Integrand size = 36, antiderivative size = 397 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=-\frac {1458 a^3 \left (A b^2+3 a (b B+3 a C)\right ) (3 a-b x) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2}}{5 b^3 (3 a+2 b x)^3}+\frac {486 a^2 \left (2 A b^2+9 a (b B+4 a C)\right ) (3 a-b x)^2 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2}}{7 b^3 (3 a+2 b x)^3}-\frac {6 a \left (4 A b^2+30 a b B+171 a^2 C\right ) (3 a-b x)^3 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2}}{b^3 (3 a+2 b x)^3}+\frac {4 \left (4 A b^2+66 a b B+603 a^2 C\right ) (3 a-b x)^4 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2}}{11 b^3 (3 a+2 b x)^3}-\frac {8 (2 b B+39 a C) (3 a-b x)^5 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2}}{13 b^3 (3 a+2 b x)^3}+\frac {16 C (3 a-b x)^6 \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2}}{15 b^3 (3 a+2 b x)^3} \] Output:
-1458/5*a^3*(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)^(3/2)/b^3/(2*b*x+3*a)^3+486/7*a^2*(2*A*b^2+9*a*(B*b+4*C*a))*(-b*x+3*a) ^2*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2)/b^3/(2*b*x+3*a)^3-6*a*(4*A*b^2+30* B*a*b+171*C*a^2)*(-b*x+3*a)^3*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2)/b^3/(2* b*x+3*a)^3+4/11*(4*A*b^2+66*B*a*b+603*C*a^2)*(-b*x+3*a)^4*(-4*b^3*x^3+27*a ^2*b*x+27*a^3)^(3/2)/b^3/(2*b*x+3*a)^3-8/13*(2*B*b+39*C*a)*(-b*x+3*a)^5*(- 4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2)/b^3/(2*b*x+3*a)^3+16/15*C*(-b*x+3*a)^6* (-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2)/b^3/(2*b*x+3*a)^3
Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.41 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=-\frac {2 (-3 a+b x)^2 \sqrt {(3 a-b x) (3 a+2 b x)^2} \left (935064 a^5 C+162 a^4 b (3333 B+4810 C x)+420 a b^4 x^2 (195 A+11 x (15 B+13 C x))+56 b^5 x^3 (195 A+11 x (15 B+13 C x))+90 a^2 b^3 x (2847 A+7 x (363 B+325 C x))+27 a^3 b^2 (14391 A+5 x (3333 B+3367 C x))\right )}{15015 b^3 (3 a+2 b x)} \] Input:
Integrate[(A + B*x + C*x^2)*(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(3/2),x]
Output:
(-2*(-3*a + b*x)^2*Sqrt[(3*a - b*x)*(3*a + 2*b*x)^2]*(935064*a^5*C + 162*a ^4*b*(3333*B + 4810*C*x) + 420*a*b^4*x^2*(195*A + 11*x*(15*B + 13*C*x)) + 56*b^5*x^3*(195*A + 11*x*(15*B + 13*C*x)) + 90*a^2*b^3*x*(2847*A + 7*x*(36 3*B + 325*C*x)) + 27*a^3*b^2*(14391*A + 5*x*(3333*B + 3367*C*x))))/(15015* b^3*(3*a + 2*b*x))
Time = 0.60 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.74, 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.
