Integrand size = 20, antiderivative size = 251 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {c \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right ) x}{e^6}-\frac {c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac {c^3 x^3}{3 e^4}-\frac {\left (c d^2-b d e+a e^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{2 e^7 (d+e x)^2}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \log (d+e x)}{e^7} \] Output:
c*(10*c^2*d^2+3*b^2*e^2-3*c*e*(-a*e+4*b*d))*x/e^6-1/2*c^2*(-3*b*e+4*c*d)*x ^2/e^5+1/3*c^3*x^3/e^4-1/3*(a*e^2-b*d*e+c*d^2)^3/e^7/(e*x+d)^3+3/2*(-b*e+2 *c*d)*(a*e^2-b*d*e+c*d^2)^2/e^7/(e*x+d)^2-3*(a*e^2-b*d*e+c*d^2)*(5*c^2*d^2 +b^2*e^2-c*e*(-a*e+5*b*d))/e^7/(e*x+d)-(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2* c*e*(-3*a*e+5*b*d))*ln(e*x+d)/e^7
Time = 0.08 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.04 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {6 c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right ) x+3 c^2 e^2 (-4 c d+3 b e) x^2+2 c^3 e^3 x^3-\frac {2 \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}+\frac {9 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )^2}{(d+e x)^2}-\frac {18 \left (5 c^3 d^4+b^2 e^3 (-b d+a e)+2 c^2 d^2 e (-5 b d+3 a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )}{d+e x}-6 (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) \log (d+e x)}{6 e^7} \] Input:
Integrate[(a + b*x + c*x^2)^3/(d + e*x)^4,x]
Output:
(6*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e))*x + 3*c^2*e^2*(-4*c *d + 3*b*e)*x^2 + 2*c^3*e^3*x^3 - (2*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e* x)^3 + (9*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x)^2 - (18*(5 *c^3*d^4 + b^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*( 6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))/(d + e*x) - 6*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(-5*b*d + 3*a*e))*Log[d + e*x])/(6*e^7)
Time = 0.53 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1140, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6 (d+e x)}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^2}+\frac {c \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}+\frac {3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 (d+e x)^4}-\frac {c^2 x (4 c d-3 b e)}{e^5}+\frac {c^3 x^2}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right ) \left (a e^2-b d e+c d^2\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \log (d+e x) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7}+\frac {c x \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )}{e^6}-\frac {\left (a e^2-b d e+c d^2\right )^3}{3 e^7 (d+e x)^3}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{2 e^7 (d+e x)^2}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}+\frac {c^3 x^3}{3 e^4}\) |
Input:
Int[(a + b*x + c*x^2)^3/(d + e*x)^4,x]
Output:
(c*(10*c^2*d^2 + 3*b^2*e^2 - 3*c*e*(4*b*d - a*e))*x)/e^6 - (c^2*(4*c*d - 3 *b*e)*x^2)/(2*e^5) + (c^3*x^3)/(3*e^4) - (c*d^2 - b*d*e + a*e^2)^3/(3*e^7* (d + e*x)^3) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(2*e^7*(d + e*x )^2) - (3*(c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e) ))/(e^7*(d + e*x)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Log[d + e*x])/e^7
