Integrand size = 20, antiderivative size = 266 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {\left (c d^2-b d e+a e^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{5 e^7 (d+e x)^5}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \] Output:
-1/6*(a*e^2-b*d*e+c*d^2)^3/e^7/(e*x+d)^6+3/5*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d ^2)^2/e^7/(e*x+d)^5-3/4*(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)^4+1/3*(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5* b*d))/e^7/(e*x+d)^3-3/2*c*(5*c^2*d^2+b^2*e^2-c*e*(-a*e+5*b*d))/e^7/(e*x+d) ^2+3*c^2*(-b*e+2*c*d)/e^7/(e*x+d)+c^3*ln(e*x+d)/e^7
Time = 0.11 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )-e^3 \left (10 a^3 e^3+6 a^2 b e^2 (d+6 e x)+3 a b^2 e \left (d^2+6 d e x+15 e^2 x^2\right )+b^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )-3 c e^2 \left (a^2 e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+2 a b e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 b^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )-6 c^2 e \left (a e \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )+5 b \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \] Input:
Integrate[(a + b*x + c*x^2)^3/(d + e*x)^7,x]
Output:
(c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350 *d*e^4*x^4 + 360*e^5*x^5) - e^3*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3* a*b^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) - 3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b*e* (d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^2*(d^4 + 6*d^3*e*x + 1 5*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) - 6*c^2*e*(a*e*(d^4 + 6*d^3*e* x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*b*(d^5 + 6*d^4*e*x + 1 5*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7*(d + e*x)^6)
Time = 0.50 (sec) , antiderivative size = 266, 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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6 (d+e x)^4}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^5}+\frac {3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)^6}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 (d+e x)^7}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac {c^3}{e^6 (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{5 e^7 (d+e x)^5}-\frac {\left (a e^2-b d e+c d^2\right )^3}{6 e^7 (d+e x)^6}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7}\) |
Input:
Int[(a + b*x + c*x^2)^3/(d + e*x)^7,x]
Output:
-1/6*(c*d^2 - b*d*e + a*e^2)^3/(e^7*(d + e*x)^6) + (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(5*e^7*(d + e*x)^5) - (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)))/(4*e^7*(d + e*x)^4) + ((2*c*d - b*e )*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(3*e^7*(d + e*x)^3) - (3 *c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(2*e^7*(d + e*x)^2) + (3*c^2 *(2*c*d - b*e))/(e^7*(d + e*x)) + (c^3*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 = 0.90 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.65
| method | result | size |
| risch | \(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{5}}{e^{2}}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}+5 b c d e -15 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-110 d^{3} c^{3}\right ) x^{3}}{3 e^{4}}-\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}+6 a b c d \,e^{3}+6 d^{2} e^{2} a \,c^{2}+b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}+30 b \,c^{2} d^{3} e -125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}-\frac {\left (6 a^{2} b \,e^{5}+3 d \,e^{4} a^{2} c +3 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+6 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}+6 b^{2} c \,d^{3} e^{2}+30 b \,c^{2} d^{4} e -137 d^{5} c^{3}\right ) x}{10 e^{6}}-\frac {10 e^{6} a^{3}+6 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+6 d^{4} e^{2} a \,c^{2}+b^{3} d^{3} e^{3}+6 b^{2} c \,d^{4} e^{2}+30 b \,c^{2} d^{5} e -147 d^{6} c^{3}}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(438\) |
