*(3*a^3 - a^2*b 
*x)^(17/2))/(17*a^12*b) - (256*B*(3*a^3 - a^2*b*x)^(19/2))/(19*a^14)))/(4* 
b^2*(3*a + 2*b*x)^5*(3*a^3 - a^2*b*x)^(5/2))
 

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 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 = 0.52 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.49

method	result	size
gosper	\(-\frac {2 \left (-b x +3 a \right ) \left (1478048 C \,b^{7} x^{7}+1633632 B \,b^{7} x^{6}[phone] C a \,b^{6} x^{6}+1825824 A \,b^{7} x^{5}[phone] B a \,b^{6} x^{5}[phone] C \,a^{2} b^{5} x^{5}[phone] A a \,b^{6} x^{4}[phone] B \,a^{2} b^{5} x^{4}[phone] C \,a^{3} b^{4} x^{4}[phone] A \,a^{2} b^{5} x^{3}[phone] B \,a^{3} b^{4} x^{3}[phone] C \,a^{4} b^{3} x^{3}[phone] A \,a^{3} b^{4} x^{2}[phone] B \,a^{4} b^{3} x^{2}[phone] C \,a^{5} b^{2} x^{2}[phone] A \,a^{4} b^{3} x [phone] B \,a^{5} b^{2} x [phone] C \,a^{6} b x [phone] A \,a^{5} b^{2}[phone] B \,a^{6} b [phone] C \,a^{7}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {5}{2}}}{969969 b^{3} \left (2 b x +3 a \right )^{5}}\)	\(263\)
default	\(-\frac {2 \left (-b x +3 a \right ) \left (1478048 C \,b^{7} x^{7}+1633632 B \,b^{7} x^{6}[phone] C a \,b^{6} x^{6}+1825824 A \,b^{7} x^{5}[phone] B a \,b^{6} x^{5}[phone] C \,a^{2} b^{5} x^{5}[phone] A a \,b^{6} x^{4}[phone] B \,a^{2} b^{5} x^{4}[phone] C \,a^{3} b^{4} x^{4}[phone] A \,a^{2} b^{5} x^{3}[phone] B \,a^{3} b^{4} x^{3}[phone] C \,a^{4} b^{3} x^{3}[phone] A \,a^{3} b^{4} x^{2}[phone] B \,a^{4} b^{3} x^{2}[phone] C \,a^{5} b^{2} x^{2}[phone] A \,a^{4} b^{3} x [phone] B \,a^{5} b^{2} x [phone] C \,a^{6} b x [phone] A \,a^{5} b^{2}[phone] B \,a^{6} b [phone] C \,a^{7}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {5}{2}}}{969969 b^{3} \left (2 b x +3 a \right )^{5}}\)	\(263\)
orering	\(-\frac {2 \left (-b x +3 a \right ) \left (1478048 C \,b^{7} x^{7}+1633632 B \,b^{7} x^{6}[phone] C a \,b^{6} x^{6}+1825824 A \,b^{7} x^{5}[phone] B a \,b^{6} x^{5}[phone] C \,a^{2} b^{5} x^{5}[phone] A a \,b^{6} x^{4}[phone] B \,a^{2} b^{5} x^{4}[phone] C \,a^{3} b^{4} x^{4}[phone] A \,a^{2} b^{5} x^{3}[phone] B \,a^{3} b^{4} x^{3}[phone] C \,a^{4} b^{3} x^{3}[phone] A \,a^{3} b^{4} x^{2}[phone] B \,a^{4} b^{3} x^{2}[phone] C \,a^{5} b^{2} x^{2}[phone] A \,a^{4} b^{3} x [phone] B \,a^{5} b^{2} x [phone] C \,a^{6} b x [phone] A \,a^{5} b^{2}[phone] B \,a^{6} b [phone] C \,a^{7}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {5}{2}}}{969969 b^{3} \left (2 b x +3 a \right )^{5}}\)	\(263\)
