Integrand size = 19, antiderivative size = 166 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {d (5 c d-2 b e) (c d-b e)^2 x}{e^6}-\frac {(c d-b e)^2 (4 c d-b e) x^2}{2 e^5}+\frac {c (c d-b e)^2 x^3}{e^4}-\frac {c^2 (2 c d-3 b e) x^4}{4 e^3}+\frac {c^3 x^5}{5 e^2}-\frac {d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7} \] Output:
d*(-2*b*e+5*c*d)*(-b*e+c*d)^2*x/e^6-1/2*(-b*e+c*d)^2*(-b*e+4*c*d)*x^2/e^5+ c*(-b*e+c*d)^2*x^3/e^4-1/4*c^2*(-3*b*e+2*c*d)*x^4/e^3+1/5*c^3*x^5/e^2-d^3* (-b*e+c*d)^3/e^7/(e*x+d)-3*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*ln(e*x+d)/e^7
Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.96 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {20 d e (5 c d-2 b e) (c d-b e)^2 x+10 e^2 (c d-b e)^2 (-4 c d+b e) x^2+20 c e^3 (c d-b e)^2 x^3-5 c^2 e^4 (2 c d-3 b e) x^4+4 c^3 e^5 x^5-\frac {20 d^3 (c d-b e)^3}{d+e x}-60 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{20 e^7} \] Input:
Integrate[(b*x + c*x^2)^3/(d + e*x)^2,x]
Output:
(20*d*e*(5*c*d - 2*b*e)*(c*d - b*e)^2*x + 10*e^2*(c*d - b*e)^2*(-4*c*d + b *e)*x^2 + 20*c*e^3*(c*d - b*e)^2*x^3 - 5*c^2*e^4*(2*c*d - 3*b*e)*x^4 + 4*c ^3*e^5*x^5 - (20*d^3*(c*d - b*e)^3)/(d + e*x) - 60*d^2*(c*d - b*e)^2*(2*c* d - b*e)*Log[d + e*x])/(20*e^7)
Time = 0.66 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (-\frac {c^2 x^3 (2 c d-3 b e)}{e^3}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^2}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)}+\frac {d (5 c d-2 b e) (c d-b e)^2}{e^6}+\frac {x (b e-4 c d) (b e-c d)^2}{e^5}+\frac {3 c x^2 (c d-b e)^2}{e^4}+\frac {c^3 x^4}{e^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c^2 x^4 (2 c d-3 b e)}{4 e^3}-\frac {d^3 (c d-b e)^3}{e^7 (d+e x)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) \log (d+e x)}{e^7}+\frac {d x (5 c d-2 b e) (c d-b e)^2}{e^6}-\frac {x^2 (c d-b e)^2 (4 c d-b e)}{2 e^5}+\frac {c x^3 (c d-b e)^2}{e^4}+\frac {c^3 x^5}{5 e^2}\) |
Input:
Int[(b*x + c*x^2)^3/(d + e*x)^2,x]
Output:
(d*(5*c*d - 2*b*e)*(c*d - b*e)^2*x)/e^6 - ((c*d - b*e)^2*(4*c*d - b*e)*x^2 )/(2*e^5) + (c*(c*d - b*e)^2*x^3)/e^4 - (c^2*(2*c*d - 3*b*e)*x^4)/(4*e^3) + (c^3*x^5)/(5*e^2) - (d^3*(c*d - b*e)^3)/(e^7*(d + e*x)) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*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 = 0.41 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.57
| method | result | size |
| norman | \(\frac {\frac {d \left (3 b^{3} d^{2} e^{3}-12 b^{2} c \,d^{3} e^{2}+15 b \,c^{2} d^{4} e -6 d^{5} c^{3}\right )}{e^{7}}+\frac {c^{3} x^{6}}{5 e}+\frac {\left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right ) x^{3}}{2 e^{4}}+\frac {c \left (4 b^{2} e^{2}-5 b c d e +2 c^{2} d^{2}\right ) x^{4}}{4 e^{3}}+\frac {3 c^{2} \left (5 b e -2 c d \right ) x^{5}}{20 e^{2}}-\frac {3 d \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right ) x^{2}}{2 e^{5}}}{e x +d}+\frac {3 d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(260\) |