| norman | \(\frac {-\frac {10 e^{6} a^{3}+6 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}+6 d^{4} e^{2} a \,c^{2}+b^{3} d^{3} e^{3}+6 b^{2} c \,d^{4} e^{2}+30 b \,c^{2} d^{5} e -147 d^{6} c^{3}}{60 e^{7}}-\frac {3 \left (b \,c^{2} e -2 d \,c^{3}\right ) x^{5}}{e^{2}}-\frac {3 \left (c^{2} a \,e^{2}+e^{2} b^{2} c +5 c^{2} d e b -15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (6 a b c \,e^{3}+6 d \,e^{2} a \,c^{2}+b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-110 d^{3} c^{3}\right ) x^{3}}{3 e^{4}}-\frac {\left (3 e^{4} a^{2} c +3 a \,b^{2} e^{4}+6 a b c d \,e^{3}+6 d^{2} e^{2} a \,c^{2}+b^{3} d \,e^{3}+6 b^{2} c \,d^{2} e^{2}+30 b \,c^{2} d^{3} e -125 d^{4} c^{3}\right ) x^{2}}{4 e^{5}}-\frac {\left (6 a^{2} b \,e^{5}+3 d \,e^{4} a^{2} c +3 a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}+6 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}+6 b^{2} c \,d^{3} e^{2}+30 b \,c^{2} d^{4} e -137 d^{5} c^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(444\) |
| default | \(-\frac {6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} a \,c^{2}-3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +15 d^{4} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} a \,c^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )}-\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{6 e^{7} \left (e x +d \right )^{6}}\) | \(459\) |
| parallelrisch | \(\frac {-6 d^{4} e^{2} a \,c^{2}+360 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}-3 d^{2} e^{4} a^{2} c +360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +147 d^{6} c^{3}-6 a^{2} b d \,e^{5}-3 a \,b^{2} d^{2} e^{4}-6 b^{2} c \,d^{4} e^{2}-10 e^{6} a^{3}+360 c^{3} d \,e^{5} x^{5}+1350 c^{3} d^{2} e^{4} x^{4}+2200 c^{3} d^{3} e^{3} x^{3}+1875 c^{3} d^{4} e^{2} x^{2}+822 c^{3} d^{5} e x -30 b \,c^{2} d^{5} e -36 x a b c \,d^{2} e^{4}+1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}+900 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-90 x^{2} a b c d \,e^{5}-180 x b \,c^{2} d^{4} e^{2}+60 \ln \left (e x +d \right ) x^{6} c^{3} e^{6}-120 x^{3} a b c \,e^{6}-120 x^{3} a \,c^{2} d \,e^{5}-120 x^{3} b^{2} c d \,e^{5}-600 x^{3} b \,c^{2} d^{2} e^{4}-90 x^{2} b^{2} c \,d^{2} e^{4}-450 x^{2} b \,c^{2} d^{3} e^{3}-18 x a \,b^{2} d \,e^{5}-36 x \,b^{2} c \,d^{3} e^{3}-450 x^{4} b \,c^{2} d \,e^{5}-90 x^{2} a \,c^{2} d^{2} e^{4}-18 x \,a^{2} c d \,e^{5}-36 x a \,c^{2} d^{3} e^{3}+900 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-180 x^{5} b \,c^{2} e^{6}-90 x^{4} a \,c^{2} e^{6}-90 x^{4} b^{2} c \,e^{6}-45 x^{2} a^{2} c \,e^{6}-45 x^{2} a \,b^{2} e^{6}-15 x^{2} b^{3} d \,e^{5}-36 x \,a^{2} b \,e^{6}-6 x \,b^{3} d^{2} e^{4}-b^{3} d^{3} e^{3}-20 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}-6 a b c \,d^{3} e^{3}}{60 e^{7} \left (e x +d \right )^{6}}\) | \(609\) |
Input:
int((c*x^2+b*x+a)^3/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
(-3*c^2*(b*e-2*c*d)/e^2*x^5-3/2*c*(a*c*e^2+b^2*e^2+5*b*c*d*e-15*c^2*d^2)/e ^3*x^4-1/3*(6*a*b*c*e^3+6*a*c^2*d*e^2+b^3*e^3+6*b^2*c*d*e^2+30*b*c^2*d^2*e -110*c^3*d^3)/e^4*x^3-1/4*(3*a^2*c*e^4+3*a*b^2*e^4+6*a*b*c*d*e^3+6*a*c^2*d ^2*e^2+b^3*d*e^3+6*b^2*c*d^2*e^2+30*b*c^2*d^3*e-125*c^3*d^4)/e^5*x^2-1/10* (6*a^2*b*e^5+3*a^2*c*d*e^4+3*a*b^2*d*e^4+6*a*b*c*d^2*e^3+6*a*c^2*d^3*e^2+b ^3*d^2*e^3+6*b^2*c*d^3*e^2+30*b*c^2*d^4*e-137*c^3*d^5)/e^6*x-1/60*(10*a^3* e^6+6*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4+6*a*b*c*d^3*e^3+6*a*c^2* d^4*e^2+b^3*d^3*e^3+6*b^2*c*d^4*e^2+30*b*c^2*d^5*e-147*c^3*d^6)/e^7)/(e*x+ d)^6+c^3*ln(e*x+d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (256) = 512\).
