Integrand size = 36, antiderivative size = 110 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=\frac {-4 c^2 C+6 B c d-9 A d^2}{81 c d^3 (2 c+3 d x)}-\frac {\left (c^2 C+3 B c d+9 A d^2\right ) \log (c-3 d x)}{243 c^2 d^3}-\frac {\left (8 c^2 C-3 B c d-9 A d^2\right ) \log (2 c+3 d x)}{243 c^2 d^3} \] Output:
1/81*(-9*A*d^2+6*B*c*d-4*C*c^2)/c/d^3/(3*d*x+2*c)-1/243*(9*A*d^2+3*B*c*d+C *c^2)*ln(-3*d*x+c)/c^2/d^3-1/243*(-9*A*d^2-3*B*c*d+8*C*c^2)*ln(3*d*x+2*c)/ c^2/d^3
Time = 0.03 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=\frac {-4 c^2 C+6 B c d-9 A d^2}{81 c d^3 (2 c+3 d x)}+\frac {\left (-c^2 C-3 B c d-9 A d^2\right ) \log (c-3 d x)}{243 c^2 d^3}+\frac {\left (-8 c^2 C+3 B c d+9 A d^2\right ) \log (2 c+3 d x)}{243 c^2 d^3} \] Input:
Integrate[(A + B*x + C*x^2)/(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3),x]
Output:
(-4*c^2*C + 6*B*c*d - 9*A*d^2)/(81*c*d^3*(2*c + 3*d*x)) + ((-(c^2*C) - 3*B *c*d - 9*A*d^2)*Log[c - 3*d*x])/(243*c^2*d^3) + ((-8*c^2*C + 3*B*c*d + 9*A *d^2)*Log[2*c + 3*d*x])/(243*c^2*d^3)
Time = 0.55 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, 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.
\(\displaystyle \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {9 A d^2-6 B c d+4 c^2 C}{27 c d^2 (2 c+3 d x)^2}+\frac {9 A d^2+3 B c d+c^2 C}{81 c^2 d^2 (c-3 d x)}+\frac {9 A d^2+3 B c d-8 c^2 C}{81 c^2 d^2 (2 c+3 d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {9 A d^2-6 B c d+4 c^2 C}{81 c d^3 (2 c+3 d x)}-\frac {\log (c-3 d x) \left (9 A d^2+3 B c d+c^2 C\right )}{243 c^2 d^3}-\frac {\log (2 c+3 d x) \left (-9 A d^2-3 B c d+8 c^2 C\right )}{243 c^2 d^3}\) |
Input:
Int[(A + B*x + C*x^2)/(4*c^3 - 27*c*d^2*x^2 - 27*d^3*x^3),x]
Output:
-1/81*(4*c^2*C - 6*B*c*d + 9*A*d^2)/(c*d^3*(2*c + 3*d*x)) - ((c^2*C + 3*B* c*d + 9*A*d^2)*Log[c - 3*d*x])/(243*c^2*d^3) - ((8*c^2*C - 3*B*c*d - 9*A*d ^2)*Log[2*c + 3*d*x])/(243*c^2*d^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 = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95
method | result | size |
norman | \(-\frac {9 A \,d^{2}-6 B c d +4 C \,c^{2}}{81 c \,d^{3} \left (3 d x +2 c \right )}+\frac {\left (9 A \,d^{2}+3 B c d -8 C \,c^{2}\right ) \ln \left (3 d x +2 c \right )}{243 c^{2} d^{3}}-\frac {\left (9 A \,d^{2}+3 B c d +C \,c^{2}\right ) \ln \left (-3 d x +c \right )}{243 c^{2} d^{3}}\) | \(105\) |
default | \(\frac {\left (-9 A \,d^{2}-3 B c d -C \,c^{2}\right ) \ln \left (-3 d x +c \right )}{243 c^{2} d^{3}}-\frac {9 A \,d^{2}-6 B c d +4 C \,c^{2}}{81 c \,d^{3} \left (3 d x +2 c \right )}+\frac {\left (9 A \,d^{2}+3 B c d -8 C \,c^{2}\right ) \ln \left (3 d x +2 c \right )}{243 c^{2} d^{3}}\) | \(106\) |