| \(\displaystyle \int \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \left (A+B x+C x^2\right ) \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle -\frac {\int -3 b \left (9 C a^2+4 A b^2+4 b^2 B x\right ) \left (27 a^3+27 b x a^2-4 b^3 x^3\right )^{3/2}dx}{12 b^3}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{30 b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \left (9 C a^2+4 A b^2+4 b^2 B x\right ) \left (27 a^3+27 b x a^2-4 b^3 x^3\right )^{3/2}dx}{4 b^2}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{30 b^3}\) |
| \(\Big \downarrow \) 2483 |
| \(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \int 1594323 \sqrt {3} a^6 (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2} \left (9 C a^2+4 A b^2+4 b^2 B x\right )dx}{6377292 \sqrt {3} a^6 b^2 (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{30 b^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \int (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2} \left (9 C a^2+4 A b^2+4 b^2 B x\right )dx}{4 b^2 (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{30 b^3}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \int \left (\frac {32 b B \left (3 a^3-a^2 b x\right )^{11/2}}{a^8}-\frac {8 \left (9 C a^2+66 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{9/2}}{a^6}+\frac {108 \left (9 C a^2+30 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{7/2}}{a^3}+486 \left (-4 A b^2-9 a (2 b B+a C)\right ) \left (3 a^3-a^2 b x\right )^{5/2}+729 a^3 \left (9 C a^2+12 b B a+4 A b^2\right ) \left (3 a^3-a^2 b x\right )^{3/2}\right )dx}{4 b^2 (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{30 b^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \left (\frac {972 \left (3 a^3-a^2 b x\right )^{7/2} \left (9 a (a C+2 b B)+4 A b^2\right )}{7 a^2 b}-\frac {1458 a \left (3 a^3-a^2 b x\right )^{5/2} \left (9 a^2 C+12 a b B+4 A b^2\right )}{5 b}-\frac {64 B \left (3 a^3-a^2 b x\right )^{13/2}}{13 a^{10}}+\frac {16 \left (3 a^3-a^2 b x\right )^{11/2} \left (9 a^2 C+66 a b B+4 A b^2\right )}{11 a^8 b}-\frac {24 \left (3 a^3-a^2 b x\right )^{9/2} \left (9 a^2 C+30 a b B+4 A b^2\right )}{a^5 b}\right )}{4 b^2 (3 a+2 b x)^3 \left (3 a^3-a^2 b x\right )^{3/2}}-\frac {C \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{5/2}}{30 b^3}\) |
Input:
Int[(A + B*x + C*x^2)*(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(3/2),x]
Output:
-1/30*(C*(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^(5/2))/b^3 + ((27*a^3 + 27*a^2* b*x - 4*b^3*x^3)^(3/2)*((-1458*a*(4*A*b^2 + 12*a*b*B + 9*a^2*C)*(3*a^3 - a ^2*b*x)^(5/2))/(5*b) + (972*(4*A*b^2 + 9*a*(2*b*B + a*C))*(3*a^3 - a^2*b*x )^(7/2))/(7*a^2*b) - (24*(4*A*b^2 + 30*a*b*B + 9*a^2*C)*(3*a^3 - a^2*b*x)^ (9/2))/(a^5*b) + (16*(4*A*b^2 + 66*a*b*B + 9*a^2*C)*(3*a^3 - a^2*b*x)^(11/ 2))/(11*a^8*b) - (64*B*(3*a^3 - a^2*b*x)^(13/2))/(13*a^10)))/(4*b^2*(3*a + 2*b*x)^3*(3*a^3 - a^2*b*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_))*((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]))))