Time = 0.08 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.05 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="fricas")
Output:
1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 10*a^3*e^6 - 6*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 3*(a*b^2 + a^2*c)*d^2*e^4 + 180*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 90*(15*c^3*d^2*e^4 - 5*b*c^2*d*e^5 - ( b^2*c + a*c^2)*e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*(b^2* c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)*e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*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 - 3*(a*b^2 + a^2*c)*e^6)*x^2 + 6*(137*c^3*d^5*e - 30*b*c^2*d^4*e^2 - 6*a^2*b*e^6 - 6* (b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - 3*(a*b^2 + a^2*c)*d*e^ 5)*x + 60*(c^3*e^6*x^6 + 6*c^3*d*e^5*x^5 + 15*c^3*d^2*e^4*x^4 + 20*c^3*d^3 *e^3*x^3 + 15*c^3*d^4*e^2*x^2 + 6*c^3*d^5*e*x + c^3*d^6)*log(e*x + d))/(e^ 13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\text {Timed out} \] Input:
integrate((c*x**2+b*x+a)**3/(e*x+d)**7,x)
Output:
Timed out
Time = 0.05 (sec) , antiderivative size = 469, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - 3 \, {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - 3 \, {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, a^{2} b e^{6} - 6 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - 3 \, {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="maxima")
Output:
1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*a^2*b*d*e^5 - 10*a^3*e^6 - 6*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - 3*(a*b^2 + a^2*c)*d^2*e^4 + 180*(2*c^3*d*e^5 - b*c^2*e^6)*x^5 + 90*(15*c^3*d^2*e^4 - 5*b*c^2*d*e^5 - ( b^2*c + a*c^2)*e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*(b^2* c + a*c^2)*d*e^5 - (b^3 + 6*a*b*c)*e^6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*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 - 3*(a*b^2 + a^2*c)*e^6)*x^2 + 6*(137*c^3*d^5*e - 30*b*c^2*d^4*e^2 - 6*a^2*b*e^6 - 6* (b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - 3*(a*b^2 + a^2*c)*d*e^ 5)*x)/(e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^ 4*e^9*x^2 + 6*d^5*e^8*x + d^6*e^7) + c^3*log(e*x + d)/e^7
Time = 0.34 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5} - a c^{2} e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - 6 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - 6 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - 3 \, a b^{2} e^{5} - 3 \, a^{2} c e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - 6 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - 3 \, a b^{2} d e^{4} - 3 \, a^{2} c d e^{4} - 6 \, a^{2} b e^{5}\right )} x + \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - 6 \, a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} - 6 \, a b c d^{3} e^{3} - 3 \, a b^{2} d^{2} e^{4} - 3 \, a^{2} c d^{2} e^{4} - 6 \, a^{2} b d e^{5} - 10 \, a^{3} e^{6}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^7,x, algorithm="giac")
