Integrand size = 20, antiderivative size = 251 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=-\frac {c^2 (5 c d-3 b e) x}{e^6}+\frac {c^3 x^2}{2 e^5}-\frac {\left (c d^2-b d e+a e^2\right )^3}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{e^7 (d+e x)^3}-\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{e^7 (d+e x)}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^7} \] Output:
-c^2*(-3*b*e+5*c*d)*x/e^6+1/2*c^3*x^2/e^5-1/4*(a*e^2-b*d*e+c*d^2)^3/e^7/(e *x+d)^4+(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)^2/e^7/(e*x+d)^3-3/2*(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)^2+(-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*c*(5*c^2*d^2+b^2*e^2- c*e*(-a*e+5*b*d))*ln(e*x+d)/e^7
Time = 0.12 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^3 \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )-e^3 \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )-c e^2 \left (a^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 a b e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-b^2 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+c^2 e \left (a d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )-b \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+12 c \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right ) (d+e x)^4 \log (d+e x)}{4 e^7 (d+e x)^4} \] Input:
Integrate[(a + b*x + c*x^2)^3/(d + e*x)^5,x]
Output:
(c^3*(57*d^6 + 168*d^5*e*x + 132*d^4*e^2*x^2 - 32*d^3*e^3*x^3 - 68*d^2*e^4 *x^4 - 12*d*e^5*x^5 + 2*e^6*x^6) - e^3*(a^3*e^3 + a^2*b*e^2*(d + 4*e*x) + a*b^2*e*(d^2 + 4*d*e*x + 6*e^2*x^2) + b^3*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) - c*e^2*(a^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*a*b*e*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - b^2*d*(25*d^3 + 88*d^2*e*x + 108*d *e^2*x^2 + 48*e^3*x^3)) + c^2*e*(a*d*e*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^ 2 + 48*e^3*x^3) - b*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x ^3 - 48*d*e^4*x^4 - 12*e^5*x^5)) + 12*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))*(d + e*x)^4*Log[d + e*x])/(4*e^7*(d + e*x)^4)
Time = 0.51 (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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac {(2 c d-b e) \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6 (d+e x)^2}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^3}+\frac {3 (b e-2 c d) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)^4}+\frac {\left (a e^2-b d e+c d^2\right )^3}{e^6 (d+e x)^5}-\frac {c^2 (5 c d-3 b e)}{e^6}+\frac {c^3 x}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (d+e x)}-\frac {3 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7}+\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{e^7 (d+e x)^3}-\frac {\left (a e^2-b d e+c d^2\right )^3}{4 e^7 (d+e x)^4}-\frac {c^2 x (5 c d-3 b e)}{e^6}+\frac {c^3 x^2}{2 e^5}\) |
Input:
Int[(a + b*x + c*x^2)^3/(d + e*x)^5,x]
Output:
