Integrand size = 34, antiderivative size = 335 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx=\frac {A b^2+3 a (b B+3 a C)}{1062882 a^6 b^3 (3 a-b x)^2}+\frac {4 A b^2+9 a (b B+2 a C)}{1594323 a^7 b^3 (3 a-b x)}-\frac {4 A b^2-6 a b B+9 a^2 C}{3645 a^3 b^3 (3 a+2 b x)^5}-\frac {4 A b^2-9 a^2 C}{8748 a^4 b^3 (3 a+2 b x)^4}-\frac {8 A b^2+6 a b B-9 a^2 C}{59049 a^5 b^3 (3 a+2 b x)^3}-\frac {40 A b^2+48 a b B+9 a^2 C}{1062882 a^6 b^3 (3 a+2 b x)^2}-\frac {20 A b^2+30 a b B+27 a^2 C}{1594323 a^7 b^3 (3 a+2 b x)}-\frac {\left (28 A b^2+48 a b B+63 a^2 C\right ) \log (3 a-b x)}{14348907 a^8 b^3}+\frac {\left (28 A b^2+48 a b B+63 a^2 C\right ) \log (3 a+2 b x)}{14348907 a^8 b^3} \] Output:
1/1062882*(A*b^2+3*a*(B*b+3*C*a))/a^6/b^3/(-b*x+3*a)^2+1/1594323*(4*A*b^2+ 9*a*(B*b+2*C*a))/a^7/b^3/(-b*x+3*a)-1/3645*(4*A*b^2-6*B*a*b+9*C*a^2)/a^3/b ^3/(2*b*x+3*a)^5-1/8748*(4*A*b^2-9*C*a^2)/a^4/b^3/(2*b*x+3*a)^4-1/59049*(8 *A*b^2+6*B*a*b-9*C*a^2)/a^5/b^3/(2*b*x+3*a)^3-1/1062882*(40*A*b^2+48*B*a*b +9*C*a^2)/a^6/b^3/(2*b*x+3*a)^2-1/1594323*(20*A*b^2+30*B*a*b+27*C*a^2)/a^7 /b^3/(2*b*x+3*a)-1/14348907*(28*A*b^2+48*B*a*b+63*C*a^2)*ln(-b*x+3*a)/a^8/ b^3+1/14348907*(28*A*b^2+48*B*a*b+63*C*a^2)*ln(2*b*x+3*a)/a^8/b^3
Time = 0.18 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx=\frac {\frac {180 a \left (4 A b^2+9 a (b B+2 a C)\right )}{3 a-b x}+\frac {270 a^2 \left (A b^2+3 a (b B+3 a C)\right )}{(-3 a+b x)^2}-\frac {78732 a^5 \left (4 A b^2-6 a b B+9 a^2 C\right )}{(3 a+2 b x)^5}+\frac {32805 a^4 \left (-4 A b^2+9 a^2 C\right )}{(3 a+2 b x)^4}+\frac {4860 a^3 \left (-8 A b^2-6 a b B+9 a^2 C\right )}{(3 a+2 b x)^3}-\frac {270 a^2 \left (40 A b^2+48 a b B+9 a^2 C\right )}{(3 a+2 b x)^2}-\frac {180 a \left (20 A b^2+3 a (10 b B+9 a C)\right )}{3 a+2 b x}-20 \left (28 A b^2+48 a b B+63 a^2 C\right ) \log (3 a-b x)+20 \left (28 A b^2+48 a b B+63 a^2 C\right ) \log (3 a+2 b x)}{286978140 a^8 b^3} \] Input:
Integrate[(A + B*x + C*x^2)/(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^3,x]
Output:
((180*a*(4*A*b^2 + 9*a*(b*B + 2*a*C)))/(3*a - b*x) + (270*a^2*(A*b^2 + 3*a *(b*B + 3*a*C)))/(-3*a + b*x)^2 - (78732*a^5*(4*A*b^2 - 6*a*b*B + 9*a^2*C) )/(3*a + 2*b*x)^5 + (32805*a^4*(-4*A*b^2 + 9*a^2*C))/(3*a + 2*b*x)^4 + (48 60*a^3*(-8*A*b^2 - 6*a*b*B + 9*a^2*C))/(3*a + 2*b*x)^3 - (270*a^2*(40*A*b^ 2 + 48*a*b*B + 9*a^2*C))/(3*a + 2*b*x)^2 - (180*a*(20*A*b^2 + 3*a*(10*b*B + 9*a*C)))/(3*a + 2*b*x) - 20*(28*A*b^2 + 48*a*b*B + 63*a^2*C)*Log[3*a - b *x] + 20*(28*A*b^2 + 48*a*b*B + 63*a^2*C)*Log[3*a + 2*b*x])/(286978140*a^8 *b^3)
Time = 0.72 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2462, 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.
