Integrand size = 19, antiderivative size = 213 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {c \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right ) x}{e^6}-\frac {c^2 (4 c d-3 b e) x^2}{2 e^5}+\frac {c^3 x^3}{3 e^4}-\frac {d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{2 e^7 (d+e x)^2}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{e^7 (d+e x)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) \log (d+e x)}{e^7} \] Output:
c*(3*b^2*e^2-12*b*c*d*e+10*c^2*d^2)*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*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^3+3/2*d^2*(-b*e+c*d)^2*(-b *e+2*c*d)/e^7/(e*x+d)^2-3*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/( e*x+d)-(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*ln(e*x+d)/e^7
Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.99 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {6 c e \left (10 c^2 d^2-12 b c d e+3 b^2 e^2\right ) x-3 c^2 e^2 (4 c d-3 b e) x^2+2 c^3 e^3 x^3-\frac {2 d^3 (c d-b e)^3}{(d+e x)^3}+\frac {9 d^2 (c d-b e)^2 (2 c d-b e)}{(d+e x)^2}+\frac {18 d \left (-5 c^3 d^3+10 b c^2 d^2 e-6 b^2 c d e^2+b^3 e^3\right )}{d+e x}+6 \left (-20 c^3 d^3+30 b c^2 d^2 e-12 b^2 c d e^2+b^3 e^3\right ) \log (d+e x)}{6 e^7} \] Input:
Integrate[(b*x + c*x^2)^3/(d + e*x)^4,x]
Output:
(6*c*e*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x - 3*c^2*e^2*(4*c*d - 3*b*e) *x^2 + 2*c^3*e^3*x^3 - (2*d^3*(c*d - b*e)^3)/(d + e*x)^3 + (9*d^2*(c*d - b *e)^2*(2*c*d - b*e))/(d + e*x)^2 + (18*d*(-5*c^3*d^3 + 10*b*c^2*d^2*e - 6* b^2*c*d*e^2 + b^3*e^3))/(d + e*x) + 6*(-20*c^3*d^3 + 30*b*c^2*d^2*e - 12*b ^2*c*d*e^2 + b^3*e^3)*Log[d + e*x])/(6*e^7)
Time = 0.74 (sec) , antiderivative size = 213, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6 (d+e x)}+\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^2}+\frac {c \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6}-\frac {c^2 x (4 c d-3 b e)}{e^5}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^4}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^3}+\frac {c^3 x^2}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (c d-b e)}{e^7 (d+e x)}-\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) \log (d+e x)}{e^7}+\frac {c x \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{e^6}-\frac {c^2 x^2 (4 c d-3 b e)}{2 e^5}-\frac {d^3 (c d-b e)^3}{3 e^7 (d+e x)^3}+\frac {3 d^2 (2 c d-b e) (c d-b e)^2}{2 e^7 (d+e x)^2}+\frac {c^3 x^3}{3 e^4}\) |
Input:
Int[(b*x + c*x^2)^3/(d + e*x)^4,x]
Output:
(c*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2)*x)/e^6 - (c^2*(4*c*d - 3*b*e)*x^2 )/(2*e^5) + (c^3*x^3)/(3*e^4) - (d^3*(c*d - b*e)^3)/(3*e^7*(d + e*x)^3) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(2*e^7*(d + e*x)^2) - (3*d*(c*d - b*e) *(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(e^7*(d + e*x)) - ((2*c*d - b*e)*(10*c ^2*d^2 - 10*b*c*d*e + b^2*e^2)*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 = 0.40 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.20
| method | result | size |
| norman | \(\frac {\frac {c^{3} x^{6}}{3 e}+\frac {d^{3} \left (11 b^{3} e^{3}-132 d \,e^{2} b^{2} c +330 d^{2} e b \,c^{2}-220 d^{3} c^{3}\right )}{6 e^{7}}+\frac {c \left (6 b^{2} e^{2}-15 b c d e +10 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}+\frac {c^{2} \left (3 b e -2 c d \right ) x^{5}}{2 e^{2}}+\frac {3 d \left (b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) x^{2}}{e^{5}}+\frac {3 d^{2} \left (3 b^{3} e^{3}-36 d \,e^{2} b^{2} c +90 d^{2} e b \,c^{2}-60 d^{3} c^{3}\right ) x}{2 e^{6}}}{\left (e x +d \right )^{3}}+\frac {\left (b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}\) | \(256\) |