-((c^2*(5*c*d - 3*b*e)*x)/e^6) + (c^3*x^2)/(2*e^5) - (c*d^2 - b*d*e + a*e^ 2)^3/(4*e^7*(d + e*x)^4) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^7* (d + e*x)^3) - (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)))/(2*e^7*(d + e*x)^2) + ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2* c*e*(5*b*d - 3*a*e)))/(e^7*(d + e*x)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5 *b*d - 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 = 430, normalized size of antiderivative = 1.71
| method | result | size |
| norman | \(\frac {-\frac {e^{6} a^{3}+a^{2} b d \,e^{5}+d^{2} e^{4} a^{2} c +a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}-25 d^{4} e^{2} a \,c^{2}+b^{3} d^{3} e^{3}-25 b^{2} c \,d^{4} e^{2}+125 b \,c^{2} d^{5} e -125 d^{6} c^{3}}{4 e^{7}}+\frac {c^{3} x^{6}}{2 e}-\frac {\left (6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +60 d^{2} e b \,c^{2}-60 d^{3} c^{3}\right ) x^{3}}{e^{4}}-\frac {3 \left (e^{4} a^{2} c +a \,b^{2} e^{4}+6 a b c d \,e^{3}-18 d^{2} e^{2} a \,c^{2}+b^{3} d \,e^{3}-18 b^{2} c \,d^{2} e^{2}+90 b \,c^{2} d^{3} e -90 d^{4} c^{3}\right ) x^{2}}{2 e^{5}}-\frac {\left (a^{2} b \,e^{5}+d \,e^{4} a^{2} c +a \,b^{2} d \,e^{4}+6 a b c \,d^{2} e^{3}-22 d^{3} e^{2} a \,c^{2}+b^{3} d^{2} e^{3}-22 b^{2} c \,d^{3} e^{2}+110 b \,c^{2} d^{4} e -110 d^{5} c^{3}\right ) x}{e^{6}}+\frac {3 c^{2} \left (b e -c d \right ) x^{5}}{e^{2}}}{\left (e x +d \right )^{4}}+\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(430\) |
| default | \(\frac {c^{2} \left (\frac {1}{2} c e \,x^{2}+3 b e x -5 c d x \right )}{e^{6}}-\frac {3 a^{2} b \,e^{5}-6 d \,e^{4} a^{2} c -6 a \,b^{2} d \,e^{4}+18 a b c \,d^{2} e^{3}-12 d^{3} e^{2} a \,c^{2}+3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}-\frac {e^{6} a^{3}-3 a^{2} b d \,e^{5}+3 d^{2} e^{4} a^{2} c +3 a \,b^{2} d^{2} e^{4}-6 a b c \,d^{3} e^{3}+3 d^{4} e^{2} a \,c^{2}-b^{3} d^{3} e^{3}+3 b^{2} c \,d^{4} e^{2}-3 b \,c^{2} d^{5} e +d^{6} c^{3}}{4 e^{7} \left (e x +d \right )^{4}}+\frac {3 c \left (a c \,e^{2}+b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {6 a b c \,e^{3}-12 d \,e^{2} a \,c^{2}+b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}}{e^{7} \left (e x +d \right )}-\frac {3 e^{4} a^{2} c +3 a \,b^{2} e^{4}-18 a b c d \,e^{3}+18 d^{2} e^{2} a \,c^{2}-3 b^{3} d \,e^{3}+18 b^{2} c \,d^{2} e^{2}-30 b \,c^{2} d^{3} e +15 d^{4} c^{3}}{2 e^{7} \left (e x +d \right )^{2}}\) | \(447\) |
| risch | \(\frac {c^{3} x^{2}}{2 e^{5}}+\frac {3 c^{2} b x}{e^{5}}-\frac {5 c^{3} d x}{e^{6}}+\frac {\left (-6 a b c \,e^{5}+12 a \,c^{2} d \,e^{4}-b^{3} e^{5}+12 b^{2} c d \,e^{4}-30 b \,c^{2} d^{2} e^{3}+20 e^{2} d^{3} c^{3}\right ) x^{3}-\frac {3 e \left (e^{4} a^{2} c +a \,b^{2} e^{4}+6 a b c d \,e^{3}-18 d^{2} e^{2} a \,c^{2}+b^{3} d \,e^{3}-18 b^{2} c \,d^{2} e^{2}+50 b \,c^{2} d^{3} e -35 d^{4} c^{3}\right ) x^{2}}{2}+\left (-a^{2} b \,e^{5}-d \,e^{4} a^{2} c -a \,b^{2} d \,e^{4}-6 a b c \,d^{2} e^{3}+22 d^{3} e^{2} a \,c^{2}-b^{3} d^{2} e^{3}+22 b^{2} c \,d^{3} e^{2}-65 b \,c^{2} d^{4} e +47 d^{5} c^{3}\right ) x -\frac {e^{6} a^{3}+a^{2} b d \,e^{5}+d^{2} e^{4} a^{2} c +a \,b^{2} d^{2} e^{4}+6 a b c \,d^{3} e^{3}-25 d^{4} e^{2} a \,c^{2}+b^{3} d^{3} e^{3}-25 b^{2} c \,d^{4} e^{2}+77 b \,c^{2} d^{5} e -57 d^{6} c^{3}}{4 e}}{e^{6} \left (e x +d \right )^{4}}+\frac {3 c^{2} \ln \left (e x +d \right ) a}{e^{5}}+\frac {3 c \ln \left (e x +d \right ) b^{2}}{e^{5}}-\frac {15 c^{2} \ln \left (e x +d \right ) b d}{e^{6}}+\frac {15 c^{3} \ln \left (e x +d \right ) d^{2}}{e^{7}}\) | \(457\) |