| default | \(-\frac {-\frac {1}{5} c^{3} x^{5} e^{4}-\frac {3}{4} b \,c^{2} e^{4} x^{4}+\frac {1}{2} c^{3} d \,e^{3} x^{4}-b^{2} c \,e^{4} x^{3}+2 b \,c^{2} d \,e^{3} x^{3}-c^{3} d^{2} e^{2} x^{3}-\frac {1}{2} b^{3} e^{4} x^{2}+3 b^{2} c d \,e^{3} x^{2}-\frac {9}{2} b \,c^{2} d^{2} e^{2} x^{2}+2 c^{3} d^{3} e \,x^{2}+2 d \,e^{3} b^{3} x -9 d^{2} e^{2} b^{2} c x +12 d^{3} e b \,c^{2} x -5 d^{4} c^{3} x}{e^{6}}+\frac {3 d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}+\frac {d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}{e^{7} \left (e x +d \right )}\) | \(276\) |
| risch | \(\frac {c^{3} x^{5}}{5 e^{2}}+\frac {3 b \,c^{2} x^{4}}{4 e^{2}}-\frac {c^{3} d \,x^{4}}{2 e^{3}}+\frac {b^{2} c \,x^{3}}{e^{2}}-\frac {2 b \,c^{2} d \,x^{3}}{e^{3}}+\frac {c^{3} d^{2} x^{3}}{e^{4}}+\frac {b^{3} x^{2}}{2 e^{2}}-\frac {3 b^{2} c d \,x^{2}}{e^{3}}+\frac {9 b \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {2 c^{3} d^{3} x^{2}}{e^{5}}-\frac {2 d \,b^{3} x}{e^{3}}+\frac {9 d^{2} b^{2} c x}{e^{4}}-\frac {12 d^{3} b \,c^{2} x}{e^{5}}+\frac {5 d^{4} c^{3} x}{e^{6}}+\frac {d^{3} b^{3}}{e^{4} \left (e x +d \right )}-\frac {3 d^{4} b^{2} c}{e^{5} \left (e x +d \right )}+\frac {3 d^{5} b \,c^{2}}{e^{6} \left (e x +d \right )}-\frac {d^{6} c^{3}}{e^{7} \left (e x +d \right )}+\frac {3 d^{2} \ln \left (e x +d \right ) b^{3}}{e^{4}}-\frac {12 d^{3} \ln \left (e x +d \right ) b^{2} c}{e^{5}}+\frac {15 d^{4} \ln \left (e x +d \right ) b \,c^{2}}{e^{6}}-\frac {6 d^{5} \ln \left (e x +d \right ) c^{3}}{e^{7}}\) | \(318\) |
| parallelrisch | \(\frac {10 x^{4} c^{3} d^{2} e^{4}-20 x^{3} c^{3} d^{3} e^{3}-30 x^{2} b^{3} d \,e^{5}-240 b^{2} c \,d^{4} e^{2}+60 \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}-120 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +300 b \,c^{2} d^{5} e -120 d^{6} c^{3}-25 x^{4} b \,c^{2} d \,e^{5}+50 x^{3} b \,c^{2} d^{2} e^{4}+120 x^{2} b^{2} c \,d^{2} e^{4}-150 x^{2} b \,c^{2} d^{3} e^{3}-240 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}+300 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -240 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}+300 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}+60 x^{2} c^{3} d^{4} e^{2}+60 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}-40 x^{3} b^{2} c d \,e^{5}+15 x^{5} b \,c^{2} e^{6}-6 x^{5} c^{3} d \,e^{5}+20 x^{4} b^{2} c \,e^{6}+10 x^{3} b^{3} e^{6}-120 \ln \left (e x +d \right ) c^{3} d^{6}+60 b^{3} d^{3} e^{3}+4 x^{6} c^{3} e^{6}}{20 e^{7} \left (e x +d \right )}\) | \(375\) |
Input:
int((c*x^2+b*x)^3/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
(d*(3*b^3*d^2*e^3-12*b^2*c*d^3*e^2+15*b*c^2*d^4*e-6*c^3*d^5)/e^7+1/5*c^3*x ^6/e+1/2*(b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)/e^4*x^3+1/4*c*(4* b^2*e^2-5*b*c*d*e+2*c^2*d^2)/e^3*x^4+3/20*c^2*(5*b*e-2*c*d)/e^2*x^5-3/2*d* (b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)/e^5*x^2)/(e*x+d)+3*d^2/e^7 *(b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (160) = 320\).