| default | \(\frac {c \left (\frac {1}{3} c^{2} e^{2} x^{3}+\frac {3}{2} b c \,e^{2} x^{2}-2 c^{2} d e \,x^{2}+3 b^{2} e^{2} x -12 b c d e x +10 c^{2} d^{2} x \right )}{e^{6}}+\frac {d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}{3 e^{7} \left (e x +d \right )^{3}}+\frac {\left (b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}\right ) \ln \left (e x +d \right )}{e^{7}}+\frac {3 d \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +10 d^{2} e b \,c^{2}-5 d^{3} c^{3}\right )}{e^{7} \left (e x +d \right )}-\frac {3 d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right )}{2 e^{7} \left (e x +d \right )^{2}}\) | \(261\) |
| risch | \(\frac {c^{3} x^{3}}{3 e^{4}}+\frac {3 c^{2} b \,x^{2}}{2 e^{4}}-\frac {2 c^{3} d \,x^{2}}{e^{5}}+\frac {3 c \,b^{2} x}{e^{4}}-\frac {12 c^{2} b d x}{e^{5}}+\frac {10 c^{3} d^{2} x}{e^{6}}+\frac {\left (3 b^{3} d \,e^{4}-18 b^{2} c \,d^{2} e^{3}+30 d^{3} b \,c^{2} e^{2}-15 c^{3} d^{4} e \right ) x^{2}+\frac {3 d^{2} \left (3 b^{3} e^{3}-20 d \,e^{2} b^{2} c +35 d^{2} e b \,c^{2}-18 d^{3} c^{3}\right ) x}{2}+\frac {d^{3} \left (11 b^{3} e^{3}-78 d \,e^{2} b^{2} c +141 d^{2} e b \,c^{2}-74 d^{3} c^{3}\right )}{6 e}}{e^{6} \left (e x +d \right )^{3}}+\frac {\ln \left (e x +d \right ) b^{3}}{e^{4}}-\frac {12 \ln \left (e x +d \right ) d \,b^{2} c}{e^{5}}+\frac {30 \ln \left (e x +d \right ) d^{2} b \,c^{2}}{e^{6}}-\frac {20 \ln \left (e x +d \right ) d^{3} c^{3}}{e^{7}}\) | \(281\) |
| parallelrisch | \(\frac {30 x^{4} c^{3} d^{2} e^{4}+18 x^{2} b^{3} d \,e^{5}-132 b^{2} c \,d^{4} e^{2}+18 \ln \left (e x +d \right ) x \,b^{3} d^{2} e^{4}-360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e +330 b \,c^{2} d^{5} e -220 d^{6} c^{3}-45 x^{4} b \,c^{2} d \,e^{5}-216 \ln \left (e x +d \right ) x^{2} b^{2} c \,d^{2} e^{4}-216 x^{2} b^{2} c \,d^{2} e^{4}+540 x^{2} b \,c^{2} d^{3} e^{3}-72 \ln \left (e x +d \right ) b^{2} c \,d^{4} e^{2}+180 \ln \left (e x +d \right ) b \,c^{2} d^{5} e -216 \ln \left (e x +d \right ) x \,b^{2} c \,d^{3} e^{3}+540 \ln \left (e x +d \right ) x b \,c^{2} d^{4} e^{2}-72 \ln \left (e x +d \right ) x^{3} b^{2} c d \,e^{5}+540 \ln \left (e x +d \right ) x^{2} b \,c^{2} d^{3} e^{3}-360 x^{2} c^{3} d^{4} e^{2}+6 \ln \left (e x +d \right ) b^{3} d^{3} e^{3}+27 x \,b^{3} d^{2} e^{4}-540 x \,c^{3} d^{5} e +180 \ln \left (e x +d \right ) x^{3} b \,c^{2} d^{2} e^{4}-324 x \,b^{2} c \,d^{3} e^{3}+810 x b \,c^{2} d^{4} e^{2}+6 \ln \left (e x +d \right ) x^{3} b^{3} e^{6}-360 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}-120 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}+9 x^{5} b \,c^{2} e^{6}-6 x^{5} c^{3} d \,e^{5}+18 x^{4} b^{2} c \,e^{6}+18 \ln \left (e x +d \right ) x^{2} b^{3} d \,e^{5}-120 \ln \left (e x +d \right ) c^{3} d^{6}+11 b^{3} d^{3} e^{3}+2 x^{6} c^{3} e^{6}}{6 e^{7} \left (e x +d \right )^{3}}\) | \(527\) |
Input:
int((c*x^2+b*x)^3/(e*x+d)^4,x,method=_RETURNVERBOSE)
Output:
(1/3*c^3*x^6/e+1/6*d^3*(11*b^3*e^3-132*b^2*c*d*e^2+330*b*c^2*d^2*e-220*c^3 *d^3)/e^7+1/2*c*(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+3*d*(b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2*e-20*c^3*d^3)/e^5*x ^2+3/2*d^2*(3*b^3*e^3-36*b^2*c*d*e^2+90*b*c^2*d^2*e-60*c^3*d^3)/e^6*x)/(e* x+d)^3+1/e^7*(b^3*e^3-12*b^2*c*d*e^2+30*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. 480 vs. \(2 (205) = 410\).