| parallelrisch | \(\frac {25 d^{4} e^{2} a \,c^{2}-d^{2} e^{4} a^{2} c +240 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +125 d^{6} c^{3}-a^{2} b d \,e^{5}-a \,b^{2} d^{2} e^{4}+25 b^{2} c \,d^{4} e^{2}-e^{6} a^{3}-12 c^{3} d \,e^{5} x^{5}+240 c^{3} d^{3} e^{3} x^{3}+540 c^{3} d^{4} e^{2} x^{2}+440 c^{3} d^{5} e x +48 \ln \left (e x +d \right ) x^{3} a \,c^{2} d \,e^{5}+48 \ln \left (e x +d \right ) x^{3} b^{2} c d \,e^{5}-125 b \,c^{2} d^{5} e -24 x a b c \,d^{2} e^{4}+72 \ln \left (e x +d \right ) x^{2} b^{2} c \,d^{2} e^{4}-60 \ln \left (e x +d \right ) x^{4} b \,c^{2} d \,e^{5}-240 \ln \left (e x +d \right ) x^{3} b \,c^{2} d^{2} e^{4}+240 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}+48 \ln \left (e x +d \right ) x a \,c^{2} d^{3} e^{3}+48 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}-240 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+12 \ln \left (e x +d \right ) x^{4} b^{2} c \,e^{6}+60 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-36 x^{2} a b c d \,e^{5}-360 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}+72 \ln \left (e x +d \right ) x^{2} a \,c^{2} d^{2} e^{4}+2 c^{3} e^{6} x^{6}-440 x b \,c^{2} d^{4} e^{2}-24 x^{3} a b c \,e^{6}+48 x^{3} a \,c^{2} d \,e^{5}+48 x^{3} b^{2} c d \,e^{5}-240 x^{3} b \,c^{2} d^{2} e^{4}+108 x^{2} b^{2} c \,d^{2} e^{4}-540 x^{2} b \,c^{2} d^{3} e^{3}-4 x a \,b^{2} d \,e^{5}+88 x \,b^{2} c \,d^{3} e^{3}+12 \ln \left (e x +d \right ) a \,c^{2} d^{4} e^{2}+12 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}-60 \ln \left (e x +d \right ) b \,c^{2} d^{5} e +108 x^{2} a \,c^{2} d^{2} e^{4}-4 x \,a^{2} c d \,e^{5}+88 x a \,c^{2} d^{3} e^{3}+12 \ln \left (e x +d \right ) x^{4} a \,c^{2} e^{6}+360 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+12 x^{5} b \,c^{2} e^{6}-6 x^{2} a^{2} c \,e^{6}-6 x^{2} a \,b^{2} e^{6}-6 x^{2} b^{3} d \,e^{5}-4 x \,a^{2} b \,e^{6}-4 x \,b^{3} d^{2} e^{4}-b^{3} d^{3} e^{3}-4 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}}{4 e^{7} \left (e x +d \right )^{4}}\) | \(820\) |
Input:
int((c*x^2+b*x+a)^3/(e*x+d)^5,x,method=_RETURNVERBOSE)
Output:
(-1/4*(a^3*e^6+a^2*b*d*e^5+a^2*c*d^2*e^4+a*b^2*d^2*e^4+6*a*b*c*d^3*e^3-25* a*c^2*d^4*e^2+b^3*d^3*e^3-25*b^2*c*d^4*e^2+125*b*c^2*d^5*e-125*c^3*d^6)/e^ 7+1/2*c^3*x^6/e-(6*a*b*c*e^3-12*a*c^2*d*e^2+b^3*e^3-12*b^2*c*d*e^2+60*b*c^ 2*d^2*e-60*c^3*d^3)/e^4*x^3-3/2*(a^2*c*e^4+a*b^2*e^4+6*a*b*c*d*e^3-18*a*c^ 2*d^2*e^2+b^3*d*e^3-18*b^2*c*d^2*e^2+90*b*c^2*d^3*e-90*c^3*d^4)/e^5*x^2-(a ^2*b*e^5+a^2*c*d*e^4+a*b^2*d*e^4+6*a*b*c*d^2*e^3-22*a*c^2*d^3*e^2+b^3*d^2* e^3-22*b^2*c*d^3*e^2+110*b*c^2*d^4*e-110*c^3*d^5)/e^6*x+3*c^2*(b*e-c*d)/e^ 2*x^5)/(e*x+d)^4+3/e^7*c*(a*c*e^2+b^2*e^2-5*b*c*d*e+5*c^2*d^2)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (245) = 490\).
