Integrand size = 19, antiderivative size = 228 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}-\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{4 e^7 (d+e x)^4}+\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right )}{3 e^7 (d+e x)^3}-\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right )}{2 e^7 (d+e x)^2}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}+\frac {c^3 \log (d+e x)}{e^7} \] Output:
-1/6*d^3*(-b*e+c*d)^3/e^7/(e*x+d)^6+3/5*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)/e^7/ (e*x+d)^5-3/4*d*(-b*e+c*d)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)/e^7/(e*x+d)^4+1/3 *(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)/e^7/(e*x+d)^3-3/2*c*(b^2*e^2 -5*b*c*d*e+5*c^2*d^2)/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 = 0.07 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.01 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {-b^3 e^3 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )-6 b^2 c e^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )-30 b c^2 e \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )+c^3 d \left (147 d^5+822 d^4 e x+1875 d^3 e^2 x^2+2200 d^2 e^3 x^3+1350 d e^4 x^4+360 e^5 x^5\right )+60 c^3 (d+e x)^6 \log (d+e x)}{60 e^7 (d+e x)^6} \] Input:
Integrate[(b*x + c*x^2)^3/(d + e*x)^7,x]
Output:
(-(b^3*e^3*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) - 6*b^2*c*e^2*(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) - 30*b*c^2*e* (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) + 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) + 60*c^3*(d + e*x)^6*Log[d + e*x])/(60*e^7 *(d + e*x)^6)
Time = 0.74 (sec) , antiderivative size = 228, 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.
| \(\displaystyle \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^3}+\frac {(2 c d-b e) \left (-b^2 e^2+10 b c d e-10 c^2 d^2\right )}{e^6 (d+e x)^4}+\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6 (d+e x)^5}-\frac {3 c^2 (2 c d-b e)}{e^6 (d+e x)^2}+\frac {d^3 (c d-b e)^3}{e^6 (d+e x)^7}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{e^6 (d+e x)^6}+\frac {c^3}{e^6 (d+e x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{2 e^7 (d+e x)^2}+\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{3 e^7 (d+e x)^3}-\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac {3 c^2 (2 c d-b e)}{e^7 (d+e x)}-\frac {d^3 (c d-b e)^3}{6 e^7 (d+e x)^6}+\frac {3 d^2 (c d-b e)^2 (2 c d-b e)}{5 e^7 (d+e x)^5}+\frac {c^3 \log (d+e x)}{e^7}\) |
Input:
Int[(b*x + c*x^2)^3/(d + e*x)^7,x]
Output:
-1/6*(d^3*(c*d - b*e)^3)/(e^7*(d + e*x)^6) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(5*e^7*(d + e*x)^5) - (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2 *e^2))/(4*e^7*(d + e*x)^4) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2 *e^2))/(3*e^7*(d + e*x)^3) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(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 = 0.40 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {-\frac {3 c^{2} \left (b e -2 c d \right ) x^{5}}{e^{2}}-\frac {3 c \left (b^{2} e^{2}+5 b c d e -15 c^{2} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-110 d^{3} c^{3}\right ) x^{3}}{3 e^{4}}-\frac {d \left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-125 d^{3} c^{3}\right ) x^{2}}{4 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-137 d^{3} c^{3}\right ) x}{10 e^{6}}-\frac {d^{3} \left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-147 d^{3} c^{3}\right )}{60 e^{7}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(252\) |