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Time = 0.92 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.76
| method | result | size |
| norman | \(\frac {-\frac {2 e^{6} a^{3}+3 a^{2} b d \,e^{5}+6 d^{2} e^{4} a^{2} c +6 a \,b^{2} d^{2} e^{4}-66 a b c \,d^{3} e^{3}+132 d^{4} e^{2} a \,c^{2}-11 b^{3} d^{3} e^{3}+132 b^{2} c \,d^{4} e^{2}-330 b \,c^{2} d^{5} e +220 d^{6} c^{3}}{6 e^{7}}+\frac {c^{3} x^{6}}{3 e}-\frac {3 \left (e^{4} a^{2} c +a \,b^{2} e^{4}-6 a b c d \,e^{3}+12 d^{2} e^{2} a \,c^{2}-b^{3} d \,e^{3}+12 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +20 d^{4} c^{3}\right ) x^{2}}{e^{5}}-\frac {3 \left (a^{2} b \,e^{5}+2 d \,e^{4} a^{2} c +2 a \,b^{2} d \,e^{4}-18 a b c \,d^{2} e^{3}+36 d^{3} e^{2} a \,c^{2}-3 b^{3} d^{2} e^{3}+36 b^{2} c \,d^{3} e^{2}-90 b \,c^{2} d^{4} e +60 d^{5} c^{3}\right ) x}{2 e^{6}}+\frac {c \left (6 a c \,e^{2}+6 b^{2} e^{2}-15 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}+\frac {c^{2} \left (3 b e -2 c d \right ) x^{5}}{2 e^{2}}}{\left (e x +d \right )^{3}}+\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(441\) |
| default | \(\frac {c \left (\frac {1}{3} c^{2} e^{2} x^{3}+\frac {3}{2} b c \,e^{2} x^{2}-2 c^{2} d e \,x^{2}+3 a c \,e^{2} x +3 b^{2} e^{2} x -12 b c d e x +10 c^{2} d^{2} x \right )}{e^{6}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} a \,c^{2}-3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +15 d^{4} c^{3}}{e^{7} \left (e x +d \right )}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} a \,c^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{2 e^{7} \left (e x +d \right )^{2}}\) | \(450\) |
| risch | \(\frac {c^{3} x^{3}}{3 e^{4}}+\frac {3 c^{2} b \,x^{2}}{2 e^{4}}-\frac {2 c^{3} d \,x^{2}}{e^{5}}+\frac {3 c^{2} a x}{e^{4}}+\frac {3 c \,b^{2} x}{e^{4}}-\frac {12 c^{2} b d x}{e^{5}}+\frac {10 c^{3} d^{2} x}{e^{6}}+\frac {\left (-3 a^{2} c \,e^{5}-3 a \,b^{2} e^{5}+18 a b c d \,e^{4}-18 e^{3} a \,c^{2} d^{2}+3 b^{3} d \,e^{4}-18 b^{2} c \,d^{2} e^{3}+30 d^{3} b \,c^{2} e^{2}-15 d^{4} e \,c^{3}\right ) x^{2}+\left (-\frac {3}{2} a^{2} b \,e^{5}-3 d \,e^{4} a^{2} c -3 a \,b^{2} d \,e^{4}+27 a b c \,d^{2} e^{3}-30 d^{3} e^{2} a \,c^{2}+\frac {9}{2} b^{3} d^{2} e^{3}-30 b^{2} c \,d^{3} e^{2}+\frac {105}{2} b \,c^{2} d^{4} e -27 d^{5} c^{3}\right ) x -\frac {2 e^{6} a^{3}+3 a^{2} b d \,e^{5}+6 d^{2} e^{4} a^{2} c +6 a \,b^{2} d^{2} e^{4}-66 a b c \,d^{3} e^{3}+78 d^{4} e^{2} a \,c^{2}-11 b^{3} d^{3} e^{3}+78 b^{2} c \,d^{4} e^{2}-141 b \,c^{2} d^{5} e +74 d^{6} c^{3}}{6 e}}{e^{6} \left (e x +d \right )^{3}}+\frac {6 \ln \left (e x +d \right ) a b c}{e^{4}}-\frac {12 \ln \left (e x +d \right ) d a \,c^{2}}{e^{5}}+\frac {\ln \left (e x +d \right ) b^{3}}{e^{4}}-\frac {12 \ln \left (e x +d \right ) d \,b^{2} c}{e^{5}}+\frac {30 \ln \left (e x +d \right ) d^{2} b \,c^{2}}{e^{6}}-\frac {20 \ln \left (e x +d \right ) d^{3} c^{3}}{e^{7}}\) | \(480\) |