Time = 0.08 (sec) , antiderivative size = 647, normalized size of antiderivative = 2.58 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {2 \, c^{3} e^{6} x^{6} + 57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} - 12 \, {\left (c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} - 4 \, {\left (17 \, c^{3} d^{2} e^{4} - 12 \, b c^{2} d e^{5}\right )} x^{4} - 4 \, {\left (8 \, c^{3} d^{3} e^{3} + 12 \, b c^{2} d^{2} e^{4} - 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} + {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \, {\left (22 \, c^{3} d^{4} e^{2} - 42 \, b c^{2} d^{3} e^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \, {\left (42 \, c^{3} d^{5} e - 62 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x + 12 \, {\left (5 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} + {\left (5 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + {\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + {\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} x^{3} + 6 \, {\left (5 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4}\right )} x^{2} + 4 \, {\left (5 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3}\right )} x\right )} \log \left (e x + d\right )}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^5,x, algorithm="fricas")
Output:
1/4*(2*c^3*e^6*x^6 + 57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - (a*b^2 + a^2*c)*d^ 2*e^4 - 12*(c^3*d*e^5 - b*c^2*e^6)*x^5 - 4*(17*c^3*d^2*e^4 - 12*b*c^2*d*e^ 5)*x^4 - 4*(8*c^3*d^3*e^3 + 12*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 + 6*(22*c^3*d^4*e^2 - 42*b*c^2*d^3*e^3 + 18*(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 + 4* (42*c^3*d^5*e - 62*b*c^2*d^4*e^2 - a^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 - (a*b^2 + a^2*c)*d*e^5)*x + 12*(5*c^3*d^6 - 5*b *c^2*d^5*e + (b^2*c + a*c^2)*d^4*e^2 + (5*c^3*d^2*e^4 - 5*b*c^2*d*e^5 + (b ^2*c + a*c^2)*e^6)*x^4 + 4*(5*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + (b^2*c + a*c ^2)*d*e^5)*x^3 + 6*(5*c^3*d^4*e^2 - 5*b*c^2*d^3*e^3 + (b^2*c + a*c^2)*d^2* e^4)*x^2 + 4*(5*c^3*d^5*e - 5*b*c^2*d^4*e^2 + (b^2*c + a*c^2)*d^3*e^3)*x)* log(e*x + d))/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9*x^2 + 4*d^3*e^8*x + d^4 *e^7)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (245) = 490\).
Time = 61.78 (sec) , antiderivative size = 520, normalized size of antiderivative = 2.07 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {c^{3} x^{2}}{2 e^{5}} + \frac {3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} + x \left (\frac {3 b c^{2}}{e^{5}} - \frac {5 c^{3} d}{e^{6}}\right ) + \frac {- a^{3} e^{6} - a^{2} b d e^{5} - a^{2} c d^{2} e^{4} - a b^{2} d^{2} e^{4} - 6 a b c d^{3} e^{3} + 25 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 25 b^{2} c d^{4} e^{2} - 77 b c^{2} d^{5} e + 57 c^{3} d^{6} + x^{3} \left (- 24 a b c e^{6} + 48 a c^{2} d e^{5} - 4 b^{3} e^{6} + 48 b^{2} c d e^{5} - 120 b c^{2} d^{2} e^{4} + 80 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 6 a^{2} c e^{6} - 6 a b^{2} e^{6} - 36 a b c d e^{5} + 108 a c^{2} d^{2} e^{4} - 6 b^{3} d e^{5} + 108 b^{2} c d^{2} e^{4} - 300 b c^{2} d^{3} e^{3} + 210 c^{3} d^{4} e^{2}\right ) + x \left (- 4 a^{2} b e^{6} - 4 a^{2} c d e^{5} - 4 a b^{2} d e^{5} - 24 a b c d^{2} e^{4} + 88 a c^{2} d^{3} e^{3} - 4 b^{3} d^{2} e^{4} + 88 b^{2} c d^{3} e^{3} - 260 b c^{2} d^{4} e^{2} + 188 c^{3} d^{5} e\right )}{4 d^{4} e^{7} + 16 d^{3} e^{8} x + 24 d^{2} e^{9} x^{2} + 16 d e^{10} x^{3} + 4 e^{11} x^{4}} \] Input:
integrate((c*x**2+b*x+a)**3/(e*x+d)**5,x)
Output:
c**3*x**2/(2*e**5) + 3*c*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)* log(d + e*x)/e**7 + x*(3*b*c**2/e**5 - 5*c**3*d/e**6) + (-a**3*e**6 - a**2 *b*d*e**5 - a**2*c*d**2*e**4 - a*b**2*d**2*e**4 - 6*a*b*c*d**3*e**3 + 25*a *c**2*d**4*e**2 - b**3*d**3*e**3 + 25*b**2*c*d**4*e**2 - 77*b*c**2*d**5*e + 57*c**3*d**6 + x**3*(-24*a*b*c*e**6 + 48*a*c**2*d*e**5 - 4*b**3*e**6 + 4 8*b**2*c*d*e**5 - 120*b*c**2*d**2*e**4 + 80*c**3*d**3*e**3) + x**2*(-6*a** 2*c*e**6 - 6*a*b**2*e**6 - 36*a*b*c*d*e**5 + 108*a*c**2*d**2*e**4 - 6*b**3 *d*e**5 + 108*b**2*c*d**2*e**4 - 300*b*c**2*d**3*e**3 + 210*c**3*d**4*e**2 ) + x*(-4*a**2*b*e**6 - 4*a**2*c*d*e**5 - 4*a*b**2*d*e**5 - 24*a*b*c*d**2* e**4 + 88*a*c**2*d**3*e**3 - 4*b**3*d**2*e**4 + 88*b**2*c*d**3*e**3 - 260* b*c**2*d**4*e**2 + 188*c**3*d**5*e))/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d* *2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4)
Time = 0.05 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=\frac {57 \, c^{3} d^{6} - 77 \, b c^{2} d^{5} e - a^{2} b d e^{5} - a^{3} e^{6} + 25 \, {\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} - {\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} - {\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 4 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, {\left (b^{2} c + a c^{2}\right )} d e^{5} - {\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 6 \, {\left (35 \, c^{3} d^{4} e^{2} - 50 \, b c^{2} d^{3} e^{3} + 18 \, {\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} - {\left (b^{3} + 6 \, a b c\right )} d e^{5} - {\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 4 \, {\left (47 \, c^{3} d^{5} e - 65 \, b c^{2} d^{4} e^{2} - a^{2} b e^{6} + 22 \, {\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} - {\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} - {\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{4 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} + \frac {c^{3} e x^{2} - 2 \, {\left (5 \, c^{3} d - 3 \, b c^{2} e\right )} x}{2 \, e^{6}} + \frac {3 \, {\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + {\left (b^{2} c + a c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^5,x, algorithm="maxima")
Output:
1/4*(57*c^3*d^6 - 77*b*c^2*d^5*e - a^2*b*d*e^5 - a^3*e^6 + 25*(b^2*c + a*c ^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 - (a*b^2 + a^2*c)*d^2*e^4 + 4*(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 + 6*(35*c^3*d^4*e^2 - 50*b*c^2*d^3*e^3 + 18*(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 + 4*(47*c^3*d^5*e - 65*b*c^2*d^4*e^2 - a^2*b*e^6 + 22*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c )*d^2*e^4 - (a*b^2 + a^2*c)*d*e^5)*x)/(e^11*x^4 + 4*d*e^10*x^3 + 6*d^2*e^9 *x^2 + 4*d^3*e^8*x + d^4*e^7) + 1/2*(c^3*e*x^2 - 2*(5*c^3*d - 3*b*c^2*e)*x )/e^6 + 3*(5*c^3*d^2 - 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*log(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (245) = 490\).