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]
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]
Time = 0.41 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.48
| method | result | size |
| gosper | \(-\frac {2 \left (-b x +3 a \right ) \left (8008 b^{5} C \,x^{5}+9240 B \,b^{5} x^{4}+60060 C a \,b^{4} x^{4}+10920 A \,b^{5} x^{3}+69300 B a \,b^{4} x^{3}+204750 C \,a^{2} b^{3} x^{3}+81900 A a \,b^{4} x^{2}+228690 B \,a^{2} b^{3} x^{2}+454545 C \,a^{3} b^{2} x^{2}+256230 A \,a^{2} b^{3} x +449955 B \,a^{3} b^{2} x +779220 C \,a^{4} b x +388557 A \,a^{3} b^{2}+539946 B \,a^{4} b +935064 C \,a^{5}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {3}{2}}}{15015 b^{3} \left (2 b x +3 a \right )^{3}}\) | \(191\) |
| default | \(-\frac {2 \left (-b x +3 a \right ) \left (8008 b^{5} C \,x^{5}+9240 B \,b^{5} x^{4}+60060 C a \,b^{4} x^{4}+10920 A \,b^{5} x^{3}+69300 B a \,b^{4} x^{3}+204750 C \,a^{2} b^{3} x^{3}+81900 A a \,b^{4} x^{2}+228690 B \,a^{2} b^{3} x^{2}+454545 C \,a^{3} b^{2} x^{2}+256230 A \,a^{2} b^{3} x +449955 B \,a^{3} b^{2} x +779220 C \,a^{4} b x +388557 A \,a^{3} b^{2}+539946 B \,a^{4} b +935064 C \,a^{5}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {3}{2}}}{15015 b^{3} \left (2 b x +3 a \right )^{3}}\) | \(191\) |
| orering | \(-\frac {2 \left (-b x +3 a \right ) \left (8008 b^{5} C \,x^{5}+9240 B \,b^{5} x^{4}+60060 C a \,b^{4} x^{4}+10920 A \,b^{5} x^{3}+69300 B a \,b^{4} x^{3}+204750 C \,a^{2} b^{3} x^{3}+81900 A a \,b^{4} x^{2}+228690 B \,a^{2} b^{3} x^{2}+454545 C \,a^{3} b^{2} x^{2}+256230 A \,a^{2} b^{3} x +449955 B \,a^{3} b^{2} x +779220 C \,a^{4} b x +388557 A \,a^{3} b^{2}+539946 B \,a^{4} b +935064 C \,a^{5}\right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{\frac {3}{2}}}{15015 b^{3} \left (2 b x +3 a \right )^{3}}\) | \(191\) |
| risch | \(-\frac {2 \sqrt {\left (-b x +3 a \right ) \left (2 b x +3 a \right )^{2}}\, \left (8008 C \,b^{7} x^{7}+9240 B \,b^{7} x^{6}+12012 C a \,b^{6} x^{6}+10920 A \,b^{7} x^{5}+13860 B a \,b^{6} x^{5}-83538 C \,a^{2} b^{5} x^{5}+16380 A a \,b^{6} x^{4}-103950 B \,a^{2} b^{5} x^{4}-233415 C \,a^{3} b^{4} x^{4}-136890 A \,a^{2} b^{5} x^{3}-298485 B \,a^{3} b^{4} x^{3}-105300 C \,a^{4} b^{3} x^{3}-411723 A \,a^{3} b^{4} x^{2}-101574 B \,a^{4} b^{3} x^{2}+350649 C \,a^{5} b^{2} x^{2}-25272 A \,a^{4} b^{3} x +809919 B \,a^{5} b^{2} x +1402596 C \,a^{6} b x +3497013 A \,a^{5} b^{2}+4859514 B \,a^{6} b +8415576 C \,a^{7}\right )}{15015 \left (2 b x +3 a \right ) b^{3}}\) | \(253\) |