| parallelrisch | \(\frac {-132 d^{4} e^{2} a \,c^{2}-6 d^{2} e^{4} a^{2} c -360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e -220 d^{6} c^{3}-3 a^{2} b d \,e^{5}-6 a \,b^{2} d^{2} e^{4}-132 b^{2} c \,d^{4} e^{2}-2 e^{6} a^{3}-6 c^{3} d \,e^{5} x^{5}+30 c^{3} d^{2} e^{4} x^{4}-360 c^{3} d^{4} e^{2} x^{2}-540 c^{3} d^{5} e x -72 \ln \left (e x +d \right ) x^{3} a \,c^{2} d \,e^{5}-72 \ln \left (e x +d \right ) x^{3} b^{2} c d \,e^{5}+330 b \,c^{2} d^{5} e +162 x a b c \,d^{2} e^{4}+108 \ln \left (e x +d \right ) x^{2} a b c d \,e^{5}+108 \ln \left (e x +d \right ) x a b c \,d^{2} e^{4}-216 \ln \left (e x +d \right ) x^{2} b^{2} c \,d^{2} e^{4}+180 \ln \left (e x +d \right ) x^{3} b \,c^{2} d^{2} e^{4}-120 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}-216 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}-216 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}+540 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+36 \ln \left (e x +d \right ) a b c \,d^{3} e^{3}+108 x^{2} a b c d \,e^{5}+540 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}-216 \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{4}+18 \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}+2 c^{3} e^{6} x^{6}+810 x b \,c^{2} d^{4} e^{2}+18 \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{5}-216 x^{2} b^{2} c \,d^{2} e^{4}+540 x^{2} b \,c^{2} d^{3} e^{3}-18 x a \,b^{2} d \,e^{5}-324 x \,b^{2} c \,d^{3} e^{3}+36 \ln \left (e x +d \right ) x^{3} a b c \,e^{6}-72 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}-72 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}+180 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -45 x^{4} b \,c^{2} d \,e^{5}-216 x^{2} a \,c^{2} d^{2} e^{4}-18 x \,a^{2} c d \,e^{5}-324 x a \,c^{2} d^{3} e^{3}+6 \ln \left (e x +d \right ) x^{3} b^{3} e^{6}-360 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+9 x^{5} b \,c^{2} e^{6}+18 x^{4} a \,c^{2} e^{6}+18 x^{4} b^{2} c \,e^{6}-18 x^{2} a^{2} c \,e^{6}-18 x^{2} a \,b^{2} e^{6}+18 x^{2} b^{3} d \,e^{5}-9 x \,a^{2} b \,e^{6}+27 x \,b^{3} d^{2} e^{4}+6 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}+11 b^{3} d^{3} e^{3}-120 \ln \left (e x +d \right ) c^{3} d^{6}+66 a b c \,d^{3} e^{3}}{6 e^{7} \left (e x +d \right )^{3}}\) | \(859\) |
Input:
int((c*x^2+b*x+a)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
(-1/6*(2*a^3*e^6+3*a^2*b*d*e^5+6*a^2*c*d^2*e^4+6*a*b^2*d^2*e^4-66*a*b*c*d^ 3*e^3+132*a*c^2*d^4*e^2-11*b^3*d^3*e^3+132*b^2*c*d^4*e^2-330*b*c^2*d^5*e+2 20*c^3*d^6)/e^7+1/3*c^3*x^6/e-3*(a^2*c*e^4+a*b^2*e^4-6*a*b*c*d*e^3+12*a*c^ 2*d^2*e^2-b^3*d*e^3+12*b^2*c*d^2*e^2-30*b*c^2*d^3*e+20*c^3*d^4)/e^5*x^2-3/ 2*(a^2*b*e^5+2*a^2*c*d*e^4+2*a*b^2*d*e^4-18*a*b*c*d^2*e^3+36*a*c^2*d^3*e^2 -3*b^3*d^2*e^3+36*b^2*c*d^3*e^2-90*b*c^2*d^4*e+60*c^3*d^5)/e^6*x+1/2*c*(6* a*c*e^2+6*b^2*e^2-15*b*c*d*e+10*c^2*d^2)/e^3*x^4+1/2*c^2*(3*b*e-2*c*d)/e^2 *x^5)/(e*x+d)^3+1/e^7*(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+3 0*b*c^2*d^2*e-20*c^3*d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 664 vs. \(2 (243) = 486\).