risch	\(-\frac {2 \sqrt {\left (-b x +3 a \right ) \left (2 b x +3 a \right )^{2}}\, \left (-1478048 C \,b^{10} x^{10}-1633632 B \,b^{10} x^{9}-2217072 C a \,b^{9} x^{9}-1825824 A \,b^{10} x^{8}-2450448 B a \,b^{9} x^{8}[phone] C \,a^{2} b^{8} x^{8}-2738736 A a \,b^{9} x^{7}[phone] B \,a^{2} b^{8} x^{7}[phone] C \,a^{3} b^{7} x^{7}[phone] A \,a^{2} b^{8} x^{6}[phone] B \,a^{3} b^{7} x^{6}-80345034 C \,a^{4} b^{6} x^{6}[phone] A \,a^{3} b^{7} x^{5}-98953974 B \,a^{4} b^{6} x^{5}-498649851 C \,a^{5} b^{5} x^{5}-127660050 A \,a^{4} b^{6} x^{4}-617927373 B \,a^{5} b^{5} x^{4}-640788813 C \,a^{6} b^{4} x^{4}-808081191 A \,a^{5} b^{5} x^{3}-805408677 B \,a^{6} b^{4} x^{3}-75668013 C \,a^{7} b^{3} x^{3}-1070613045 A \,a^{6} b^{4} x^{2}[phone] B \,a^{7} b^{3} x^{2}[phone] C \,a^{8} b^{2} x^{2}[phone] A \,a^{7} b^{3} x [phone] B \,a^{8} b^{2} x [phone] C \,a^{9} b x [phone] A \,a^{8} b^{2}[phone] B \,a^{9} b [phone] C \,a^{10}\right )}{969969 \left (2 b x +3 a \right ) b^{3}}\)	\(361\)
trager	\(-\frac {2 \left (-1478048 C \,b^{10} x^{10}-1633632 B \,b^{10} x^{9}-2217072 C a \,b^{9} x^{9}-1825824 A \,b^{10} x^{8}-2450448 B a \,b^{9} x^{8}[phone] C \,a^{2} b^{8} x^{8}-2738736 A a \,b^{9} x^{7}[phone] B \,a^{2} b^{8} x^{7}[phone] C \,a^{3} b^{7} x^{7}[phone] A \,a^{2} b^{8} x^{6}[phone] B \,a^{3} b^{7} x^{6}-80345034 C \,a^{4} b^{6} x^{6}[phone] A \,a^{3} b^{7} x^{5}-98953974 B \,a^{4} b^{6} x^{5}-498649851 C \,a^{5} b^{5} x^{5}-127660050 A \,a^{4} b^{6} x^{4}-617927373 B \,a^{5} b^{5} x^{4}-640788813 C \,a^{6} b^{4} x^{4}-808081191 A \,a^{5} b^{5} x^{3}-805408677 B \,a^{6} b^{4} x^{3}-75668013 C \,a^{7} b^{3} x^{3}-1070613045 A \,a^{6} b^{4} x^{2}[phone] B \,a^{7} b^{3} x^{2}[phone] C \,a^{8} b^{2} x^{2}[phone] A \,a^{7} b^{3} x [phone] B \,a^{8} b^{2} x [phone] C \,a^{9} b x [phone] A \,a^{8} b^{2}[phone] B \,a^{9} b [phone] C \,a^{10}\right ) \sqrt {-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}}}{969969 \left (2 b x +3 a \right ) b^{3}}\)	\(363\)