| \(\displaystyle \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {9 a (2 a C+b B)+4 A b^2}{1594323 a^7 b^2 (3 a-b x)^2}+\frac {3 a (3 a C+b B)+A b^2}{531441 a^6 b^2 (3 a-b x)^3}+\frac {63 a^2 C+48 a b B+28 A b^2}{14348907 a^8 b^2 (3 a-b x)}+\frac {2 \left (63 a^2 C+48 a b B+28 A b^2\right )}{14348907 a^8 b^2 (3 a+2 b x)}+\frac {2 \left (27 a^2 C+30 a b B+20 A b^2\right )}{1594323 a^7 b^2 (3 a+2 b x)^2}+\frac {2 \left (9 a^2 C+48 a b B+40 A b^2\right )}{531441 a^6 b^2 (3 a+2 b x)^3}-\frac {2 \left (9 a^2 C-6 a b B-8 A b^2\right )}{19683 a^5 b^2 (3 a+2 b x)^4}-\frac {2 \left (9 a^2 C-4 A b^2\right )}{2187 a^4 b^2 (3 a+2 b x)^5}+\frac {2 \left (9 a^2 C-6 a b B+4 A b^2\right )}{729 a^3 b^2 (3 a+2 b x)^6}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {9 a (2 a C+b B)+4 A b^2}{1594323 a^7 b^3 (3 a-b x)}+\frac {3 a (3 a C+b B)+A b^2}{1062882 a^6 b^3 (3 a-b x)^2}-\frac {\log (3 a-b x) \left (63 a^2 C+48 a b B+28 A b^2\right )}{14348907 a^8 b^3}+\frac {\log (3 a+2 b x) \left (63 a^2 C+48 a b B+28 A b^2\right )}{14348907 a^8 b^3}-\frac {27 a^2 C+30 a b B+20 A b^2}{1594323 a^7 b^3 (3 a+2 b x)}-\frac {9 a^2 C+48 a b B+40 A b^2}{1062882 a^6 b^3 (3 a+2 b x)^2}-\frac {-9 a^2 C+6 a b B+8 A b^2}{59049 a^5 b^3 (3 a+2 b x)^3}-\frac {4 A b^2-9 a^2 C}{8748 a^4 b^3 (3 a+2 b x)^4}-\frac {9 a^2 C-6 a b B+4 A b^2}{3645 a^3 b^3 (3 a+2 b x)^5}\) |
Input:
Int[(A + B*x + C*x^2)/(27*a^3 + 27*a^2*b*x - 4*b^3*x^3)^3,x]
Output:
(A*b^2 + 3*a*(b*B + 3*a*C))/(1062882*a^6*b^3*(3*a - b*x)^2) + (4*A*b^2 + 9 *a*(b*B + 2*a*C))/(1594323*a^7*b^3*(3*a - b*x)) - (4*A*b^2 - 6*a*b*B + 9*a ^2*C)/(3645*a^3*b^3*(3*a + 2*b*x)^5) - (4*A*b^2 - 9*a^2*C)/(8748*a^4*b^3*( 3*a + 2*b*x)^4) - (8*A*b^2 + 6*a*b*B - 9*a^2*C)/(59049*a^5*b^3*(3*a + 2*b* x)^3) - (40*A*b^2 + 48*a*b*B + 9*a^2*C)/(1062882*a^6*b^3*(3*a + 2*b*x)^2) - (20*A*b^2 + 30*a*b*B + 27*a^2*C)/(1594323*a^7*b^3*(3*a + 2*b*x)) - ((28* A*b^2 + 48*a*b*B + 63*a^2*C)*Log[3*a - b*x])/(14348907*a^8*b^3) + ((28*A*b ^2 + 48*a*b*B + 63*a^2*C)*Log[3*a + 2*b*x])/(14348907*a^8*b^3)
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.24 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.87
| method | result | size |