Time = 0.08 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.65 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} - 2 \, a^{3} e^{6} - 78 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + 11 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 3 \, {\left (2 \, c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (10 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 6 \, {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + {\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \, {\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 3 \, {\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} + 6 \, {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 6 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} - 3 \, {\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 3 \, a^{2} b e^{6} + 54 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 9 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 6 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x - 6 \, {\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 3 \, {\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} x^{2} + 3 \, {\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="fricas")
Output:
1/6*(2*c^3*e^6*x^6 - 74*c^3*d^6 + 141*b*c^2*d^5*e - 3*a^2*b*d*e^5 - 2*a^3* e^6 - 78*(b^2*c + a*c^2)*d^4*e^2 + 11*(b^3 + 6*a*b*c)*d^3*e^3 - 6*(a*b^2 + a^2*c)*d^2*e^4 - 3*(2*c^3*d*e^5 - 3*b*c^2*e^6)*x^5 + 3*(10*c^3*d^2*e^4 - 15*b*c^2*d*e^5 + 6*(b^2*c + a*c^2)*e^6)*x^4 + (146*c^3*d^3*e^3 - 189*b*c^2 *d^2*e^4 + 54*(b^2*c + a*c^2)*d*e^5)*x^3 + 3*(26*c^3*d^4*e^2 - 9*b*c^2*d^3 *e^3 - 18*(b^2*c + a*c^2)*d^2*e^4 + 6*(b^3 + 6*a*b*c)*d*e^5 - 6*(a*b^2 + a ^2*c)*e^6)*x^2 - 3*(34*c^3*d^5*e - 81*b*c^2*d^4*e^2 + 3*a^2*b*e^6 + 54*(b^ 2*c + a*c^2)*d^3*e^3 - 9*(b^3 + 6*a*b*c)*d^2*e^4 + 6*(a*b^2 + a^2*c)*d*e^5 )*x - 6*(20*c^3*d^6 - 30*b*c^2*d^5*e + 12*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + (20*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 + 12*(b^2*c + a*c^2 )*d*e^5 - (b^3 + 6*a*b*c)*e^6)*x^3 + 3*(20*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 + 12*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5)*x^2 + 3*(20*c^3*d^5* e - 30*b*c^2*d^4*e^2 + 12*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^ 4)*x)*log(e*x + d))/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (243) = 486\).
Time = 14.46 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {c^{3} x^{3}}{3 e^{4}} + x^{2} \cdot \left (\frac {3 b c^{2}}{2 e^{4}} - \frac {2 c^{3} d}{e^{5}}\right ) + x \left (\frac {3 a c^{2}}{e^{4}} + \frac {3 b^{2} c}{e^{4}} - \frac {12 b c^{2} d}{e^{5}} + \frac {10 c^{3} d^{2}}{e^{6}}\right ) + \frac {- 2 a^{3} e^{6} - 3 a^{2} b d e^{5} - 6 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 66 a b c d^{3} e^{3} - 78 a c^{2} d^{4} e^{2} + 11 b^{3} d^{3} e^{3} - 78 b^{2} c d^{4} e^{2} + 141 b