Input:

int((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x,method=_RETURNVER 
BOSE)
 

Output:

-2/969969*(-b*x+3*a)*(1478048*C*b^7*x^7+1633632*B*b^7*x^6+15519504*C*a*b^6 
*x^6+1825824*A*b^7*x^5+17153136*B*a*b^6*x^5+73945872*C*a^2*b^5*x^5+1917115 
2*A*a*b^6*x^4+80864784*B*a^2*b^5*x^4+217729512*C*a^3*b^4*x^4+89114256*A*a^ 
2*b^5*x^3+229870872*B*a^3*b^4*x^3+462398706*C*a^4*b^3*x^3+241056648*A*a^3* 
b^4*x^2+447577326*B*a^4*b^3*x^2+778079925*C*a^5*b^2*x^2+408706074*A*a^4*b^ 
3*x[phone]*B*a^5*b^2*x[phone]*C*a^6*b*x[phone]*A*a^5*b^2+5339647 
98*B*a^6*b[phone]*C*a^7)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2)/b^3/(2*b* 
x+3*a)^5
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.64 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\frac {2 \, {\left (1478048 \, C b^{10} x^{10} - 24009323400 \, C a^{10} - 14417049546 \, B a^{9} b - 10367764371 \, A a^{8} b^{2} + 116688 \, {\left (19 \, C a b^{9} + 14 \, B b^{10}\right )} x^{9} - 6864 \, {\left (3762 \, C a^{2} b^{8} - 357 \, B a b^{9} - 266 \, A b^{10}\right )} x^{8} - 10296 \, {\left (6669 \, C a^{3} b^{7} + 2856 \, B a^{2} b^{8} - 266 \, A a b^{9}\right )} x^{7} + 37422 \, {\left (2147 \, C a^{4} b^{6} - 2108 \, B a^{3} b^{7} - 912 \, A a^{2} b^{8}\right )} x^{6} + 567 \, {\left (879453 \, C a^{5} b^{5} + 174522 \, B a^{4} b^{6} - 163400 \, A a^{3} b^{7}\right )} x^{5} + 1701 \, {\left (376713 \, C a^{6} b^{4} + 363273 \, B a^{5} b^{5} + 75050 \, A a^{4} b^{6}\right )} x^{4} + 2187 \, {\left (34599 \, C a^{7} b^{3} + 368271 \, B a^{6} b^{4} + 369493 \, A a^{5} b^{5}\right )} x^{3} - 2187 \, {\left (457425 \, C a^{8} b^{2} + 32181 \, B a^{7} b^{3} - 489535 \, A a^{6} b^{4}\right )} x^{2} - 6561 \, {\left (609900 \, C a^{9} b + 366231 \, B a^{8} b^{2} + 101707 \, A a^{7} b^{3}\right )} x\right )} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}}}{969969 \, {\left (2 \, b^{4} x + 3 \, a b^{3}\right )}} \] Input:

integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x, algorithm= 
"fricas")
 

Output:

2/969969*(1478048*C*b^10*x^10 - 24009323400*C*a^10 - 14417049546*B*a^9*b - 
 10367764371*A*a^8*b^2 + 116688*(19*C*a*b^9 + 14*B*b^10)*x^9 - 6864*(3762* 
C*a^2*b^8 - 357*B*a*b^9 - 266*A*b^10)*x^8 - 10296*(6669*C*a^3*b^7 + 2856*B 
*a^2*b^8 - 266*A*a*b^9)*x^7 + 37422*(2147*C*a^4*b^6 - 2108*B*a^3*b^7 - 912 
*A*a^2*b^8)*x^6 + 567*(879453*C*a^5*b^5 + 174522*B*a^4*b^6 - 163400*A*a^3* 
b^7)*x^5 + 1701*(376713*C*a^6*b^4 + 363273*B*a^5*b^5 + 75050*A*a^4*b^6)*x^ 
4 + 2187*(34599*C*a^7*b^3 + 368271*B*a^6*b^4 + 369493*A*a^5*b^5)*x^3 - 218 
7*(457425*C*a^8*b^2 + 32181*B*a^7*b^3 - 489535*A*a^6*b^4)*x^2 - 6561*(6099 
00*C*a^9*b + 366231*B*a^8*b^2 + 101707*A*a^7*b^3)*x)*sqrt(-4*b^3*x^3 + 27* 
a^2*b*x + 27*a^3)/(2*b^4*x + 3*a*b^3)
 

Sympy [F]

\[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\int \left (- \left (- 3 a + b x\right ) \left (3 a + 2 b x\right )^{2}\right )^{\frac {5}{2}} \left (A + B x + C x^{2}\right )\, dx \] Input:

integrate((C*x**2+B*x+A)*(-4*b**3*x**3+27*a**2*b*x+27*a**3)**(5/2),x)
 

Output:

Integral((-(-3*a + b*x)*(3*a + 2*b*x)**2)**(5/2)*(A + B*x + C*x**2), x)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.62 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\frac {2 \, {\left (32032 \, b^{8} x^{8} + 48048 \, a b^{7} x^{7} - 598752 \, a^{2} b^{6} x^{6} - 1625400 \, a^{3} b^{5} x^{5} + 2239650 \, a^{4} b^{4} x^{4} + 14176863 \, a^{5} b^{3} x^{3} + 18782685 \, a^{6} b^{2} x^{2} - 11707011 \, a^{7} b x - 181890603 \, a^{8}\right )} \sqrt {-b x + 3 \, a} A}{17017 \, b} + \frac {2 \, {\left (32032 \, b^{9} x^{9} + 48048 \, a b^{8} x^{8} - 576576 \, a^{2} b^{7} x^{7} - 1546776 \, a^{3} b^{6} x^{6} + 1940274 \, a^{4} b^{5} x^{5} + 12116223 \, a^{5} b^{4} x^{4} + 15792327 \, a^{6} b^{3} x^{3} - 1379997 \, a^{7} b^{2} x^{2} - 47114541 \, a^{8} b x - 282687246 \, a^{9}\right )} \sqrt {-b x + 3 \, a} B}{19019 \, b^{2}} + \frac {2 \, {\left (77792 \, b^{10} x^{10} + 116688 \, a b^{9} x^{9} - 1359072 \, a^{2} b^{8} x^{8} - 3613896 \, a^{3} b^{7} x^{7} + 4228686 \, a^{4} b^{6} x^{6} + 26244729 \, a^{5} b^{5} x^{5} + 33725727 \, a^{6} b^{4} x^{4} + 3982527 \, a^{7} b^{3} x^{3} - 52652025 \, a^{8} b^{2} x^{2} - 210608100 \, a^{9} b x - 1263648600 \, a^{10}\right )} \sqrt {-b x + 3 \, a} C}{51051 \, b^{3}} \] Input:

integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x, algorithm= 
"maxima")
 

Output:

2/17017*(32032*b^8*x^8 + 48048*a*b^7*x^7 - 598752*a^2*b^6*x^6 - 1625400*a^ 
3*b^5*x^5 + 2239650*a^4*b^4*x^4 + 14176863*a^5*b^3*x^3 + 18782685*a^6*b^2* 
x^2 - 11707011*a^7*b*x - 181890603*a^8)*sqrt(-b*x + 3*a)*A/b + 2/19019*(32 
032*b^9*x^9 + 48048*a*b^8*x^8 - 576576*a^2*b^7*x^7 - 1546776*a^3*b^6*x^6 + 
 1940274*a^4*b^5*x^5 + 12116223*a^5*b^4*x^4 + 15792327*a^6*b^3*x^3 - 13799 
97*a^7*b^2*x^2 - 47114541*a^8*b*x - 282687246*a^9)*sqrt(-b*x + 3*a)*B/b^2 
+ 2/51051*(77792*b^10*x^10 + 116688*a*b^9*x^9 - 1359072*a^2*b^8*x^8 - 3613 
896*a^3*b^7*x^7 + 4228686*a^4*b^6*x^6 + 26244729*a^5*b^5*x^5 + 33725727*a^ 
6*b^4*x^4 + 3982527*a^7*b^3*x^3 - 52652025*a^8*b^2*x^2 - 210608100*a^9*b*x 
 - 1263648600*a^10)*sqrt(-b*x + 3*a)*C/b^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3745 vs. \(2 (505) = 1010\).

Time = 0.19 (sec) , antiderivative size = 3745, normalized size of antiderivative = 6.92 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x, algorithm= 
"giac")
 

Output:

2/4849845*(31819833045*sqrt(-b*x + 3*a)*A*a^8*sgn(-2*b*x - 3*a) - 24748759 
035*((-b*x + 3*a)^(3/2) - 9*sqrt(-b*x + 3*a)*a)*A*a^7*sgn(-2*b*x - 3*a) - 
10606611015*((-b*x + 3*a)^(3/2) - 9*sqrt(-b*x + 3*a)*a)*B*a^8*sgn(-2*b*x - 
 3*a)/b + 9192396213*((b*x - 3*a)^2*sqrt(-b*x + 3*a) - 10*(-b*x + 3*a)^(3/ 
2)*a + 45*sqrt(-b*x + 3*a)*a^2)*A*a^6*sgn(-2*b*x - 3*a) + 6363966609*((b*x 
 - 3*a)^2*sqrt(-b*x + 3*a) - 10*(-b*x + 3*a)^(3/2)*a + 45*sqrt(-b*x + 3*a) 
*a^2)*C*a^8*sgn(-2*b*x - 3*a)/b^2 + 14849255421*((b*x - 3*a)^2*sqrt(-b*x + 
 3*a) - 10*(-b*x + 3*a)^(3/2)*a + 45*sqrt(-b*x + 3*a)*a^2)*B*a^7*sgn(-2*b* 
x - 3*a)/b - 370389591*(5*(b*x - 3*a)^3*sqrt(-b*x + 3*a) + 63*(b*x - 3*a)^ 
2*sqrt(-b*x + 3*a)*a - 315*(-b*x + 3*a)^(3/2)*a^2 + 945*sqrt(-b*x + 3*a)*a 
^3)*A*a^5*sgn(-2*b*x - 3*a) + 2121322203*(5*(b*x - 3*a)^3*sqrt(-b*x + 3*a) 
 + 63*(b*x - 3*a)^2*sqrt(-b*x + 3*a)*a - 315*(-b*x + 3*a)^(3/2)*a^2 + 945* 
sqrt(-b*x + 3*a)*a^3)*C*a^7*sgn(-2*b*x - 3*a)/b^2 + 1313199459*(5*(b*x - 3 
*a)^3*sqrt(-b*x + 3*a) + 63*(b*x - 3*a)^2*sqrt(-b*x + 3*a)*a - 315*(-b*x + 
 3*a)^(3/2)*a^2 + 945*sqrt(-b*x + 3*a)*a^3)*B*a^6*sgn(-2*b*x - 3*a)/b - 62 
355150*(35*(b*x - 3*a)^4*sqrt(-b*x + 3*a) + 540*(b*x - 3*a)^3*sqrt(-b*x + 
3*a)*a + 3402*(b*x - 3*a)^2*sqrt(-b*x + 3*a)*a^2 - 11340*(-b*x + 3*a)^(3/2 
)*a^3 + 25515*sqrt(-b*x + 3*a)*a^4)*A*a^4*sgn(-2*b*x - 3*a) + 145911051*(3 
5*(b*x - 3*a)^4*sqrt(-b*x + 3*a) + 540*(b*x - 3*a)^3*sqrt(-b*x + 3*a)*a + 
3402*(b*x - 3*a)^2*sqrt(-b*x + 3*a)*a^2 - 11340*(-b*x + 3*a)^(3/2)*a^3 ...
 

Mupad [B] (verification not implemented)

Time = 12.56 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.82 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\frac {2\,A\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}\,\left (-9611865\,a^7+2505573\,a^6\,b\,x+4590513\,a^5\,b^2\,x^2+1665279\,a^4\,b^3\,x^3-363636\,a^3\,b^4\,x^4-299376\,a^2\,b^5\,x^5+16016\,b^7\,x^7\right )}{17017\,b}-\frac {2\,B\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}\,\left (17701578\,a^8+3903795\,a^7\,b\,x-2142531\,a^6\,b^2\,x^2-3835755\,a^5\,b^3\,x^3-1481571\,a^4\,b^4\,x^4+340956\,a^3\,b^5\,x^5+288288\,a^2\,b^6\,x^6-16016\,b^8\,x^8\right )}{19019\,b^2}-\frac {2\,C\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}\,\left (76842432\,a^9+18974412\,a^8\,b\,x+4901067\,a^7\,b^2\,x^2-4594887\,a^6\,b^3\,x^3-8178651\,a^5\,b^4\,x^4-3295809\,a^4\,b^5\,x^5+787644\,a^3\,b^6\,x^6+679536\,a^2\,b^7\,x^7-38896\,b^9\,x^9\right )}{51051\,b^3}-\frac {306110016\,A\,a^8\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}}{17017\,b\,\left (3\,a+2\,b\,x\right )}-\frac {459165024\,B\,a^9\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}}{19019\,b^2\,\left (3\,a+2\,b\,x\right )}-\frac {688747536\,C\,a^{10}\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}}{17017\,b^3\,\left (3\,a+2\,b\,x\right )} \] Input:

int((A + B*x + C*x^2)*(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(5/2),x)
 