Output:
c^3*log(abs(e*x + d))/e^7 + 1/60*(180*(2*c^3*d*e^4 - b*c^2*e^5)*x^5 + 90*( 15*c^3*d^2*e^3 - 5*b*c^2*d*e^4 - b^2*c*e^5 - a*c^2*e^5)*x^4 + 20*(110*c^3* d^3*e^2 - 30*b*c^2*d^2*e^3 - 6*b^2*c*d*e^4 - 6*a*c^2*d*e^4 - b^3*e^5 - 6*a *b*c*e^5)*x^3 + 15*(125*c^3*d^4*e - 30*b*c^2*d^3*e^2 - 6*b^2*c*d^2*e^3 - 6 *a*c^2*d^2*e^3 - b^3*d*e^4 - 6*a*b*c*d*e^4 - 3*a*b^2*e^5 - 3*a^2*c*e^5)*x^ 2 + 6*(137*c^3*d^5 - 30*b*c^2*d^4*e - 6*b^2*c*d^3*e^2 - 6*a*c^2*d^3*e^2 - b^3*d^2*e^3 - 6*a*b*c*d^2*e^3 - 3*a*b^2*d*e^4 - 3*a^2*c*d*e^4 - 6*a^2*b*e^ 5)*x + (147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - 6*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 - 3*a*b^2*d^2*e^4 - 3*a^2*c*d^2*e^4 - 6*a^2 *b*d*e^5 - 10*a^3*e^6)/e)/((e*x + d)^6*e^6)
Time = 0.11 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.87 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {\frac {10\,a^3\,e^6+6\,a^2\,b\,d\,e^5+3\,a^2\,c\,d^2\,e^4+3\,a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3+6\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3+6\,b^2\,c\,d^4\,e^2+30\,b\,c^2\,d^5\,e-147\,c^3\,d^6}{60\,e^7}+\frac {3\,x^4\,\left (b^2\,c\,e^2+5\,b\,c^2\,d\,e-15\,c^3\,d^2+a\,c^2\,e^2\right )}{2\,e^3}+\frac {x^3\,\left (b^3\,e^3+6\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e+6\,a\,b\,c\,e^3-110\,c^3\,d^3+6\,a\,c^2\,d\,e^2\right )}{3\,e^4}+\frac {x^2\,\left (3\,a^2\,c\,e^4+3\,a\,b^2\,e^4+6\,a\,b\,c\,d\,e^3+6\,a\,c^2\,d^2\,e^2+b^3\,d\,e^3+6\,b^2\,c\,d^2\,e^2+30\,b\,c^2\,d^3\,e-125\,c^3\,d^4\right )}{4\,e^5}+\frac {x\,\left (6\,a^2\,b\,e^5+3\,a^2\,c\,d\,e^4+3\,a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3+6\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3+6\,b^2\,c\,d^3\,e^2+30\,b\,c^2\,d^4\,e-137\,c^3\,d^5\right )}{10\,e^6}+\frac {3\,c^2\,x^5\,\left (b\,e-2\,c\,d\right )}{e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:
int((a + b*x + c*x^2)^3/(d + e*x)^7,x)
Output:
(c^3*log(d + e*x))/e^7 - ((10*a^3*e^6 - 147*c^3*d^6 + b^3*d^3*e^3 + 3*a*b^ 2*d^2*e^4 + 6*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + 6*b^2*c*d^4*e^2 + 6*a^2*b* d*e^5 + 30*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(60*e^7) + (3*x^4*(a*c^2*e^2 - 1 5*c^3*d^2 + b^2*c*e^2 + 5*b*c^2*d*e))/(2*e^3) + (x^3*(b^3*e^3 - 110*c^3*d^ 3 + 6*a*b*c*e^3 + 6*a*c^2*d*e^2 + 30*b*c^2*d^2*e + 6*b^2*c*d*e^2))/(3*e^4) + (x^2*(3*a*b^2*e^4 - 125*c^3*d^4 + 3*a^2*c*e^4 + b^3*d*e^3 + 6*a*c^2*d^2 *e^2 + 6*b^2*c*d^2*e^2 + 30*b*c^2*d^3*e + 6*a*b*c*d*e^3))/(4*e^5) + (x*(6* a^2*b*e^5 - 137*c^3*d^5 + b^3*d^2*e^3 + 6*a*c^2*d^3*e^2 + 6*b^2*c*d^3*e^2 + 3*a*b^2*d*e^4 + 3*a^2*c*d*e^4 + 30*b*c^2*d^4*e + 6*a*b*c*d^2*e^3))/(10*e ^6) + (3*c^2*x^5*(b*e - 2*c*d))/e^2)/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4 *e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)