c^{2} d^{5} e - 74 c^{3} d^{6} + x^{2} \left (- 18 a^{2} c e^{6} - 18 a b^{2} e^{6} + 108 a b c d e^{5} - 108 a c^{2} d^{2} e^{4} + 18 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 180 b c^{2} d^{3} e^{3} - 90 c^{3} d^{4} e^{2}\right ) + x \left (- 9 a^{2} b e^{6} - 18 a^{2} c d e^{5} - 18 a b^{2} d e^{5} + 162 a b c d^{2} e^{4} - 180 a c^{2} d^{3} e^{3} + 27 b^{3} d^{2} e^{4} - 180 b^{2} c d^{3} e^{3} + 315 b c^{2} d^{4} e^{2} - 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac {\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \] Input:
integrate((c*x**2+b*x+a)**3/(e*x+d)**4,x)
Output:
c**3*x**3/(3*e**4) + x**2*(3*b*c**2/(2*e**4) - 2*c**3*d/e**5) + x*(3*a*c** 2/e**4 + 3*b**2*c/e**4 - 12*b*c**2*d/e**5 + 10*c**3*d**2/e**6) + (-2*a**3* e**6 - 3*a**2*b*d*e**5 - 6*a**2*c*d**2*e**4 - 6*a*b**2*d**2*e**4 + 66*a*b* c*d**3*e**3 - 78*a*c**2*d**4*e**2 + 11*b**3*d**3*e**3 - 78*b**2*c*d**4*e** 2 + 141*b*c**2*d**5*e - 74*c**3*d**6 + x**2*(-18*a**2*c*e**6 - 18*a*b**2*e **6 + 108*a*b*c*d*e**5 - 108*a*c**2*d**2*e**4 + 18*b**3*d*e**5 - 108*b**2* c*d**2*e**4 + 180*b*c**2*d**3*e**3 - 90*c**3*d**4*e**2) + x*(-9*a**2*b*e** 6 - 18*a**2*c*d*e**5 - 18*a*b**2*d*e**5 + 162*a*b*c*d**2*e**4 - 180*a*c**2 *d**3*e**3 + 27*b**3*d**2*e**4 - 180*b**2*c*d**3*e**3 + 315*b*c**2*d**4*e* *2 - 162*c**3*d**5*e))/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6* e**10*x**3) + (b*e - 2*c*d)*(6*a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2 *d**2)*log(d + e*x)/e**7
Time = 0.04 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.71 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 78 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - 11 \, {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 6 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} + {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + a^{2} b e^{6} + 20 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - 3 \, {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 2 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {2 \, c^{3} e^{2} x^{3} - 3 \, {\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \, {\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \, {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} x}{6 \, e^{6}} - \frac {{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{2} - {\left (b^{3} + 6 \, a b c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="maxima")
Output:
-1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 3*a^2*b*d*e^5 + 2*a^3*e^6 + 78*(b^2*c + a*c^2)*d^4*e^2 - 11*(b^3 + 6*a*b*c)*d^3*e^3 + 6*(a*b^2 + a^2*c)*d^2*e^4 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5 + (a*b^2 + a^2*c)*e^6)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^2* d^4*e^2 + a^2*b*e^6 + 20*(b^2*c + a*c^2)*d^3*e^3 - 3*(b^3 + 6*a*b*c)*d^2*e ^4 + 2*(a*b^2 + a^2*c)*d*e^5)*x)/(e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d ^3*e^7) + 1/6*(2*c^3*e^2*x^3 - 3*(4*c^3*d*e - 3*b*c^2*e^2)*x^2 + 6*(10*c^3 *d^2 - 12*b*c^2*d*e + 3*(b^2*c + a*c^2)*e^2)*x)/e^6 - (20*c^3*d^3 - 30*b*c ^2*d^2*e + 12*(b^2*c + a*c^2)*d*e^2 - (b^3 + 6*a*b*c)*e^3)*log(e*x + d)/e^ 7