| trager | \(-\frac {2 \left (8008 C \,b^{7} x^{7}+9240 B \,b^{7} x^{6}+12012 C a \,b^{6} x^{6}+10920 A \,b^{7} x^{5}+13860 B a \,b^{6} x^{5}-83538 C \,a^{2} b^{5} x^{5}+16380 A a \,b^{6} x^{4}-103950 B \,a^{2} b^{5} x^{4}-233415 C \,a^{3} b^{4} x^{4}-136890 A \,a^{2} b^{5} x^{3}-298485 B \,a^{3} b^{4} x^{3}-105300 C \,a^{4} b^{3} x^{3}-411723 A \,a^{3} b^{4} x^{2}-101574 B \,a^{4} b^{3} x^{2}+350649 C \,a^{5} b^{2} x^{2}-25272 A \,a^{4} b^{3} x +809919 B \,a^{5} b^{2} x +1402596 C \,a^{6} b x +3497013 A \,a^{5} b^{2}+4859514 B \,a^{6} b +8415576 C \,a^{7}\right ) \sqrt {-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}}}{15015 \left (2 b x +3 a \right ) b^{3}}\) | \(255\) |
Input:
int((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2),x,method=_RETURNVER BOSE)
Output:
-2/15015*(-b*x+3*a)*(8008*C*b^5*x^5+9240*B*b^5*x^4+60060*C*a*b^4*x^4+10920 *A*b^5*x^3+69300*B*a*b^4*x^3+204750*C*a^2*b^3*x^3+81900*A*a*b^4*x^2+228690 *B*a^2*b^3*x^2+454545*C*a^3*b^2*x^2+256230*A*a^2*b^3*x+449955*B*a^3*b^2*x+ 779220*C*a^4*b*x+388557*A*a^3*b^2+539946*B*a^4*b+935064*C*a^5)*(-4*b^3*x^3 +27*a^2*b*x+27*a^3)^(3/2)/b^3/(2*b*x+3*a)^3
Time = 0.09 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.62 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (8008 \, C b^{7} x^{7} + 8415576 \, C a^{7} + 4859514 \, B a^{6} b + 3497013 \, A a^{5} b^{2} + 924 \, {\left (13 \, C a b^{6} + 10 \, B b^{7}\right )} x^{6} - 42 \, {\left (1989 \, C a^{2} b^{5} - 330 \, B a b^{6} - 260 \, A b^{7}\right )} x^{5} - 315 \, {\left (741 \, C a^{3} b^{4} + 330 \, B a^{2} b^{5} - 52 \, A a b^{6}\right )} x^{4} - 405 \, {\left (260 \, C a^{4} b^{3} + 737 \, B a^{3} b^{4} + 338 \, A a^{2} b^{5}\right )} x^{3} + 81 \, {\left (4329 \, C a^{5} b^{2} - 1254 \, B a^{4} b^{3} - 5083 \, A a^{3} b^{4}\right )} x^{2} + 243 \, {\left (5772 \, C a^{6} b + 3333 \, B a^{5} b^{2} - 104 \, A a^{4} b^{3}\right )} x\right )} \sqrt {-4 \, b^{3} x^{3} + 27 \, a^{2} b x + 27 \, a^{3}}}{15015 \, {\left (2 \, b^{4} x + 3 \, a b^{3}\right )}} \] Input:
integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2),x, algorithm= "fricas")
Output:
-2/15015*(8008*C*b^7*x^7 + 8415576*C*a^7 + 4859514*B*a^6*b + 3497013*A*a^5 *b^2 + 924*(13*C*a*b^6 + 10*B*b^7)*x^6 - 42*(1989*C*a^2*b^5 - 330*B*a*b^6 - 260*A*b^7)*x^5 - 315*(741*C*a^3*b^4 + 330*B*a^2*b^5 - 52*A*a*b^6)*x^4 - 405*(260*C*a^4*b^3 + 737*B*a^3*b^4 + 338*A*a^2*b^5)*x^3 + 81*(4329*C*a^5*b ^2 - 1254*B*a^4*b^3 - 5083*A*a^3*b^4)*x^2 + 243*(5772*C*a^6*b + 3333*B*a^5 *b^2 - 104*A*a^4*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)
\[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=\int \left (- \left (- 3 a + b x\right ) \left (3 a + 2 b x\right )^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((C*x**2+B*x+A)*(-4*b**3*x**3+27*a**2*b*x+27*a**3)**(3/2),x)