Time = 0.09 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.25 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {2 \, c^{3} e^{6} x^{6} - 74 \, c^{3} d^{6} + 141 \, b c^{2} d^{5} e - 78 \, b^{2} c d^{4} e^{2} + 11 \, b^{3} d^{3} e^{3} - 3 \, {\left (2 \, c^{3} d e^{5} - 3 \, b c^{2} e^{6}\right )} x^{5} + 3 \, {\left (10 \, c^{3} d^{2} e^{4} - 15 \, b c^{2} d e^{5} + 6 \, b^{2} c e^{6}\right )} x^{4} + {\left (146 \, c^{3} d^{3} e^{3} - 189 \, b c^{2} d^{2} e^{4} + 54 \, b^{2} c d e^{5}\right )} x^{3} + 3 \, {\left (26 \, c^{3} d^{4} e^{2} - 9 \, b c^{2} d^{3} e^{3} - 18 \, b^{2} c d^{2} e^{4} + 6 \, b^{3} d e^{5}\right )} x^{2} - 3 \, {\left (34 \, c^{3} d^{5} e - 81 \, b c^{2} d^{4} e^{2} + 54 \, b^{2} c d^{3} e^{3} - 9 \, b^{3} d^{2} e^{4}\right )} x - 6 \, {\left (20 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + {\left (20 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 3 \, {\left (20 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 3 \, {\left (20 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x\right )} \log \left (e x + d\right )}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x)^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 - 78*b^2*c*d^4*e^2 + 11* b^3*d^3*e^3 - 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*e^6)*x^4 + (146*c^3*d^3*e^3 - 189*b*c^2*d^2*e^4 + 54* b^2*c*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*d^2*e^4 + 6*b^3*d*e^5)*x^2 - 3*(34*c^3*d^5*e - 81*b*c^2*d^4*e^2 + 54*b^2*c*d^3*e^3 - 9*b^3*d^2*e^4)*x - 6*(20*c^3*d^6 - 30*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 - b^3*d^3*e^3 + (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)*x^3 + 3*(20*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4 - b^3*d*e ^5)*x^2 + 3*(20*c^3*d^5*e - 30*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 - b^3*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)
Time = 1.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.41 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {c^{3} x^{3}}{3 e^{4}} + x^{2} \cdot \left (\frac {3 b c^{2}}{2 e^{4}} - \frac {2 c^{3} d}{e^{5}}\right ) + x \left (\frac {3 b^{2} c}{e^{4}} - \frac {12 b c^{2} d}{e^{5}} + \frac {10 c^{3} d^{2}}{e^{6}}\right ) + \frac {11 b^{3} d^{3} e^{3} - 78 b^{2} c d^{4} e^{2} + 141 b c^{2} d^{5} e - 74 c^{3} d^{6} + x^{2} \cdot \left (18 b^{3} d e^{5} - 108 b^{2} c d^{2} e^{4} + 180 b c^{2} d^{3} e^{3} - 90 c^{3} d^{4} e^{2}\right ) + x \left (27 b^{3} d^{2} e^{4} - 180 b^{2} c d^{3} e^{3} + 315 b c^{2} d^{4} e^{2} - 162 c^{3} d^{5} e\right )}{6 d^{3} e^{7} + 18 d^{2} e^{8} x + 18 d e^{9} x^{2} + 6 e^{10} x^{3}} + \frac {\left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \] Input:
integrate((c*x**2+b*x)**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*b**2* c/e**4 - 12*b*c**2*d/e**5 + 10*c**3*d**2/e**6) + (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*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*(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)*(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 = 294, normalized size of antiderivative = 1.38 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4}\right )} x}{6 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )}} + \frac {2 \, c^{3} e^{2} x^{3} - 3 \, {\left (4 \, c^{3} d e - 3 \, b c^{2} e^{2}\right )} x^{2} + 6 \, {\left (10 \, c^{3} d^{2} - 12 \, b c^{2} d e + 3 \, b^{2} c e^{2}\right )} x}{6 \, e^{6}} - \frac {{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^4,x, algorithm="maxima")
Output:
-1/6*(74*c^3*d^6 - 141*b*c^2*d^5*e + 78*b^2*c*d^4*e^2 - 11*b^3*d^3*e^3 + 1 8*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 - b^3*d*e^5)*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 - 3*b^3*d^2*e^4)*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*e^2) *x)/e^6 - (20*c^3*d^3 - 30*b*c^2*d^2*e + 12*b^2*c*d*e^2 - b^3*e^3)*log(e*x + d)/e^7
Time = 0.12 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.30 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=-\frac {{\left (20 \, c^{3} d^{3} - 30 \, b c^{2} d^{2} e + 12 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} \log \left ({\left | e x + d \right |}\right )}{e^{7}} - \frac {74 \, c^{3} d^{6} - 141 \, b c^{2} d^{5} e + 78 \, b^{2} c d^{4} e^{2} - 11 \, b^{3} d^{3} e^{3} + 18 \, {\left (5 \, c^{3} d^{4} e^{2} - 10 \, b c^{2} d^{3} e^{3} + 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 9 \, {\left (18 \, c^{3} d^{5} e - 35 \, b c^{2} d^{4} e^{2} + 20 \, b^{2} c d^{3} e^{3} - 3 \, b^{3} d^{2} e^{4}\right )} x}{6 \, {\left (e x + d\right )}^{3} e^{7}} + \frac {2 \, c^{3} e^{8} x^{3} - 12 \, c^{3} d e^{7} x^{2} + 9 \, b c^{2} e^{8} x^{2} + 60 \, c^{3} d^{2} e^{6} x - 72 \, b c^{2} d e^{7} x + 18 \, b^{2} c e^{8} x}{6 \, e^{12}} \] Input:
integrate((c*x^2+b*x)^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 - b^3*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 - 11*b^3*d^3* e^3 + 18*(5*c^3*d^4*e^2 - 10*b*c^2*d^3*e^3 + 6*b^2*c*d^2*e^4 - b^3*d*e^5)* 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 - 3*b^3*d^2*e^ 4)*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)/e^12
Time = 8.71 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.44 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=x^2\,\left (\frac {3\,b\,c^2}{2\,e^4}-\frac {2\,c^3\,d}{e^5}\right )-x\,\left (\frac {4\,d\,\left (\frac {3\,b\,c^2}{e^4}-\frac {4\,c^3\,d}{e^5}\right )}{e}-\frac {3\,b^2\,c}{e^4}+\frac {6\,c^3\,d^2}{e^6}\right )-\frac {x\,\left (-\frac {9\,b^3\,d^2\,e^3}{2}+30\,b^2\,c\,d^3\,e^2-\frac {105\,b\,c^2\,d^4\,e}{2}+27\,c^3\,d^5\right )-x^2\,\left (3\,b^3\,d\,e^4-18\,b^2\,c\,d^2\,e^3+30\,b\,c^2\,d^3\,e^2-15\,c^3\,d^4\,e\right )+\frac {-11\,b^3\,d^3\,e^3+78\,b^2\,c\,d^4\,e^2-141\,b\,c^2\,d^5\,e+74\,c^3\,d^6}{6\,e}}{d^3\,e^6+3\,d^2\,e^7\,x+3\,d\,e^8\,x^2+e^9\,x^3}+\frac {\ln \left (d+e\,x\right )\,\left (b^3\,e^3-12\,b^2\,c\,d\,e^2+30\,b\,c^2\,d^2\,e-20\,c^3\,d^3\right )}{e^7}+\frac {c^3\,x^3}{3\,e^4} \] Input:
int((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 - (3*b^2*c)/e^4 + (6*c^3*d^2)/e^6) - (x*(27*c^3*d^5 - (9*b^3*d^ 2*e^3)/2 + 30*b^2*c*d^3*e^2 - (105*b*c^2*d^4*e)/2) - x^2*(3*b^3*d*e^4 - 15 *c^3*d^4*e + 30*b*c^2*d^3*e^2 - 18*b^2*c*d^2*e^3) + (74*c^3*d^6 - 11*b^3*d ^3*e^3 + 78*b^2*c*d^4*e^2 - 141*b*c^2*d^5*e)/(6*e))/(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 + 30*b*c^2 *d^2*e - 12*b^2*c*d*e^2))/e^7 + (c^3*x^3)/(3*e^4)
Time = 0.22 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.56 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^4} \, dx=\frac {-100 c^{3} d^{6}-120 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{3} e^{3} x^{3}+6 \,\mathrm {log}\left (e x +d \right ) b^{3} e^{6} x^{3}-60 b^{2} c \,d^{4} e^{2}+150 b \,c^{2} d^{5} e +5 b^{3} d^{3} e^{3}-216 \,\mathrm {log}\left (e x +d \right ) b^{2} c \,d^{2} e^{4} x^{2}+540 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{3} e^{3} x^{2}-72 \,\mathrm {log}\left (e x +d \right ) b^{2} c \,d^{4} e^{2}+180 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{5} e -108 b^{2} c \,d^{3} e^{3} x +72 b^{2} c d \,e^{5} x^{3}+270 b \,c^{2} d^{4} e^{2} x -180 b \,c^{2} d^{2} e^{4} x^{3}-45 b \,c^{2} d \,e^{5} x^{4}+6 \,\mathrm {log}\left (e x +d \right ) b^{3} d^{3} e^{3}+9 b^{3} d^{2} e^{4} x +18 b^{2} c \,e^{6} x^{4}+9 b \,c^{2} e^{6} x^{5}-180 c^{3} d^{5} e x +120 c^{3} d^{3} e^{3} x^{3}+30 c^{3} d^{2} e^{4} x^{4}-120 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{6}-6 b^{3} e^{6} x^{3}+2 c^{3} e^{6} x^{6}-216 \,\mathrm {log}\left (e x +d \right ) b^{2} c \,d^{3} e^{3} x +540 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{4} e^{2} x +18 \,\mathrm {log}\left (e x +d \right ) b^{3} d \,e^{5} x^{2}-360 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{4} e^{2} x^{2}-6 c^{3} d \,e^{5} x^{5}-72 \,\mathrm {log}\left (e x +d \right ) b^{2} c d \,e^{5} x^{3}+180 \,\mathrm {log}\left (e x +d \right ) b \,c^{2} d^{2} e^{4} x^{3}+18 \,\mathrm {log}\left (e x +d \right ) b^{3} d^{2} e^{4} x -360 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{5} e x}{6 e^{7} \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right )} \] Input:
int((c*x^2+b*x)^3/(e*x+d)^4,x)
Output:
(6*log(d + e*x)*b**3*d**3*e**3 + 18*log(d + e*x)*b**3*d**2*e**4*x + 18*log (d + e*x)*b**3*d*e**5*x**2 + 6*log(d + e*x)*b**3*e**6*x**3 - 72*log(d + e* x)*b**2*c*d**4*e**2 - 216*log(d + e*x)*b**2*c*d**3*e**3*x - 216*log(d + e* x)*b**2*c*d**2*e**4*x**2 - 72*log(d + e*x)*b**2*c*d*e**5*x**3 + 180*log(d + e*x)*b*c**2*d**5*e + 540*log(d + e*x)*b*c**2*d**4*e**2*x + 540*log(d + e *x)*b*c**2*d**3*e**3*x**2 + 180*log(d + e*x)*b*c**2*d**2*e**4*x**3 - 120*l og(d + e*x)*c**3*d**6 - 360*log(d + e*x)*c**3*d**5*e*x - 360*log(d + e*x)* c**3*d**4*e**2*x**2 - 120*log(d + e*x)*c**3*d**3*e**3*x**3 + 5*b**3*d**3*e **3 + 9*b**3*d**2*e**4*x - 6*b**3*e**6*x**3 - 60*b**2*c*d**4*e**2 - 108*b* *2*c*d**3*e**3*x + 72*b**2*c*d*e**5*x**3 + 18*b**2*c*e**6*x**4 + 150*b*c** 2*d**5*e + 270*b*c**2*d**4*e**2*x - 180*b*c**2*d**2*e**4*x**3 - 45*b*c**2* d*e**5*x**4 + 9*b*c**2*e**6*x**5 - 100*c**3*d**6 - 180*c**3*d**5*e*x + 120 *c**3*d**3*e**3*x**3 + 30*c**3*d**2*e**4*x**4 - 6*c**3*d*e**5*x**5 + 2*c** 3*e**6*x**6)/(6*e**7*(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3))