| norman | \(\frac {\frac {\left (187 A \,b^{2}+147 a b B +117 C \,a^{2}\right ) x^{3}}{6561 a^{4}}+\frac {\left (1561 A \,b^{2}+489 a b B -315 C \,a^{2}\right ) x^{2}}{13122 b \,a^{3}}-\frac {20 b \left (191 A \,b^{2}+15 a b B -117 C \,a^{2}\right ) x^{4}}{177147 a^{5}}-\frac {8 b^{2} \left (1151 A \,b^{2}+411 a b B -144 C \,a^{2}\right ) x^{5}}{885735 a^{6}}+\frac {4 b^{3} \left (178 A \,b^{2}-42 a b B -207 C \,a^{2}\right ) x^{6}}{885735 a^{7}}+\frac {8 b^{4} \left (2162 A \,b^{2}+582 a b B -603 C \,a^{2}\right ) x^{7}}{23914845 a^{8}}+\frac {\left (701 A \,b^{2}-48 a b B -63 C \,a^{2}\right ) x}{6561 a^{2} b^{2}}}{\left (2 b x +3 a \right )^{5} \left (-b x +3 a \right )^{2}}-\frac {\left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) \ln \left (-b x +3 a \right )}{14348907 a^{8} b^{3}}+\frac {\left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) \ln \left (2 b x +3 a \right )}{14348907 a^{8} b^{3}}\) | \(290\) |
| default | \(\frac {4 A \,b^{2}+9 a b B +18 C \,a^{2}}{1594323 b^{3} a^{7} \left (-b x +3 a \right )}+\frac {\left (-28 A \,b^{2}-48 a b B -63 C \,a^{2}\right ) \ln \left (-b x +3 a \right )}{14348907 a^{8} b^{3}}-\frac {-A \,b^{2}-3 a b B -9 C \,a^{2}}{1062882 a^{6} b^{3} \left (-b x +3 a \right )^{2}}-\frac {8 A \,b^{2}-18 C \,a^{2}}{17496 a^{4} b^{3} \left (2 b x +3 a \right )^{4}}-\frac {8 A \,b^{2}-12 a b B +18 C \,a^{2}}{7290 a^{3} b^{3} \left (2 b x +3 a \right )^{5}}-\frac {16 A \,b^{2}+12 a b B -18 C \,a^{2}}{118098 a^{5} b^{3} \left (2 b x +3 a \right )^{3}}-\frac {40 A \,b^{2}+60 a b B +54 C \,a^{2}}{3188646 a^{7} b^{3} \left (2 b x +3 a \right )}+\frac {\left (56 A \,b^{2}+96 a b B +126 C \,a^{2}\right ) \ln \left (2 b x +3 a \right )}{28697814 a^{8} b^{3}}-\frac {80 A \,b^{2}+96 a b B +18 C \,a^{2}}{2125764 a^{6} b^{3} \left (2 b x +3 a \right )^{2}}\) | \(319\) |
| risch | \(\frac {-\frac {16 b^{3} \left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) x^{6}}{1594323 a^{7}}-\frac {4 \left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) b^{2} x^{5}}{177147 a^{6}}+\frac {2 \left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) b \,x^{4}}{19683 a^{5}}+\frac {43 \left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) x^{3}}{118098 a^{4}}+\frac {17 \left (28 A \,b^{2}+48 a b B +63 C \,a^{2}\right ) x^{2}}{131220 a^{3} b}-\frac {\left (91 A \,b^{2}+156 a b B -99 C \,a^{2}\right ) x}{3645 a^{2} b^{2}}-\frac {2162 A \,b^{2}+582 a b B -603 C \,a^{2}}{43740 b^{3} a}}{\left (2 b x +3 a \right ) \left (-4 b^{3} x^{3}+27 b \,a^{2} x +27 a^{3}\right )^{2}}-\frac {28 \ln \left (-b x +3 a \right ) A}{14348907 a^{8} b}-\frac {16 \ln \left (-b x +3 a \right ) B}{4782969 a^{7} b^{2}}-\frac {7 \ln \left (-b x +3 a \right ) C}{1594323 a^{6} b^{3}}+\frac {28 \ln \left (2 b x +3 a \right ) A}{14348907 a^{8} b}+\frac {16 \ln \left (2 b x +3 a \right ) B}{4782969 a^{7} b^{2}}+\frac {7 \ln \left (2 b x +3 a \right ) C}{1594323 a^{6} b^{3}}\) | \(339\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1182\) |