Output:
Integral((-(-3*a + b*x)*(3*a + 2*b*x)**2)**(3/2)*(A + B*x + C*x**2), x)
Time = 0.07 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.60 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=-\frac {2 \, {\left (280 \, b^{5} x^{5} + 420 \, a b^{4} x^{4} - 3510 \, a^{2} b^{3} x^{3} - 10557 \, a^{3} b^{2} x^{2} - 648 \, a^{4} b x + 89667 \, a^{5}\right )} \sqrt {-b x + 3 \, a} A}{385 \, b} - \frac {2 \, {\left (280 \, b^{6} x^{6} + 420 \, a b^{5} x^{5} - 3150 \, a^{2} b^{4} x^{4} - 9045 \, a^{3} b^{3} x^{3} - 3078 \, a^{4} b^{2} x^{2} + 24543 \, a^{5} b x + 147258 \, a^{6}\right )} \sqrt {-b x + 3 \, a} B}{455 \, b^{2}} - \frac {2 \, {\left (616 \, b^{7} x^{7} + 924 \, a b^{6} x^{6} - 6426 \, a^{2} b^{5} x^{5} - 17955 \, a^{3} b^{4} x^{4} - 8100 \, a^{4} b^{3} x^{3} + 26973 \, a^{5} b^{2} x^{2} + 107892 \, a^{6} b x + 647352 \, a^{7}\right )} \sqrt {-b x + 3 \, a} C}{1155 \, b^{3}} \] Input:
integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2),x, algorithm= "maxima")
Output:
-2/385*(280*b^5*x^5 + 420*a*b^4*x^4 - 3510*a^2*b^3*x^3 - 10557*a^3*b^2*x^2 - 648*a^4*b*x + 89667*a^5)*sqrt(-b*x + 3*a)*A/b - 2/455*(280*b^6*x^6 + 42 0*a*b^5*x^5 - 3150*a^2*b^4*x^4 - 9045*a^3*b^3*x^3 - 3078*a^4*b^2*x^2 + 245 43*a^5*b*x + 147258*a^6)*sqrt(-b*x + 3*a)*B/b^2 - 2/1155*(616*b^7*x^7 + 92 4*a*b^6*x^6 - 6426*a^2*b^5*x^5 - 17955*a^3*b^4*x^4 - 8100*a^4*b^3*x^3 + 26 973*a^5*b^2*x^2 + 107892*a^6*b*x + 647352*a^7)*sqrt(-b*x + 3*a)*C/b^3
Leaf count of result is larger than twice the leaf count of optimal. 1846 vs. \(2 (369) = 738\).
Time = 0.15 (sec) , antiderivative size = 1846, normalized size of antiderivative = 4.65 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2),x, algorithm= "giac")
Output:
2/15015*(3648645*sqrt(-b*x + 3*a)*A*a^5*sgn(-2*b*x - 3*a) - 1621620*((-b*x + 3*a)^(3/2) - 9*sqrt(-b*x + 3*a)*a)*A*a^4*sgn(-2*b*x - 3*a) - 1216215*(( -b*x + 3*a)^(3/2) - 9*sqrt(-b*x + 3*a)*a)*B*a^5*sgn(-2*b*x - 3*a)/b + 8108 1*((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^3*sgn(-2*b*x - 3*a) + 729729*((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^5*sgn(-2*b*x - 3*a)/b^2 + 972972*((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^4*sgn(-2*b*x - 3*a)/b - 38610*(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^2*sgn(-2*b*x - 3*a) + 13 8996*(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^4*sgn(-2*b *x - 3*a)/b^2 + 11583*(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^3*sgn(-2*b*x - 3*a)/b - 572*(35*(b*x - 3*a)^4*sqrt(-b*x + 3*a) + 54 0*(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*sgn(-2 *b*x - 3*a) + 1287*(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)*C*a^3*sgn(-2*b*x - 3*a)...