Time = 0.30 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {900 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{3} e^{4} x^{4}+30 b \,c^{2} e^{7} x^{6}-10 a^{3} d \,e^{6}-b^{3} d^{4} e^{3}+60 \,\mathrm {log}\left (e x +d \right ) c^{3} d \,e^{6} x^{6}-6 a^{2} b \,d^{2} e^{5}+900 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{5} e^{2} x^{2}-6 a b c \,d^{4} e^{3}-36 a b c \,d^{3} e^{4} x -90 a b c \,d^{2} e^{5} x^{2}-120 a b c d \,e^{6} x^{3}+60 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{7}+360 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{2} e^{5} x^{5}+87 c^{3} d^{7}+360 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{6} e x -36 a^{2} b d \,e^{6} x -18 a^{2} c \,d^{2} e^{5} x -45 a^{2} c d \,e^{6} x^{2}-18 a \,b^{2} d^{2} e^{5} x -45 a \,b^{2} d \,e^{6} x^{2}-36 a \,c^{2} d^{4} e^{3} x -90 a \,c^{2} d^{3} e^{4} x^{2}-120 a \,c^{2} d^{2} e^{5} x^{3}-90 a \,c^{2} d \,e^{6} x^{4}-36 b^{2} c \,d^{4} e^{3} x -90 b^{2} c \,d^{3} e^{4} x^{2}-120 b^{2} c \,d^{2} e^{5} x^{3}-90 b^{2} c d \,e^{6} x^{4}+1200 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{4} e^{3} x^{3}-6 b^{3} d^{3} e^{4} x -15 b^{3} d^{2} e^{5} x^{2}-20 b^{3} d \,e^{6} x^{3}+462 c^{3} d^{6} e x +975 c^{3} d^{5} e^{2} x^{2}+1000 c^{3} d^{4} e^{3} x^{3}+450 c^{3} d^{3} e^{4} x^{4}-60 c^{3} d \,e^{6} x^{6}-3 a^{2} c \,d^{3} e^{4}-3 a \,b^{2} d^{3} e^{4}-6 a \,c^{2} d^{5} e^{2}-6 b^{2} c \,d^{5} e^{2}}{60 d \,e^{7} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int((c*x^2+b*x+a)^3/(e*x+d)^7,x)
Output:
(60*log(d + e*x)*c**3*d**7 + 360*log(d + e*x)*c**3*d**6*e*x + 900*log(d + e*x)*c**3*d**5*e**2*x**2 + 1200*log(d + e*x)*c**3*d**4*e**3*x**3 + 900*log (d + e*x)*c**3*d**3*e**4*x**4 + 360*log(d + e*x)*c**3*d**2*e**5*x**5 + 60* log(d + e*x)*c**3*d*e**6*x**6 - 10*a**3*d*e**6 - 6*a**2*b*d**2*e**5 - 36*a **2*b*d*e**6*x - 3*a**2*c*d**3*e**4 - 18*a**2*c*d**2*e**5*x - 45*a**2*c*d* e**6*x**2 - 3*a*b**2*d**3*e**4 - 18*a*b**2*d**2*e**5*x - 45*a*b**2*d*e**6* x**2 - 6*a*b*c*d**4*e**3 - 36*a*b*c*d**3*e**4*x - 90*a*b*c*d**2*e**5*x**2 - 120*a*b*c*d*e**6*x**3 - 6*a*c**2*d**5*e**2 - 36*a*c**2*d**4*e**3*x - 90* a*c**2*d**3*e**4*x**2 - 120*a*c**2*d**2*e**5*x**3 - 90*a*c**2*d*e**6*x**4 - b**3*d**4*e**3 - 6*b**3*d**3*e**4*x - 15*b**3*d**2*e**5*x**2 - 20*b**3*d *e**6*x**3 - 6*b**2*c*d**5*e**2 - 36*b**2*c*d**4*e**3*x - 90*b**2*c*d**3*e **4*x**2 - 120*b**2*c*d**2*e**5*x**3 - 90*b**2*c*d*e**6*x**4 + 30*b*c**2*e **7*x**6 + 87*c**3*d**7 + 462*c**3*d**6*e*x + 975*c**3*d**5*e**2*x**2 + 10 00*c**3*d**4*e**3*x**3 + 450*c**3*d**3*e**4*x**4 - 60*c**3*d*e**6*x**6)/(6 0*d*e**7*(d**6 + 6*d**5*e*x + 15*d**4*e**2*x**2 + 20*d**3*e**3*x**3 + 15*d **2*e**4*x**4 + 6*d*e**5*x**5 + e**6*x**6))