risch | \(-\frac {A}{18 c d \left (\frac {3 d x}{2}+c \right )}+\frac {B}{27 d^{2} \left (\frac {3 d x}{2}+c \right )}-\frac {2 c C}{81 d^{3} \left (\frac {3 d x}{2}+c \right )}+\frac {\ln \left (3 d x +2 c \right ) A}{27 c^{2} d}+\frac {\ln \left (3 d x +2 c \right ) B}{81 c \,d^{2}}-\frac {8 \ln \left (3 d x +2 c \right ) C}{243 d^{3}}-\frac {\ln \left (-3 d x +c \right ) A}{27 c^{2} d}-\frac {\ln \left (-3 d x +c \right ) B}{81 c \,d^{2}}-\frac {\ln \left (-3 d x +c \right ) C}{243 d^{3}}\) | \(144\) |
parallelrisch | \(-\frac {27 A \ln \left (d x -\frac {c}{3}\right ) x \,d^{3}-27 A \ln \left (d x +\frac {2 c}{3}\right ) x \,d^{3}+9 B \ln \left (d x -\frac {c}{3}\right ) x c \,d^{2}-9 B \ln \left (d x +\frac {2 c}{3}\right ) x c \,d^{2}+3 C \ln \left (d x -\frac {c}{3}\right ) x \,c^{2} d +24 C \ln \left (d x +\frac {2 c}{3}\right ) x \,c^{2} d +18 A \ln \left (d x -\frac {c}{3}\right ) c \,d^{2}-18 A \ln \left (d x +\frac {2 c}{3}\right ) c \,d^{2}+6 B \ln \left (d x -\frac {c}{3}\right ) c^{2} d -6 B \ln \left (d x +\frac {2 c}{3}\right ) c^{2} d +2 C \ln \left (d x -\frac {c}{3}\right ) c^{3}+16 C \ln \left (d x +\frac {2 c}{3}\right ) c^{3}+27 A c \,d^{2}-18 d \,c^{2} B +12 C \,c^{3}}{243 c^{2} d^{3} \left (3 d x +2 c \right )}\) | \(222\) |
Input:
int((C*x^2+B*x+A)/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3),x,method=_RETURNVERBOSE )
Output:
-1/81*(9*A*d^2-6*B*c*d+4*C*c^2)/c/d^3/(3*d*x+2*c)+1/243*(9*A*d^2+3*B*c*d-8 *C*c^2)/c^2/d^3*ln(3*d*x+2*c)-1/243*(9*A*d^2+3*B*c*d+C*c^2)*ln(-3*d*x+c)/c ^2/d^3
Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.38 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=-\frac {12 \, C c^{3} - 18 \, B c^{2} d + 27 \, A c d^{2} + {\left (16 \, C c^{3} - 6 \, B c^{2} d - 18 \, A c d^{2} + 3 \, {\left (8 \, C c^{2} d - 3 \, B c d^{2} - 9 \, A d^{3}\right )} x\right )} \log \left (3 \, d x + 2 \, c\right ) + {\left (2 \, C c^{3} + 6 \, B c^{2} d + 18 \, A c d^{2} + 3 \, {\left (C c^{2} d + 3 \, B c d^{2} + 9 \, A d^{3}\right )} x\right )} \log \left (3 \, d x - c\right )}{243 \, {\left (3 \, c^{2} d^{4} x + 2 \, c^{3} d^{3}\right )}} \] Input:
integrate((C*x^2+B*x+A)/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3),x, algorithm="fri cas")
Output:
-1/243*(12*C*c^3 - 18*B*c^2*d + 27*A*c*d^2 + (16*C*c^3 - 6*B*c^2*d - 18*A* c*d^2 + 3*(8*C*c^2*d - 3*B*c*d^2 - 9*A*d^3)*x)*log(3*d*x + 2*c) + (2*C*c^3 + 6*B*c^2*d + 18*A*c*d^2 + 3*(C*c^2*d + 3*B*c*d^2 + 9*A*d^3)*x)*log(3*d*x - c))/(3*c^2*d^4*x + 2*c^3*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (110) = 220\).