Time = 0.09 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.23 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {4 \, c^{3} e^{6} x^{6} - 20 \, c^{3} d^{6} + 60 \, b c^{2} d^{5} e - 60 \, b^{2} c d^{4} e^{2} + 20 \, b^{3} d^{3} e^{3} - 3 \, {\left (2 \, c^{3} d e^{5} - 5 \, b c^{2} e^{6}\right )} x^{5} + 5 \, {\left (2 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} + 4 \, b^{2} c e^{6}\right )} x^{4} - 10 \, {\left (2 \, c^{3} d^{3} e^{3} - 5 \, b c^{2} d^{2} e^{4} + 4 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 30 \, {\left (2 \, c^{3} d^{4} e^{2} - 5 \, b c^{2} d^{3} e^{3} + 4 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{5} e - 12 \, b c^{2} d^{4} e^{2} + 9 \, b^{2} c d^{3} e^{3} - 2 \, b^{3} d^{2} e^{4}\right )} x - 60 \, {\left (2 \, c^{3} d^{6} - 5 \, b c^{2} d^{5} e + 4 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + {\left (2 \, c^{3} d^{5} e - 5 \, b c^{2} d^{4} e^{2} + 4 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{8} x + d e^{7}\right )}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^2,x, algorithm="fricas")
Output:
1/20*(4*c^3*e^6*x^6 - 20*c^3*d^6 + 60*b*c^2*d^5*e - 60*b^2*c*d^4*e^2 + 20* b^3*d^3*e^3 - 3*(2*c^3*d*e^5 - 5*b*c^2*e^6)*x^5 + 5*(2*c^3*d^2*e^4 - 5*b*c ^2*d*e^5 + 4*b^2*c*e^6)*x^4 - 10*(2*c^3*d^3*e^3 - 5*b*c^2*d^2*e^4 + 4*b^2* c*d*e^5 - b^3*e^6)*x^3 + 30*(2*c^3*d^4*e^2 - 5*b*c^2*d^3*e^3 + 4*b^2*c*d^2 *e^4 - b^3*d*e^5)*x^2 + 20*(5*c^3*d^5*e - 12*b*c^2*d^4*e^2 + 9*b^2*c*d^3*e ^3 - 2*b^3*d^2*e^4)*x - 60*(2*c^3*d^6 - 5*b*c^2*d^5*e + 4*b^2*c*d^4*e^2 - b^3*d^3*e^3 + (2*c^3*d^5*e - 5*b*c^2*d^4*e^2 + 4*b^2*c*d^3*e^3 - b^3*d^2*e ^4)*x)*log(e*x + d))/(e^8*x + d*e^7)
Time = 0.43 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.55 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {c^{3} x^{5}}{5 e^{2}} + \frac {3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2} \log {\left (d + e x \right )}}{e^{7}} + x^{4} \cdot \left (\frac {3 b c^{2}}{4 e^{2}} - \frac {c^{3} d}{2 e^{3}}\right ) + x^{3} \left (\frac {b^{2} c}{e^{2}} - \frac {2 b c^{2} d}{e^{3}} + \frac {c^{3} d^{2}}{e^{4}}\right ) + x^{2} \left (\frac {b^{3}}{2 