Time = 0.34 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.75 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^3/(e*x+d)^5,x, algorithm="giac")
Output:
1/2*(c^3 - 6*(2*c^3*d*e - b*c^2*e^2)/((e*x + d)*e))*(e*x + d)^2/e^7 - 3*(5 *c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2 + a*c^2*e^2)*log(abs(e*x + d)/((e*x + d )^2*abs(e)))/e^7 + 1/4*(80*c^3*d^3*e^29/(e*x + d) - 30*c^3*d^4*e^29/(e*x + d)^2 + 8*c^3*d^5*e^29/(e*x + d)^3 - c^3*d^6*e^29/(e*x + d)^4 - 120*b*c^2* d^2*e^30/(e*x + d) + 60*b*c^2*d^3*e^30/(e*x + d)^2 - 20*b*c^2*d^4*e^30/(e* x + d)^3 + 3*b*c^2*d^5*e^30/(e*x + d)^4 + 48*b^2*c*d*e^31/(e*x + d) + 48*a *c^2*d*e^31/(e*x + d) - 36*b^2*c*d^2*e^31/(e*x + d)^2 - 36*a*c^2*d^2*e^31/ (e*x + d)^2 + 16*b^2*c*d^3*e^31/(e*x + d)^3 + 16*a*c^2*d^3*e^31/(e*x + d)^ 3 - 3*b^2*c*d^4*e^31/(e*x + d)^4 - 3*a*c^2*d^4*e^31/(e*x + d)^4 - 4*b^3*e^ 32/(e*x + d) - 24*a*b*c*e^32/(e*x + d) + 6*b^3*d*e^32/(e*x + d)^2 + 36*a*b *c*d*e^32/(e*x + d)^2 - 4*b^3*d^2*e^32/(e*x + d)^3 - 24*a*b*c*d^2*e^32/(e* x + d)^3 + b^3*d^3*e^32/(e*x + d)^4 + 6*a*b*c*d^3*e^32/(e*x + d)^4 - 6*a*b ^2*e^33/(e*x + d)^2 - 6*a^2*c*e^33/(e*x + d)^2 + 8*a*b^2*d*e^33/(e*x + d)^ 3 + 8*a^2*c*d*e^33/(e*x + d)^3 - 3*a*b^2*d^2*e^33/(e*x + d)^4 - 3*a^2*c*d^ 2*e^33/(e*x + d)^4 - 4*a^2*b*e^34/(e*x + d)^3 + 3*a^2*b*d*e^34/(e*x + d)^4 - a^3*e^35/(e*x + d)^4)/e^36
Time = 5.71 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.89 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx=x\,\left (\frac {3\,b\,c^2}{e^5}-\frac {5\,c^3\,d}{e^6}\right )-\frac {x\,\left (a^2\,b\,e^5+a^2\,c\,d\,e^4+a\,b^2\,d\,e^4+6\,a\,b\,c\,d^2\,e^3-22\,a\,c^2\,d^3\,e^2+b^3\,d^2\,e^3-22\,b^2\,c\,d^3\,e^2+65\,b\,c^2\,d^4\,e-47\,c^3\,d^5\right )+\frac {a^3\,e^6+a^2\,b\,d\,e^5+a^2\,c\,d^2\,e^4+a\,b^2\,d^2\,e^4+6\,a\,b\,c\,d^3\,e^3-25\,a\,c^2\,d^4\,e^2+b^3\,d^3\,e^3-25\,b^2\,c\,d^4\,e^2+77\,b\,c^2\,d^5\,e-57\,c^3\,d^6}{4\,e}+x^2\,\left (\frac {3\,a^2\,c\,e^5}{2}+\frac {3\,a\,b^2\,e^5}{2}+9\,a\,b\,c\,d\,e^4-27\,a\,c^2\,d^2\,e^3+\frac {3\,b^3\,d\,e^4}{2}-27\,b^2\,c\,d^2\,e^3+75\,b\,c^2\,d^3\,e^2-\frac {105\,c^3\,d^4\,e}{2}\right )+x^3\,\left (b^3\,e^5-12\,b^2\,c\,d\,e^4+30\,b\,c^2\,d^2\,e^3+6\,a\,b\,c\,e^5-20\,c^3\,d^3\,e^2-12\,a\,c^2\,d\,e^4\right )}{d^4\,e^6+4\,d^3\,e^7\,x+6\,d^2\,e^8\,x^2+4\,d\,e^9\,x^3+e^{10}\,x^4}+\frac {\ln \left (d+e\,x\right )\,\left (3\,b^2\,c\,e^2-15\,b\,c^2\,d\,e+15\,c^3\,d^2+3\,a\,c^2\,e^2\right )}{e^7}+\frac {c^3\,x^2}{2\,e^5} \] Input:
int((a + b*x + c*x^2)^3/(d + e*x)^5,x)