Time = 0.75 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.19 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=- \frac {9 A d^{2} - 6 B c d + 4 C c^{2}}{162 c^{2} d^{3} + 243 c d^{4} x} - \frac {\left (- 9 A d^{2} - 3 B c d + 8 C c^{2}\right ) \log {\left (x + \frac {- 9 A c d^{2} - 3 B c^{2} d - 10 C c^{3} + 3 c \left (- 9 A d^{2} - 3 B c d + 8 C c^{2}\right )}{- 54 A d^{3} - 18 B c d^{2} + 21 C c^{2} d} \right )}}{243 c^{2} d^{3}} - \frac {\left (9 A d^{2} + 3 B c d + C c^{2}\right ) \log {\left (x + \frac {- 9 A c d^{2} - 3 B c^{2} d - 10 C c^{3} + 3 c \left (9 A d^{2} + 3 B c d + C c^{2}\right )}{- 54 A d^{3} - 18 B c d^{2} + 21 C c^{2} d} \right )}}{243 c^{2} d^{3}} \] Input:
integrate((C*x**2+B*x+A)/(-27*d**3*x**3-27*c*d**2*x**2+4*c**3),x)
Output:
-(9*A*d**2 - 6*B*c*d + 4*C*c**2)/(162*c**2*d**3 + 243*c*d**4*x) - (-9*A*d* *2 - 3*B*c*d + 8*C*c**2)*log(x + (-9*A*c*d**2 - 3*B*c**2*d - 10*C*c**3 + 3 *c*(-9*A*d**2 - 3*B*c*d + 8*C*c**2))/(-54*A*d**3 - 18*B*c*d**2 + 21*C*c**2 *d))/(243*c**2*d**3) - (9*A*d**2 + 3*B*c*d + C*c**2)*log(x + (-9*A*c*d**2 - 3*B*c**2*d - 10*C*c**3 + 3*c*(9*A*d**2 + 3*B*c*d + C*c**2))/(-54*A*d**3 - 18*B*c*d**2 + 21*C*c**2*d))/(243*c**2*d**3)
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=-\frac {4 \, C c^{2} - 6 \, B c d + 9 \, A d^{2}}{81 \, {\left (3 \, c d^{4} x + 2 \, c^{2} d^{3}\right )}} - \frac {{\left (8 \, C c^{2} - 3 \, B c d - 9 \, A d^{2}\right )} \log \left (3 \, d x + 2 \, c\right )}{243 \, c^{2} d^{3}} - \frac {{\left (C c^{2} + 3 \, B c d + 9 \, A d^{2}\right )} \log \left (3 \, d x - c\right )}{243 \, c^{2} d^{3}} \] Input:
integrate((C*x^2+B*x+A)/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3),x, algorithm="max ima")
Output:
-1/81*(4*C*c^2 - 6*B*c*d + 9*A*d^2)/(3*c*d^4*x + 2*c^2*d^3) - 1/243*(8*C*c ^2 - 3*B*c*d - 9*A*d^2)*log(3*d*x + 2*c)/(c^2*d^3) - 1/243*(C*c^2 + 3*B*c* d + 9*A*d^2)*log(3*d*x - c)/(c^2*d^3)
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=-\frac {{\left (8 \, C c^{2} - 3 \, B c d - 9 \, A d^{2}\right )} \log \left ({\left | 3 \, d x + 2 \, c \right |}\right )}{243 \, c^{2} d^{3}} - \frac {{\left (C c^{2} + 3 \, B c d + 9 \, A d^{2}\right )} \log \left ({\left | 3 \, d x - c \right |}\right )}{243 \, c^{2} d^{3}} - \frac {4 \, C c^{3} - 6 \, B c^{2} d + 9 \, A c d^{2}}{81 \, {\left (3 \, d x + 2 \, c\right )} c^{2} d^{3}} \] Input:
integrate((C*x^2+B*x+A)/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3),x, algorithm="gia c")
Output:
-1/243*(8*C*c^2 - 3*B*c*d - 9*A*d^2)*log(abs(3*d*x + 2*c))/(c^2*d^3) - 1/2 43*(C*c^2 + 3*B*c*d + 9*A*d^2)*log(abs(3*d*x - c))/(c^2*d^3) - 1/81*(4*C*c ^3 - 6*B*c^2*d + 9*A*c*d^2)/((3*d*x + 2*c)*c^2*d^3)
Time = 12.44 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=\frac {\ln \left (2\,c+3\,d\,x\right )\,\left (-8\,C\,c^2+3\,B\,c\,d+9\,A\,d^2\right )}{243\,c^2\,d^3}-\frac {4\,C\,c^2-6\,B\,c\,d+9\,A\,d^2}{81\,c\,d^3\,\left (2\,c+3\,d\,x\right )}-\frac {\ln \left (c-3\,d\,x\right )\,\left (C\,c^2+3\,B\,c\,d+9\,A\,d^2\right )}{243\,c^2\,d^3} \] Input:
int(-(A + B*x + C*x^2)/(27*d^3*x^3 - 4*c^3 + 27*c*d^2*x^2),x)
Output:
(log(2*c + 3*d*x)*(9*A*d^2 - 8*C*c^2 + 3*B*c*d))/(243*c^2*d^3) - (9*A*d^2 + 4*C*c^2 - 6*B*c*d)/(81*c*d^3*(2*c + 3*d*x)) - (log(c - 3*d*x)*(9*A*d^2 + C*c^2 + 3*B*c*d))/(243*c^2*d^3)
Time = 0.18 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.99 \[ \int \frac {A+B x+C x^2}{4 c^3-27 c d^2 x^2-27 d^3 x^3} \, dx=\frac {36 \,\mathrm {log}\left (3 d x +2 c \right ) a c \,d^{2}+54 \,\mathrm {log}\left (3 d x +2 c \right ) a \,d^{3} x +12 \,\mathrm {log}\left (3 d x +2 c \right ) b \,c^{2} d +18 \,\mathrm {log}\left (3 d x +2 c \right ) b c \,d^{2} x -32 \,\mathrm {log}\left (3 d x +2 c \right ) c^{4}-48 \,\mathrm {log}\left (3 d x +2 c \right ) c^{3} d x -36 \,\mathrm {log}\left (-3 d x +c \right ) a c \,d^{2}-54 \,\mathrm {log}\left (-3 d x +c \right ) a \,d^{3} x -12 \,\mathrm {log}\left (-3 d x +c \right ) b \,c^{2} d -18 \,\mathrm {log}\left (-3 d x +c \right ) b c \,d^{2} x -4 \,\mathrm {log}\left (-3 d x +c \right ) c^{4}-6 \,\mathrm {log}\left (-3 d x +c \right ) c^{3} d x +81 a \,d^{3} x -54 b c \,d^{2} x +36 c^{3} d x}{486 c^{2} d^{3} \left (3 d x +2 c \right )} \] Input:
int((C*x^2+B*x+A)/(-27*d^3*x^3-27*c*d^2*x^2+4*c^3),x)
Output:
(36*log(2*c + 3*d*x)*a*c*d**2 + 54*log(2*c + 3*d*x)*a*d**3*x + 12*log(2*c + 3*d*x)*b*c**2*d + 18*log(2*c + 3*d*x)*b*c*d**2*x - 32*log(2*c + 3*d*x)*c **4 - 48*log(2*c + 3*d*x)*c**3*d*x - 36*log(c - 3*d*x)*a*c*d**2 - 54*log(c - 3*d*x)*a*d**3*x - 12*log(c - 3*d*x)*b*c**2*d - 18*log(c - 3*d*x)*b*c*d* *2*x - 4*log(c - 3*d*x)*c**4 - 6*log(c - 3*d*x)*c**3*d*x + 81*a*d**3*x - 5 4*b*c*d**2*x + 36*c**3*d*x)/(486*c**2*d**3*(2*c + 3*d*x))