e^{2}} - \frac {3 b^{2} c d}{e^{3}} + \frac {9 b c^{2} d^{2}}{2 e^{4}} - \frac {2 c^{3} d^{3}}{e^{5}}\right ) + x \left (- \frac {2 b^{3} d}{e^{3}} + \frac {9 b^{2} c d^{2}}{e^{4}} - \frac {12 b c^{2} d^{3}}{e^{5}} + \frac {5 c^{3} d^{4}}{e^{6}}\right ) + \frac {b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}}{d e^{7} + e^{8} x} \] Input:
integrate((c*x**2+b*x)**3/(e*x+d)**2,x)
Output:
c**3*x**5/(5*e**2) + 3*d**2*(b*e - 2*c*d)*(b*e - c*d)**2*log(d + e*x)/e**7 + x**4*(3*b*c**2/(4*e**2) - c**3*d/(2*e**3)) + x**3*(b**2*c/e**2 - 2*b*c* *2*d/e**3 + c**3*d**2/e**4) + x**2*(b**3/(2*e**2) - 3*b**2*c*d/e**3 + 9*b* c**2*d**2/(2*e**4) - 2*c**3*d**3/e**5) + x*(-2*b**3*d/e**3 + 9*b**2*c*d**2 /e**4 - 12*b*c**2*d**3/e**5 + 5*c**3*d**4/e**6) + (b**3*d**3*e**3 - 3*b**2 *c*d**4*e**2 + 3*b*c**2*d**5*e - c**3*d**6)/(d*e**7 + e**8*x)
Time = 0.04 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.64 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=-\frac {c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}}{e^{8} x + d e^{7}} + \frac {4 \, c^{3} e^{4} x^{5} - 5 \, {\left (2 \, c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} x^{4} + 20 \, {\left (c^{3} d^{2} e^{2} - 2 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )} x^{3} - 10 \, {\left (4 \, c^{3} d^{3} e - 9 \, b c^{2} d^{2} e^{2} + 6 \, b^{2} c d e^{3} - b^{3} e^{4}\right )} x^{2} + 20 \, {\left (5 \, c^{3} d^{4} - 12 \, b c^{2} d^{3} e + 9 \, b^{2} c d^{2} e^{2} - 2 \, b^{3} d e^{3}\right )} x}{20 \, e^{6}} - \frac {3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^2,x, algorithm="maxima")
Output:
-(c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 - b^3*d^3*e^3)/(e^8*x + d*e^7) + 1/20*(4*c^3*e^4*x^5 - 5*(2*c^3*d*e^3 - 3*b*c^2*e^4)*x^4 + 20*(c^3*d^2*e ^2 - 2*b*c^2*d*e^3 + b^2*c*e^4)*x^3 - 10*(4*c^3*d^3*e - 9*b*c^2*d^2*e^2 + 6*b^2*c*d*e^3 - b^3*e^4)*x^2 + 20*(5*c^3*d^4 - 12*b*c^2*d^3*e + 9*b^2*c*d^ 2*e^2 - 2*b^3*d*e^3)*x)/e^6 - 3*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e ^2 - b^3*d^2*e^3)*log(e*x + d)/e^7
Leaf count of result is larger than twice the leaf count of optimal. 346 vs. \(2 (160) = 320\).