Output:
x*((3*b*c^2)/e^5 - (5*c^3*d)/e^6) - (x*(a^2*b*e^5 - 47*c^3*d^5 + b^3*d^2*e ^3 - 22*a*c^2*d^3*e^2 - 22*b^2*c*d^3*e^2 + a*b^2*d*e^4 + a^2*c*d*e^4 + 65* b*c^2*d^4*e + 6*a*b*c*d^2*e^3) + (a^3*e^6 - 57*c^3*d^6 + b^3*d^3*e^3 + a*b ^2*d^2*e^4 - 25*a*c^2*d^4*e^2 + a^2*c*d^2*e^4 - 25*b^2*c*d^4*e^2 + a^2*b*d *e^5 + 77*b*c^2*d^5*e + 6*a*b*c*d^3*e^3)/(4*e) + x^2*((3*a*b^2*e^5)/2 + (3 *a^2*c*e^5)/2 + (3*b^3*d*e^4)/2 - (105*c^3*d^4*e)/2 - 27*a*c^2*d^2*e^3 + 7 5*b*c^2*d^3*e^2 - 27*b^2*c*d^2*e^3 + 9*a*b*c*d*e^4) + x^3*(b^3*e^5 - 20*c^ 3*d^3*e^2 + 30*b*c^2*d^2*e^3 + 6*a*b*c*e^5 - 12*a*c^2*d*e^4 - 12*b^2*c*d*e ^4))/(d^4*e^6 + e^10*x^4 + 4*d^3*e^7*x + 4*d*e^9*x^3 + 6*d^2*e^8*x^2) + (l og(d + e*x)*(15*c^3*d^2 + 3*a*c^2*e^2 + 3*b^2*c*e^2 - 15*b*c^2*d*e))/e^7 + (c^3*x^2)/(2*e^5)
Time = 0.27 (sec) , antiderivative size = 806, normalized size of antiderivative = 3.21 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^3/(e*x+d)^5,x)
Output:
(12*log(d + e*x)*a*c**2*d**5*e**2 + 48*log(d + e*x)*a*c**2*d**4*e**3*x + 7 2*log(d + e*x)*a*c**2*d**3*e**4*x**2 + 48*log(d + e*x)*a*c**2*d**2*e**5*x* *3 + 12*log(d + e*x)*a*c**2*d*e**6*x**4 + 12*log(d + e*x)*b**2*c*d**5*e**2 + 48*log(d + e*x)*b**2*c*d**4*e**3*x + 72*log(d + e*x)*b**2*c*d**3*e**4*x **2 + 48*log(d + e*x)*b**2*c*d**2*e**5*x**3 + 12*log(d + e*x)*b**2*c*d*e** 6*x**4 - 60*log(d + e*x)*b*c**2*d**6*e - 240*log(d + e*x)*b*c**2*d**5*e**2 *x - 360*log(d + e*x)*b*c**2*d**4*e**3*x**2 - 240*log(d + e*x)*b*c**2*d**3 *e**4*x**3 - 60*log(d + e*x)*b*c**2*d**2*e**5*x**4 + 60*log(d + e*x)*c**3* d**7 + 240*log(d + e*x)*c**3*d**6*e*x + 360*log(d + e*x)*c**3*d**5*e**2*x* *2 + 240*log(d + e*x)*c**3*d**4*e**3*x**3 + 60*log(d + e*x)*c**3*d**3*e**4 *x**4 - a**3*d*e**6 - a**2*b*d**2*e**5 - 4*a**2*b*d*e**6*x - a**2*c*d**3*e **4 - 4*a**2*c*d**2*e**5*x - 6*a**2*c*d*e**6*x**2 - a*b**2*d**3*e**4 - 4*a *b**2*d**2*e**5*x - 6*a*b**2*d*e**6*x**2 + 6*a*b*c*e**7*x**4 + 13*a*c**2*d **5*e**2 + 40*a*c**2*d**4*e**3*x + 36*a*c**2*d**3*e**4*x**2 - 12*a*c**2*d* e**6*x**4 + b**3*e**7*x**4 + 13*b**2*c*d**5*e**2 + 40*b**2*c*d**4*e**3*x + 36*b**2*c*d**3*e**4*x**2 - 12*b**2*c*d*e**6*x**4 - 65*b*c**2*d**6*e - 200 *b*c**2*d**5*e**2*x - 180*b*c**2*d**4*e**3*x**2 + 60*b*c**2*d**2*e**5*x**4 + 12*b*c**2*d*e**6*x**5 + 65*c**3*d**7 + 200*c**3*d**6*e*x + 180*c**3*d** 5*e**2*x**2 - 60*c**3*d**3*e**4*x**4 - 12*c**3*d**2*e**5*x**5 + 2*c**3*d*e **6*x**6)/(4*d*e**7*(d**4 + 4*d**3*e*x + 6*d**2*e**2*x**2 + 4*d*e**3*x*...