Time = 12.67 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.84 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=-\frac {\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}\,\left (\frac {89667\,A\,a^5}{385\,b^2}-\frac {10557\,A\,a^3\,x^2}{385}+\frac {8\,A\,b^3\,x^5}{11}-\frac {702\,A\,a^2\,b\,x^3}{77}-\frac {648\,A\,a^4\,x}{385\,b}+\frac {12\,A\,a\,b^2\,x^4}{11}\right )}{x+\frac {3\,a}{2\,b}}-\frac {2\,B\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}\,\left (9720\,a^5+1701\,a^4\,b\,x-2160\,a^3\,b^2\,x^2-1575\,a^2\,b^3\,x^3+140\,b^5\,x^5\right )}{455\,b^2}-\frac {2\,C\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}\,\left (38637\,a^6+10206\,a^5\,b\,x+2187\,a^4\,b^2\,x^2-4158\,a^3\,b^3\,x^3-3213\,a^2\,b^4\,x^4+308\,b^6\,x^6\right )}{1155\,b^3}-\frac {236196\,B\,a^6\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}}{455\,b^2\,\left (3\,a+2\,b\,x\right )}-\frac {354294\,C\,a^7\,\sqrt {27\,a^3+27\,a^2\,b\,x-4\,b^3\,x^3}}{385\,b^3\,\left (3\,a+2\,b\,x\right )} \] Input:
int((A + B*x + C*x^2)*(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(3/2),x)
Output:
- ((27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2)*((89667*A*a^5)/(385*b^2) - (105 57*A*a^3*x^2)/385 + (8*A*b^3*x^5)/11 - (702*A*a^2*b*x^3)/77 - (648*A*a^4*x )/(385*b) + (12*A*a*b^2*x^4)/11))/(x + (3*a)/(2*b)) - (2*B*(27*a^3 - 4*b^3 *x^3 + 27*a^2*b*x)^(1/2)*(9720*a^5 + 140*b^5*x^5 - 2160*a^3*b^2*x^2 - 1575 *a^2*b^3*x^3 + 1701*a^4*b*x))/(455*b^2) - (2*C*(27*a^3 - 4*b^3*x^3 + 27*a^ 2*b*x)^(1/2)*(38637*a^6 + 308*b^6*x^6 + 2187*a^4*b^2*x^2 - 4158*a^3*b^3*x^ 3 - 3213*a^2*b^4*x^4 + 10206*a^5*b*x))/(1155*b^3) - (236196*B*a^6*(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2))/(455*b^2*(3*a + 2*b*x)) - (354294*C*a^7*( 27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^(1/2))/(385*b^3*(3*a + 2*b*x))
Time = 0.16 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.41 \[ \int \left (A+B x+C x^2\right ) \left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^{3/2} \, dx=\frac {2 \sqrt {-b x +3 a}\, \left (-8008 b^{7} c \,x^{7}-12012 a \,b^{6} c \,x^{6}-9240 b^{8} x^{6}+83538 a^{2} b^{5} c \,x^{5}-24780 a \,b^{7} x^{5}+233415 a^{3} b^{4} c \,x^{4}+87570 a^{2} b^{6} x^{4}+105300 a^{4} b^{3} c \,x^{3}+435375 a^{3} b^{5} x^{3}-350649 a^{5} b^{2} c \,x^{2}+513297 a^{4} b^{4} x^{2}-1402596 a^{6} b c x -784647 a^{5} b^{3} x -8415576 a^{7} c -8356527 a^{6} b^{2}\right )}{15015 b^{3}} \] Input:
int((C*x^2+B*x+A)*(-4*b^3*x^3+27*a^2*b*x+27*a^3)^(3/2),x)
Output:
(2*sqrt(3*a - b*x)*( - 8415576*a**7*c - 8356527*a**6*b**2 - 1402596*a**6*b *c*x - 784647*a**5*b**3*x - 350649*a**5*b**2*c*x**2 + 513297*a**4*b**4*x** 2 + 105300*a**4*b**3*c*x**3 + 435375*a**3*b**5*x**3 + 233415*a**3*b**4*c*x **4 + 87570*a**2*b**6*x**4 + 83538*a**2*b**5*c*x**5 - 24780*a*b**7*x**5 - 12012*a*b**6*c*x**6 - 9240*b**8*x**6 - 8008*b**7*c*x**7))/(15015*b**3)