Input:
int((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x,method=_RETURNVERBOSE )
Output:
(1/6561*(187*A*b^2+147*B*a*b+117*C*a^2)/a^4*x^3+1/13122/b*(1561*A*b^2+489* B*a*b-315*C*a^2)/a^3*x^2-20/177147*b*(191*A*b^2+15*B*a*b-117*C*a^2)/a^5*x^ 4-8/885735*b^2*(1151*A*b^2+411*B*a*b-144*C*a^2)/a^6*x^5+4/885735*b^3*(178* A*b^2-42*B*a*b-207*C*a^2)/a^7*x^6+8/23914845*b^4*(2162*A*b^2+582*B*a*b-603 *C*a^2)/a^8*x^7+1/6561*(701*A*b^2-48*B*a*b-63*C*a^2)/a^2/b^2*x)/(2*b*x+3*a )^5/(-b*x+3*a)^2-1/14348907*(28*A*b^2+48*B*a*b+63*C*a^2)*ln(-b*x+3*a)/a^8/ b^3+1/14348907*(28*A*b^2+48*B*a*b+63*C*a^2)*ln(2*b*x+3*a)/a^8/b^3
Leaf count of result is larger than twice the leaf count of optimal. 810 vs. \(2 (315) = 630\).
Time = 0.10 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.42 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx =\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x, algorithm="fri cas")
Output:
1/286978140*(3956283*C*a^9 - 3818502*B*a^8*b - 14184882*A*a^7*b^2 - 2880*( 63*C*a^3*b^6 + 48*B*a^2*b^7 + 28*A*a*b^8)*x^6 - 6480*(63*C*a^4*b^5 + 48*B* a^3*b^6 + 28*A*a^2*b^7)*x^5 + 29160*(63*C*a^5*b^4 + 48*B*a^4*b^5 + 28*A*a^ 3*b^6)*x^4 + 104490*(63*C*a^6*b^3 + 48*B*a^5*b^4 + 28*A*a^4*b^5)*x^3 + 371 79*(63*C*a^7*b^2 + 48*B*a^6*b^3 + 28*A*a^5*b^4)*x^2 + 78732*(99*C*a^8*b - 156*B*a^7*b^2 - 91*A*a^6*b^3)*x + 20*(137781*C*a^9 + 104976*B*a^8*b + 6123 6*A*a^7*b^2 + 32*(63*C*a^2*b^7 + 48*B*a*b^8 + 28*A*b^9)*x^7 + 48*(63*C*a^3 *b^6 + 48*B*a^2*b^7 + 28*A*a*b^8)*x^6 - 432*(63*C*a^4*b^5 + 48*B*a^3*b^6 + 28*A*a^2*b^7)*x^5 - 1080*(63*C*a^5*b^4 + 48*B*a^4*b^5 + 28*A*a^3*b^6)*x^4 + 810*(63*C*a^6*b^3 + 48*B*a^5*b^4 + 28*A*a^4*b^5)*x^3 + 5103*(63*C*a^7*b ^2 + 48*B*a^6*b^3 + 28*A*a^5*b^4)*x^2 + 5832*(63*C*a^8*b + 48*B*a^7*b^2 + 28*A*a^6*b^3)*x)*log(2*b*x + 3*a) - 20*(137781*C*a^9 + 104976*B*a^8*b + 61 236*A*a^7*b^2 + 32*(63*C*a^2*b^7 + 48*B*a*b^8 + 28*A*b^9)*x^7 + 48*(63*C*a ^3*b^6 + 48*B*a^2*b^7 + 28*A*a*b^8)*x^6 - 432*(63*C*a^4*b^5 + 48*B*a^3*b^6 + 28*A*a^2*b^7)*x^5 - 1080*(63*C*a^5*b^4 + 48*B*a^4*b^5 + 28*A*a^3*b^6)*x ^4 + 810*(63*C*a^6*b^3 + 48*B*a^5*b^4 + 28*A*a^4*b^5)*x^3 + 5103*(63*C*a^7 *b^2 + 48*B*a^6*b^3 + 28*A*a^5*b^4)*x^2 + 5832*(63*C*a^8*b + 48*B*a^7*b^2 + 28*A*a^6*b^3)*x)*log(b*x - 3*a))/(32*a^8*b^10*x^7 + 48*a^9*b^9*x^6 - 432 *a^10*b^8*x^5 - 1080*a^11*b^7*x^4 + 810*a^12*b^6*x^3 + 5103*a^13*b^5*x^2 + 5832*a^14*b^4*x + 2187*a^15*b^3)