Time = 0.29 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.82 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} + 12 \, a c^{2} d e^{2} - b^{3} e^{3} - 6 \, a b c e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} + 78 \, a c^{2} d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} - 66 \, a b c d^{3} e^{3} + 6 \, a b^{2} d^{2} e^{4} + 6 \, a^{2} c d^{2} e^{4} + 3 \, a^{2} b d e^{5} + 2 \, a^{3} e^{6} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} + 6 \, a c^{2} d^{2} e^{4} - b^{3} d e^{5} - 6 \, a b c d e^{5} + a b^{2} e^{6} + a^{2} c e^{6}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} + 20 \, a c^{2} d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4} - 18 \, a b c d^{2} e^{4} + 2 \, a b^{2} d e^{5} + 2 \, a^{2} c d e^{5} + a^{2} b e^{6}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{7}} + \frac {2 \, c^{3} e^{8} x^{3} - 12 \, c^{3} d e^{7} x^{2} + 9 \, b c^{2} e^{8} x^{2} + 60 \, c^{3} d^{2} e^{6} x - 72 \, b c^{2} d e^{7} x + 18 \, b^{2} c e^{8} x + 18 \, a c^{2} e^{8} x}{6 \, e^{12}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^4,x, algorithm="giac")
Output:
-(20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 12*a*c^2*d*e^2 - b^3*e^3 - 6*a*b*c*e^3)*log(abs(e*x + d))/e^7 - 1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 + 78*a*c^2*d^4*e^2 - 11*b^3*d^3*e^3 - 66*a*b*c*d^3*e^3 + 6*a*b^2*d^2*e^4 + 6*a^2*c*d^2*e^4 + 3*a^2*b*d*e^5 + 2*a^3*e^6 + 18*(5*c^3 *d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 + 6*a*c^2*d^2*e^4 - b^3*d*e^ 5 - 6*a*b*c*d*e^5 + a*b^2*e^6 + a^2*c*e^6)*x^2 + 9*(18*c^3*d^5*e - 35*b*c^ 2*d^4*e^2 + 20*b^2*c*d^3*e^3 + 20*a*c^2*d^3*e^3 - 3*b^3*d^2*e^4 - 18*a*b*c *d^2*e^4 + 2*a*b^2*d*e^5 + 2*a^2*c*d*e^5 + a^2*b*e^6)*x)/((e*x + d)^3*e^7) + 1/6*(2*c^3*e^8*x^3 - 12*c^3*d*e^7*x^2 + 9*b*c^2*e^8*x^2 + 60*c^3*d^2*e^ 6*x - 72*b*c^2*d*e^7*x + 18*b^2*c*e^8*x + 18*a*c^2*e^8*x)/e^12
Time = 5.43 (sec) , antiderivative size = 484, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx=x^2\,\left (\frac {3\,b\,c^2}{2\,e^4}-\frac {2\,c^3\,d}{e^5}\right )-x\,\left (\frac {4\,d\,\left (\frac {3\,b\,c^2}{e^4}-\frac {4\,c^3\,d}{e^5}\right )}{e}+\frac {6\,c^3\,d^2}{e^6}-\frac {3\,c\,\left (b^2+a\,c\right )}{e^4}\right )-\frac {x\,\left (\frac {3\,a^2\,b\,e^5}{2}+3\,a^2\,c\,d\,e^4+3\,a\,b^2\,d\,e^4-27\,a\,b\,c\,d^2\,e^3+30\,a\,c^2\,d^3\,e^2-\frac {9\,b^3\,d^2\,e^3}{2}+30\,b^2\,c\,d^3\,e^2-\frac {105\,b\,c^2\,d^4\,e}{2}+27\,c^3\,d^5\right )+\frac {2\,a^3\,e^6+3\,a^2\,b\,d\,e^5+6\,a^2\,c\,d^2\,e^4+6\,a\,b^2\,d^2\,e^4-66\,a\,b\,c\,d^3\,e^3+78\,a\,c^2\,d^4\,e^2-11\,b^3\,d^3\,e^3+78\,b^2\,c\,d^4\,e^2-141\,b\,c^2\,d^5\,e+74\,c^3\,d^6}{6\,e}+x^2\,\left (3\,a^2\,c\,e^5+3\,a\,b^2\,e^5-18\,a\,b\,c\,d\,e^4+18\,a\,c^2\,d^2\,e^3-3\,b^3\,d\,e^4+18\,b^2\,c\,d^2\,e^3-30\,b\,c^2\,d^3\,e^2+15\,c^3\,d^4\,e\right )}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-20\,c^3\,d^3-12\,a\,c^2\,d\,e^2\right )}{e^7}+\frac {c^3\,x^3}{3\,e^4} \] Input:
int((a + b*x + c*x^2)^3/(d + e*x)^4,x)
Output:
x^2*((3*b*c^2)/(2*e^4) - (2*c^3*d)/e^5) - x*((4*d*((3*b*c^2)/e^4 - (4*c^3* d)/e^5))/e + (6*c^3*d^2)/e^6 - (3*c*(a*c + b^2))/e^4) - (x*(27*c^3*d^5 + ( 3*a^2*b*e^5)/2 - (9*b^3*d^2*e^3)/2 + 30*a*c^2*d^3*e^2 + 30*b^2*c*d^3*e^2 + 3*a*b^2*d*e^4 + 3*a^2*c*d*e^4 - (105*b*c^2*d^4*e)/2 - 27*a*b*c*d^2*e^3) + (2*a^3*e^6 + 74*c^3*d^6 - 11*b^3*d^3*e^3 + 6*a*b^2*d^2*e^4 + 78*a*c^2*d^4 *e^2 + 6*a^2*c*d^2*e^4 + 78*b^2*c*d^4*e^2 + 3*a^2*b*d*e^5 - 141*b*c^2*d^5* e - 66*a*b*c*d^3*e^3)/(6*e) + x^2*(3*a*b^2*e^5 + 3*a^2*c*e^5 - 3*b^3*d*e^4 + 15*c^3*d^4*e + 18*a*c^2*d^2*e^3 - 30*b*c^2*d^3*e^2 + 18*b^2*c*d^2*e^3 - 18*a*b*c*d*e^4))/(d^3*e^6 + e^9*x^3 + 3*d^2*e^7*x + 3*d*e^8*x^2) + (log(d + e*x)*(b^3*e^3 - 20*c^3*d^3 + 6*a*b*c*e^3 - 12*a*c^2*d*e^2 + 30*b*c^2*d^ 2*e - 12*b^2*c*d*e^2))/e^7 + (c^3*x^3)/(3*e^4)
Time = 0.20 (sec) , antiderivative size = 859, normalized size of antiderivative = 3.42 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^3/(e*x+d)^4,x)
Output:
(36*log(d + e*x)*a*b*c*d**4*e**3 + 108*log(d + e*x)*a*b*c*d**3*e**4*x + 10 8*log(d + e*x)*a*b*c*d**2*e**5*x**2 + 36*log(d + e*x)*a*b*c*d*e**6*x**3 - 72*log(d + e*x)*a*c**2*d**5*e**2 - 216*log(d + e*x)*a*c**2*d**4*e**3*x - 2 16*log(d + e*x)*a*c**2*d**3*e**4*x**2 - 72*log(d + e*x)*a*c**2*d**2*e**5*x **3 + 6*log(d + e*x)*b**3*d**4*e**3 + 18*log(d + e*x)*b**3*d**3*e**4*x + 1 8*log(d + e*x)*b**3*d**2*e**5*x**2 + 6*log(d + e*x)*b**3*d*e**6*x**3 - 72* log(d + e*x)*b**2*c*d**5*e**2 - 216*log(d + e*x)*b**2*c*d**4*e**3*x - 216* log(d + e*x)*b**2*c*d**3*e**4*x**2 - 72*log(d + e*x)*b**2*c*d**2*e**5*x**3 + 180*log(d + e*x)*b*c**2*d**6*e + 540*log(d + e*x)*b*c**2*d**5*e**2*x + 540*log(d + e*x)*b*c**2*d**4*e**3*x**2 + 180*log(d + e*x)*b*c**2*d**3*e**4 *x**3 - 120*log(d + e*x)*c**3*d**7 - 360*log(d + e*x)*c**3*d**6*e*x - 360* log(d + e*x)*c**3*d**5*e**2*x**2 - 120*log(d + e*x)*c**3*d**4*e**3*x**3 - 2*a**3*d*e**6 - 3*a**2*b*d**2*e**5 - 9*a**2*b*d*e**6*x + 6*a**2*c*e**7*x** 3 + 6*a*b**2*e**7*x**3 + 30*a*b*c*d**4*e**3 + 54*a*b*c*d**3*e**4*x - 36*a* b*c*d*e**6*x**3 - 60*a*c**2*d**5*e**2 - 108*a*c**2*d**4*e**3*x + 72*a*c**2 *d**2*e**5*x**3 + 18*a*c**2*d*e**6*x**4 + 5*b**3*d**4*e**3 + 9*b**3*d**3*e **4*x - 6*b**3*d*e**6*x**3 - 60*b**2*c*d**5*e**2 - 108*b**2*c*d**4*e**3*x + 72*b**2*c*d**2*e**5*x**3 + 18*b**2*c*d*e**6*x**4 + 150*b*c**2*d**6*e + 2 70*b*c**2*d**5*e**2*x - 180*b*c**2*d**3*e**4*x**3 - 45*b*c**2*d**2*e**5*x* *4 + 9*b*c**2*d*e**6*x**5 - 100*c**3*d**7 - 180*c**3*d**6*e*x + 120*c**...