Output:

(2*A*(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2)*(16016*b^7*x^7 - 9611865*a^7 
+ 4590513*a^5*b^2*x^2 + 1665279*a^4*b^3*x^3 - 363636*a^3*b^4*x^4 - 299376* 
a^2*b^5*x^5 + 2505573*a^6*b*x))/(17017*b) - (2*B*(27*a^3 - 4*b^3*x^3 + 27* 
a^2*b*x)^(1/2)*(17701578*a^8 - 16016*b^8*x^8 - 2142531*a^6*b^2*x^2 - 38357 
55*a^5*b^3*x^3 - 1481571*a^4*b^4*x^4 + 340956*a^3*b^5*x^5 + 288288*a^2*b^6 
*x^6 + 3903795*a^7*b*x))/(19019*b^2) - (2*C*(27*a^3 - 4*b^3*x^3 + 27*a^2*b 
*x)^(1/2)*(76842432*a^9 - 38896*b^9*x^9 + 4901067*a^7*b^2*x^2 - 4594887*a^ 
6*b^3*x^3 - 8178651*a^5*b^4*x^4 - 3295809*a^4*b^5*x^5 + 787644*a^3*b^6*x^6 
 + 679536*a^2*b^7*x^7 + 18974412*a^8*b*x))/(51051*b^3) - (306110016*A*a^8* 
(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2))/(17017*b*(3*a + 2*b*x)) - (459165 
024*B*a^9*(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2))/(19019*b^2*(3*a + 2*b*x 
)) - (688747536*C*a^10*(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2))/(17017*b^3 
*(3*a + 2*b*x))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.43 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2} \, dx=\frac {2 \sqrt {-b x +3 a}\, \left (1478048 b^{10} c \,x^{10}+2217072 a \,b^{9} c \,x^{9}+1633632 b^{11} x^{9}-25822368 a^{2} b^{8} c \,x^{8}+4276272 a \,b^{10} x^{8}-68664024 a^{3} b^{7} c \,x^{7}-26666640 a^{2} b^{9} x^{7}[phone] a^{4} b^{6} c \,x^{6}-113014440 a^{3} b^{8} x^{6}[phone] a^{5} b^{5} c \,x^{5}+6306174 a^{4} b^{7} x^{5}[phone] a^{6} b^{4} c \,x^{4}[phone] a^{5} b^{6} x^{4}[phone] a^{7} b^{3} c \,x^{3}[phone] a^{6} b^{5} x^{3}-1000388475 a^{8} b^{2} c \,x^{2}[phone] a^{7} b^{4} x^{2}-4001553900 a^{9} b c x -3070141218 a^{8} b^{3} x -24009323400 a^{10} c -24784813917 a^{9} b^{2}\right )}{969969 b^{3}} \] Input:

int((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(5/2),x)
 

Output:

(2*sqrt(3*a - b*x)*( - 24009323400*a**10*c - 24784813917*a**9*b**2 - 40015 
53900*a**9*b*c*x - 3070141218*a**8*b**3*x - 1000388475*a**8*b**2*c*x**2 + 
1000233198*a**7*b**4*x**2 + 75668013*a**7*b**3*c*x**3 + 1613489868*a**6*b* 
*5*x**3 + 640788813*a**6*b**4*c*x**4 + 745587423*a**5*b**6*x**4 + 49864985 
1*a**5*b**5*c*x**5 + 6306174*a**4*b**7*x**5 + 80345034*a**4*b**6*c*x**6 - 
113014440*a**3*b**8*x**6 - 68664024*a**3*b**7*c*x**7 - 26666640*a**2*b**9* 
x**7 - 25822368*a**2*b**8*c*x**8 + 4276272*a*b**10*x**8 + 2217072*a*b**9*c 
*x**9 + 1633632*b**11*x**9 + 1478048*b**10*c*x**10))/(969969*b**3)