Time = 3.26 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.56 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx=- \frac {1576098 A a^{6} b^{2} + 424278 B a^{7} b - 439587 C a^{8} + x^{6} \cdot \left (8960 A b^{8} + 15360 B a b^{7} + 20160 C a^{2} b^{6}\right ) + x^{5} \cdot \left (20160 A a b^{7} + 34560 B a^{2} b^{6} + 45360 C a^{3} b^{5}\right ) + x^{4} \left (- 90720 A a^{2} b^{6} - 155520 B a^{3} b^{5} - 204120 C a^{4} b^{4}\right ) + x^{3} \left (- 325080 A a^{3} b^{5} - 557280 B a^{4} b^{4} - 731430 C a^{5} b^{3}\right ) + x^{2} \left (- 115668 A a^{4} b^{4} - 198288 B a^{5} b^{3} - 260253 C a^{6} b^{2}\right ) + x \left (796068 A a^{5} b^{3} + 1364688 B a^{6} b^{2} - 866052 C a^{7} b\right )}{69735688020 a^{14} b^{3} + 185961834720 a^{13} b^{4} x + 162716605380 a^{12} b^{5} x^{2} + 25828032600 a^{11} b^{6} x^{3} - 34437376800 a^{10} b^{7} x^{4} - 13774950720 a^{9} b^{8} x^{5} + 1530550080 a^{8} b^{9} x^{6} + 1020366720 a^{7} b^{10} x^{7}} - \frac {\left (28 A b^{2} + 48 B a b + 63 C a^{2}\right ) \log {\left (x + \frac {- 84 A a b^{2} - 144 B a^{2} b - 189 C a^{3} - 9 a \left (28 A b^{2} + 48 B a b + 63 C a^{2}\right )}{112 A b^{3} + 192 B a b^{2} + 252 C a^{2} b} \right )}}{14348907 a^{8} b^{3}} + \frac {\left (28 A b^{2} + 48 B a b + 63 C a^{2}\right ) \log {\left (x + \frac {- 84 A a b^{2} - 144 B a^{2} b - 189 C a^{3} + 9 a \left (28 A b^{2} + 48 B a b + 63 C a^{2}\right )}{112 A b^{3} + 192 B a b^{2} + 252 C a^{2} b} \right )}}{14348907 a^{8} b^{3}} \] Input:
integrate((C*x**2+B*x+A)/(-4*b**3*x**3+27*a**2*b*x+27*a**3)**3,x)
Output:
-(1576098*A*a**6*b**2 + 424278*B*a**7*b - 439587*C*a**8 + x**6*(8960*A*b** 8 + 15360*B*a*b**7 + 20160*C*a**2*b**6) + x**5*(20160*A*a*b**7 + 34560*B*a **2*b**6 + 45360*C*a**3*b**5) + x**4*(-90720*A*a**2*b**6 - 155520*B*a**3*b **5 - 204120*C*a**4*b**4) + x**3*(-325080*A*a**3*b**5 - 557280*B*a**4*b**4 - 731430*C*a**5*b**3) + x**2*(-115668*A*a**4*b**4 - 198288*B*a**5*b**3 - 260253*C*a**6*b**2) + x*(796068*A*a**5*b**3 + 1364688*B*a**6*b**2 - 866052 *C*a**7*b))/(69735688020*a**14*b**3 + 185961834720*a**13*b**4*x + 16271660 5380*a**12*b**5*x**2 + 25828032600*a**11*b**6*x**3 - 34437376800*a**10*b** 7*x**4 - 13774950720*a**9*b**8*x**5 + 1530550080*a**8*b**9*x**6 + 10203667 20*a**7*b**10*x**7) - (28*A*b**2 + 48*B*a*b + 63*C*a**2)*log(x + (-84*A*a* b**2 - 144*B*a**2*b - 189*C*a**3 - 9*a*(28*A*b**2 + 48*B*a*b + 63*C*a**2)) /(112*A*b**3 + 192*B*a*b**2 + 252*C*a**2*b))/(14348907*a**8*b**3) + (28*A* b**2 + 48*B*a*b + 63*C*a**2)*log(x + (-84*A*a*b**2 - 144*B*a**2*b - 189*C* a**3 + 9*a*(28*A*b**2 + 48*B*a*b + 63*C*a**2))/(112*A*b**3 + 192*B*a*b**2 + 252*C*a**2*b))/(14348907*a**8*b**3)