Time = 0.13 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.08 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {{\left (4 \, c^{3} - \frac {15 \, {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )}}{{\left (e x + d\right )} e} + \frac {20 \, {\left (5 \, c^{3} d^{2} e^{2} - 5 \, b c^{2} d e^{3} + b^{2} c e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {10 \, {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} - b^{3} e^{6}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {60 \, {\left (5 \, c^{3} d^{4} e^{4} - 10 \, b c^{2} d^{3} e^{5} + 6 \, b^{2} c d^{2} e^{6} - b^{3} d e^{7}\right )}}{{\left (e x + d\right )}^{4} e^{4}}\right )} {\left (e x + d\right )}^{5}}{20 \, e^{7}} + \frac {3 \, {\left (2 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e + 4 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{7}} - \frac {\frac {c^{3} d^{6} e^{5}}{e x + d} - \frac {3 \, b c^{2} d^{5} e^{6}}{e x + d} + \frac {3 \, b^{2} c d^{4} e^{7}}{e x + d} - \frac {b^{3} d^{3} e^{8}}{e x + d}}{e^{12}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^2,x, algorithm="giac")
Output:
1/20*(4*c^3 - 15*(2*c^3*d*e - b*c^2*e^2)/((e*x + d)*e) + 20*(5*c^3*d^2*e^2 - 5*b*c^2*d*e^3 + b^2*c*e^4)/((e*x + d)^2*e^2) - 10*(20*c^3*d^3*e^3 - 30* b*c^2*d^2*e^4 + 12*b^2*c*d*e^5 - b^3*e^6)/((e*x + d)^3*e^3) + 60*(5*c^3*d^ 4*e^4 - 10*b*c^2*d^3*e^5 + 6*b^2*c*d^2*e^6 - b^3*d*e^7)/((e*x + d)^4*e^4)) *(e*x + d)^5/e^7 + 3*(2*c^3*d^5 - 5*b*c^2*d^4*e + 4*b^2*c*d^3*e^2 - b^3*d^ 2*e^3)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^7 - (c^3*d^6*e^5/(e*x + d) - 3*b*c^2*d^5*e^6/(e*x + d) + 3*b^2*c*d^4*e^7/(e*x + d) - b^3*d^3*e^8/(e* x + d))/e^12
Time = 8.95 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.62 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=x^4\,\left (\frac {3\,b\,c^2}{4\,e^2}-\frac {c^3\,d}{2\,e^3}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{3\,e}-\frac {b^2\,c}{e^2}+\frac {c^3\,d^2}{3\,e^4}\right )+x^2\,\left (\frac {b^3}{2\,e^2}+\frac {d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}-\frac {3\,b^2\,c}{e^2}+\frac {c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{2\,e^2}\right )+x\,\left (\frac {d^2\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}-\frac {3\,b^2\,c}{e^2}+\frac {c^3\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {b^3}{e^2}+\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e}-\frac {3\,b^2\,c}{e^2}+\frac {c^3\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {3\,b\,c^2}{e^2}-\frac {2\,c^3\,d}{e^3}\right )}{e^2}\right )}{e}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-3\,b^3\,d^2\,e^3+12\,b^2\,c\,d^3\,e^2-15\,b\,c^2\,d^4\,e+6\,c^3\,d^5\right )}{e^7}-\frac {-b^3\,d^3\,e^3+3\,b^2\,c\,d^4\,e^2-3\,b\,c^2\,d^5\,e+c^3\,d^6}{e\,\left (x\,e^7+d\,e^6\right )}+\frac {c^3\,x^5}{5\,e^2} \] Input:
int((b*x + c*x^2)^3/(d + e*x)^2,x)
Output:
x^4*((3*b*c^2)/(4*e^2) - (c^3*d)/(2*e^3)) - x^3*((2*d*((3*b*c^2)/e^2 - (2* c^3*d)/e^3))/(3*e) - (b^2*c)/e^2 + (c^3*d^2)/(3*e^4)) + x^2*(b^3/(2*e^2) + (d*((2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e - (3*b^2*c)/e^2 + (c^3*d^2)/e ^4))/e - (d^2*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/(2*e^2)) + x*((d^2*((2*d*(( 3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e - (3*b^2*c)/e^2 + (c^3*d^2)/e^4))/e^2 - ( 2*d*(b^3/e^2 + (2*d*((2*d*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e - (3*b^2*c)/e ^2 + (c^3*d^2)/e^4))/e - (d^2*((3*b*c^2)/e^2 - (2*c^3*d)/e^3))/e^2))/e) - (log(d + e*x)*(6*c^3*d^5 - 3*b^3*d^2*e^3 + 12*b^2*c*d^3*e^2 - 15*b*c^2*d^4 *e))/e^7 - (c^3*d^6 - b^3*d^3*e^3 + 3*b^2*c*d^4*e^2 - 3*b*c^2*d^5*e)/(e*(d *e^6 + e^7*x)) + (c^3*x^5)/(5*e^2)
Time = 0.24 (sec) , antiderivative size = 381, normalized size of antiderivative = 2.30 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^2} \, dx=\frac {-240 \,\mathrm {log}\left (e x +d \right ) b^{2} c \,d^{4} e^{2}+300 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{5} e +240 b^{2} c \,d^{3} e^{3} x +120 b^{2} c \,d^{2} e^{4} x^{2}-40 b^{2} c d \,e^{5} x^{3}-300 b \,c^{2} d^{4} e^{2} x -150 b \,c^{2} d^{3} e^{3} x^{2}+50 b \,c^{2} d^{2} e^{4} x^{3}-25 b \,c^{2} d \,e^{5} x^{4}+60 \,\mathrm {log}\left (e x +d \right ) b^{3} d^{3} e^{3}-60 b^{3} d^{2} e^{4} x -30 b^{3} d \,e^{5} x^{2}+20 b^{2} c \,e^{6} x^{4}+15 b \,c^{2} e^{6} x^{5}+120 c^{3} d^{5} e x +60 c^{3} d^{4} e^{2} x^{2}-20 c^{3} d^{3} e^{3} x^{3}+10 c^{3} d^{2} e^{4} x^{4}-120 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{6}+10 b^{3} e^{6} x^{3}+4 c^{3} e^{6} x^{6}-240 \,\mathrm {log}\left (e x +d \right ) b^{2} c \,d^{3} e^{3} x +300 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{4} e^{2} x -6 c^{3} d \,e^{5} x^{5}+60 \,\mathrm {log}\left (e x +d \right ) b^{3} d^{2} e^{4} x -120 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{5} e x}{20 e^{7} \left (e x +d \right )} \] Input:
int((c*x^2+b*x)^3/(e*x+d)^2,x)
Output:
(60*log(d + e*x)*b**3*d**3*e**3 + 60*log(d + e*x)*b**3*d**2*e**4*x - 240*l og(d + e*x)*b**2*c*d**4*e**2 - 240*log(d + e*x)*b**2*c*d**3*e**3*x + 300*l og(d + e*x)*b*c**2*d**5*e + 300*log(d + e*x)*b*c**2*d**4*e**2*x - 120*log( d + e*x)*c**3*d**6 - 120*log(d + e*x)*c**3*d**5*e*x - 60*b**3*d**2*e**4*x - 30*b**3*d*e**5*x**2 + 10*b**3*e**6*x**3 + 240*b**2*c*d**3*e**3*x + 120*b **2*c*d**2*e**4*x**2 - 40*b**2*c*d*e**5*x**3 + 20*b**2*c*e**6*x**4 - 300*b *c**2*d**4*e**2*x - 150*b*c**2*d**3*e**3*x**2 + 50*b*c**2*d**2*e**4*x**3 - 25*b*c**2*d*e**5*x**4 + 15*b*c**2*e**6*x**5 + 120*c**3*d**5*e*x + 60*c**3 *d**4*e**2*x**2 - 20*c**3*d**3*e**3*x**3 + 10*c**3*d**2*e**4*x**4 - 6*c**3 *d*e**5*x**5 + 4*c**3*e**6*x**6)/(20*e**7*(d + e*x))