Time = 0.05 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx=\frac {439587 \, C a^{8} - 424278 \, B a^{7} b - 1576098 \, A a^{6} b^{2} - 320 \, {\left (63 \, C a^{2} b^{6} + 48 \, B a b^{7} + 28 \, A b^{8}\right )} x^{6} - 720 \, {\left (63 \, C a^{3} b^{5} + 48 \, B a^{2} b^{6} + 28 \, A a b^{7}\right )} x^{5} + 3240 \, {\left (63 \, C a^{4} b^{4} + 48 \, B a^{3} b^{5} + 28 \, A a^{2} b^{6}\right )} x^{4} + 11610 \, {\left (63 \, C a^{5} b^{3} + 48 \, B a^{4} b^{4} + 28 \, A a^{3} b^{5}\right )} x^{3} + 4131 \, {\left (63 \, C a^{6} b^{2} + 48 \, B a^{5} b^{3} + 28 \, A a^{4} b^{4}\right )} x^{2} + 8748 \, {\left (99 \, C a^{7} b - 156 \, B a^{6} b^{2} - 91 \, A a^{5} b^{3}\right )} x}{31886460 \, {\left (32 \, a^{7} b^{10} x^{7} + 48 \, a^{8} b^{9} x^{6} - 432 \, a^{9} b^{8} x^{5} - 1080 \, a^{10} b^{7} x^{4} + 810 \, a^{11} b^{6} x^{3} + 5103 \, a^{12} b^{5} x^{2} + 5832 \, a^{13} b^{4} x + 2187 \, a^{14} b^{3}\right )}} + \frac {{\left (63 \, C a^{2} + 48 \, B a b + 28 \, A b^{2}\right )} \log \left (2 \, b x + 3 \, a\right )}{14348907 \, a^{8} b^{3}} - \frac {{\left (63 \, C a^{2} + 48 \, B a b + 28 \, A b^{2}\right )} \log \left (b x - 3 \, a\right )}{14348907 \, a^{8} b^{3}} \] Input:
integrate((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x, algorithm="max ima")
Output:
1/31886460*(439587*C*a^8 - 424278*B*a^7*b - 1576098*A*a^6*b^2 - 320*(63*C* a^2*b^6 + 48*B*a*b^7 + 28*A*b^8)*x^6 - 720*(63*C*a^3*b^5 + 48*B*a^2*b^6 + 28*A*a*b^7)*x^5 + 3240*(63*C*a^4*b^4 + 48*B*a^3*b^5 + 28*A*a^2*b^6)*x^4 + 11610*(63*C*a^5*b^3 + 48*B*a^4*b^4 + 28*A*a^3*b^5)*x^3 + 4131*(63*C*a^6*b^ 2 + 48*B*a^5*b^3 + 28*A*a^4*b^4)*x^2 + 8748*(99*C*a^7*b - 156*B*a^6*b^2 - 91*A*a^5*b^3)*x)/(32*a^7*b^10*x^7 + 48*a^8*b^9*x^6 - 432*a^9*b^8*x^5 - 108 0*a^10*b^7*x^4 + 810*a^11*b^6*x^3 + 5103*a^12*b^5*x^2 + 5832*a^13*b^4*x + 2187*a^14*b^3) + 1/14348907*(63*C*a^2 + 48*B*a*b + 28*A*b^2)*log(2*b*x + 3 *a)/(a^8*b^3) - 1/14348907*(63*C*a^2 + 48*B*a*b + 28*A*b^2)*log(b*x - 3*a) /(a^8*b^3)
Time = 0.14 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx=\frac {{\left (63 \, C a^{2} + 48 \, B a b + 28 \, A b^{2}\right )} \log \left ({\left | 2 \, b x + 3 \, a \right |}\right )}{14348907 \, a^{8} b^{3}} - \frac {{\left (63 \, C a^{2} + 48 \, B a b + 28 \, A b^{2}\right )} \log \left ({\left | b x - 3 \, a \right |}\right )}{14348907 \, a^{8} b^{3}} + \frac {439587 \, C a^{9} - 424278 \, B a^{8} b - 1576098 \, A a^{7} b^{2} - 320 \, {\left (63 \, C a^{3} b^{6} + 48 \, B a^{2} b^{7} + 28 \, A a b^{8}\right )} x^{6} - 720 \, {\left (63 \, C a^{4} b^{5} + 48 \, B a^{3} b^{6} + 28 \, A a^{2} b^{7}\right )} x^{5} + 3240 \, {\left (63 \, C a^{5} b^{4} + 48 \, B a^{4} b^{5} + 28 \, A a^{3} b^{6}\right )} x^{4} + 11610 \, {\left (63 \, C a^{6} b^{3} + 48 \, B a^{5} b^{4} + 28 \, A a^{4} b^{5}\right )} x^{3} + 4131 \, {\left (63 \, C a^{7} b^{2} + 48 \, B a^{6} b^{3} + 28 \, A a^{5} b^{4}\right )} x^{2} + 8748 \, {\left (99 \, C a^{8} b - 156 \, B a^{7} b^{2} - 91 \, A a^{6} b^{3}\right )} x}{31886460 \, {\left (2 \, b x + 3 \, a\right )}^{5} {\left (b x - 3 \, a\right )}^{2} a^{8} b^{3}} \] Input:
integrate((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x, algorithm="gia c")
Output:
1/14348907*(63*C*a^2 + 48*B*a*b + 28*A*b^2)*log(abs(2*b*x + 3*a))/(a^8*b^3 ) - 1/14348907*(63*C*a^2 + 48*B*a*b + 28*A*b^2)*log(abs(b*x - 3*a))/(a^8*b ^3) + 1/31886460*(439587*C*a^9 - 424278*B*a^8*b - 1576098*A*a^7*b^2 - 320* (63*C*a^3*b^6 + 48*B*a^2*b^7 + 28*A*a*b^8)*x^6 - 720*(63*C*a^4*b^5 + 48*B* a^3*b^6 + 28*A*a^2*b^7)*x^5 + 3240*(63*C*a^5*b^4 + 48*B*a^4*b^5 + 28*A*a^3 *b^6)*x^4 + 11610*(63*C*a^6*b^3 + 48*B*a^5*b^4 + 28*A*a^4*b^5)*x^3 + 4131* (63*C*a^7*b^2 + 48*B*a^6*b^3 + 28*A*a^5*b^4)*x^2 + 8748*(99*C*a^8*b - 156* B*a^7*b^2 - 91*A*a^6*b^3)*x)/((2*b*x + 3*a)^5*(b*x - 3*a)^2*a^8*b^3)
Time = 12.08 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {4\,b\,x}{9\,a}-\frac {1}{3}\right )\,\left (63\,C\,a^2+48\,B\,a\,b+28\,A\,b^2\right )}{14348907\,a^8\,b^3}-\frac {\frac {-603\,C\,a^2+582\,B\,a\,b+2162\,A\,b^2}{43740\,a\,b^3}-\frac {43\,x^3\,\left (63\,C\,a^2+48\,B\,a\,b+28\,A\,b^2\right )}{118098\,a^4}-\frac {17\,x^2\,\left (63\,C\,a^2+48\,B\,a\,b+28\,A\,b^2\right )}{131220\,a^3\,b}+\frac {4\,b^2\,x^5\,\left (63\,C\,a^2+48\,B\,a\,b+28\,A\,b^2\right )}{177147\,a^6}+\frac {16\,b^3\,x^6\,\left (63\,C\,a^2+48\,B\,a\,b+28\,A\,b^2\right )}{1594323\,a^7}-\frac {2\,b\,x^4\,\left (63\,C\,a^2+48\,B\,a\,b+28\,A\,b^2\right )}{19683\,a^5}+\frac {x\,\left (-99\,C\,a^2+156\,B\,a\,b+91\,A\,b^2\right )}{3645\,a^2\,b^2}}{2187\,a^7+5832\,a^6\,b\,x+5103\,a^5\,b^2\,x^2+810\,a^4\,b^3\,x^3-1080\,a^3\,b^4\,x^4-432\,a^2\,b^5\,x^5+48\,a\,b^6\,x^6+32\,b^7\,x^7} \] Input:
int((A + B*x + C*x^2)/(27*a^3 - 4*b^3*x^3 + 27*a^2*b*x)^3,x)
Output:
(2*atanh((4*b*x)/(9*a) - 1/3)*(28*A*b^2 + 63*C*a^2 + 48*B*a*b))/(14348907* a^8*b^3) - ((2162*A*b^2 - 603*C*a^2 + 582*B*a*b)/(43740*a*b^3) - (43*x^3*( 28*A*b^2 + 63*C*a^2 + 48*B*a*b))/(118098*a^4) - (17*x^2*(28*A*b^2 + 63*C*a ^2 + 48*B*a*b))/(131220*a^3*b) + (4*b^2*x^5*(28*A*b^2 + 63*C*a^2 + 48*B*a* b))/(177147*a^6) + (16*b^3*x^6*(28*A*b^2 + 63*C*a^2 + 48*B*a*b))/(1594323* a^7) - (2*b*x^4*(28*A*b^2 + 63*C*a^2 + 48*B*a*b))/(19683*a^5) + (x*(91*A*b ^2 - 99*C*a^2 + 156*B*a*b))/(3645*a^2*b^2))/(2187*a^7 + 32*b^7*x^7 + 48*a* b^6*x^6 + 5103*a^5*b^2*x^2 + 810*a^4*b^3*x^3 - 1080*a^3*b^4*x^4 - 432*a^2* b^5*x^5 + 5832*a^6*b*x)
Time = 0.17 (sec) , antiderivative size = 838, normalized size of antiderivative = 2.50 \[ \int \frac {A+B x+C x^2}{\left (27 a^3+27 a^2 b x-4 b^3 x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((C*x^2+B*x+A)/(-4*b^3*x^3+27*a^2*b*x+27*a^3)^3,x)
Output:
( - 2755620*log(3*a - b*x)*a**8*c - 3324240*log(3*a - b*x)*a**7*b**2 - 734 8320*log(3*a - b*x)*a**7*b*c*x - 8864640*log(3*a - b*x)*a**6*b**3*x - 6429 780*log(3*a - b*x)*a**6*b**2*c*x**2 - 7756560*log(3*a - b*x)*a**5*b**4*x** 2 - 1020600*log(3*a - b*x)*a**5*b**3*c*x**3 - 1231200*log(3*a - b*x)*a**4* b**5*x**3 + 1360800*log(3*a - b*x)*a**4*b**4*c*x**4 + 1641600*log(3*a - b* x)*a**3*b**6*x**4 + 544320*log(3*a - b*x)*a**3*b**5*c*x**5 + 656640*log(3* a - b*x)*a**2*b**7*x**5 - 60480*log(3*a - b*x)*a**2*b**6*c*x**6 - 72960*lo g(3*a - b*x)*a*b**8*x**6 - 40320*log(3*a - b*x)*a*b**7*c*x**7 - 48640*log( 3*a - b*x)*b**9*x**7 + 2755620*log(3*a + 2*b*x)*a**8*c + 3324240*log(3*a + 2*b*x)*a**7*b**2 + 7348320*log(3*a + 2*b*x)*a**7*b*c*x + 8864640*log(3*a + 2*b*x)*a**6*b**3*x + 6429780*log(3*a + 2*b*x)*a**6*b**2*c*x**2 + 7756560 *log(3*a + 2*b*x)*a**5*b**4*x**2 + 1020600*log(3*a + 2*b*x)*a**5*b**3*c*x* *3 + 1231200*log(3*a + 2*b*x)*a**4*b**5*x**3 - 1360800*log(3*a + 2*b*x)*a* *4*b**4*c*x**4 - 1641600*log(3*a + 2*b*x)*a**3*b**6*x**4 - 544320*log(3*a + 2*b*x)*a**3*b**5*c*x**5 - 656640*log(3*a + 2*b*x)*a**2*b**7*x**5 + 60480 *log(3*a + 2*b*x)*a**2*b**6*c*x**6 + 72960*log(3*a + 2*b*x)*a*b**8*x**6 + 40320*log(3*a + 2*b*x)*a*b**7*c*x**7 + 48640*log(3*a + 2*b*x)*b**9*x**7 + 12223143*a**8*c - 8030664*a**7*b**2 + 29839428*a**7*b*c*x + 7147116*a**6*b **3*x + 21631617*a**6*b**2*c*x**2 + 26095284*a**5*b**4*x**2 + 9644670*a**5 *b**3*c*x**3 + 11634840*a**4*b**5*x**3 - 2245320*a**4*b**4*c*x**4 - 270...