| norman | \(\frac {-\frac {d^{3} \left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-147 d^{3} c^{3}\right )}{60 e^{7}}-\frac {3 \left (b \,c^{2} e -2 d \,c^{3}\right ) x^{5}}{e^{2}}-\frac {3 \left (e^{2} b^{2} c +5 c^{2} d e b -15 c^{3} d^{2}\right ) x^{4}}{2 e^{3}}-\frac {\left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-110 d^{3} c^{3}\right ) x^{3}}{3 e^{4}}-\frac {d \left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-125 d^{3} c^{3}\right ) x^{2}}{4 e^{5}}-\frac {d^{2} \left (b^{3} e^{3}+6 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-137 d^{3} c^{3}\right ) x}{10 e^{6}}}{\left (e x +d \right )^{6}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}\) | \(256\) |
| default | \(-\frac {b^{3} e^{3}-12 d \,e^{2} b^{2} c +30 d^{2} e b \,c^{2}-20 d^{3} c^{3}}{3 e^{7} \left (e x +d \right )^{3}}+\frac {3 d \left (b^{3} e^{3}-6 d \,e^{2} b^{2} c +10 d^{2} e b \,c^{2}-5 d^{3} c^{3}\right )}{4 e^{7} \left (e x +d \right )^{4}}+\frac {c^{3} \ln \left (e x +d \right )}{e^{7}}-\frac {3 d^{2} \left (b^{3} e^{3}-4 d \,e^{2} b^{2} c +5 d^{2} e b \,c^{2}-2 d^{3} c^{3}\right )}{5 e^{7} \left (e x +d \right )^{5}}-\frac {3 c^{2} \left (b e -2 c d \right )}{e^{7} \left (e x +d \right )}-\frac {3 c \left (b^{2} e^{2}-5 b c d e +5 c^{2} d^{2}\right )}{2 e^{7} \left (e x +d \right )^{2}}+\frac {d^{3} \left (b^{3} e^{3}-3 d \,e^{2} b^{2} c +3 d^{2} e b \,c^{2}-d^{3} c^{3}\right )}{6 e^{7} \left (e x +d \right )^{6}}\) | \(272\) |
| parallelrisch | \(\frac {1350 x^{4} c^{3} d^{2} e^{4}+2200 x^{3} c^{3} d^{3} e^{3}-15 x^{2} b^{3} d \,e^{5}-6 b^{2} c \,d^{4} e^{2}+360 \ln \left (e x +d \right ) x \,c^{3} d^{5} e -30 b \,c^{2} d^{5} e +147 d^{6} c^{3}+60 \ln \left (e x +d \right ) x^{6} c^{3} e^{6}-450 x^{4} b \,c^{2} d \,e^{5}-600 x^{3} b \,c^{2} d^{2} e^{4}-90 x^{2} b^{2} c \,d^{2} e^{4}-450 x^{2} b \,c^{2} d^{3} e^{3}+1875 x^{2} c^{3} d^{4} e^{2}-6 x \,b^{3} d^{2} e^{4}+822 x \,c^{3} d^{5} e -120 x^{3} b^{2} c d \,e^{5}+900 \ln \left (e x +d \right ) x^{4} c^{3} d^{2} e^{4}-36 x \,b^{2} c \,d^{3} e^{3}-180 x b \,c^{2} d^{4} e^{2}+900 \ln \left (e x +d \right ) x^{2} c^{3} d^{4} e^{2}+1200 \ln \left (e x +d \right ) x^{3} c^{3} d^{3} e^{3}-180 x^{5} b \,c^{2} e^{6}+360 x^{5} c^{3} d \,e^{5}-90 x^{4} b^{2} c \,e^{6}+360 \ln \left (e x +d \right ) x^{5} c^{3} d \,e^{5}-20 x^{3} b^{3} e^{6}+60 \ln \left (e x +d \right ) c^{3} d^{6}-b^{3} d^{3} e^{3}}{60 e^{7} \left (e x +d \right )^{6}}\) | \(400\) |
Input:
int((c*x^2+b*x)^3/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
(-3*c^2*(b*e-2*c*d)/e^2*x^5-3/2*c*(b^2*e^2+5*b*c*d*e-15*c^2*d^2)/e^3*x^4-1 /3*(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*d*(b^3*e ^3+6*b^2*c*d*e^2+30*b*c^2*d^2*e-125*c^3*d^3)/e^5*x^2-1/10*d^2*(b^3*e^3+6*b ^2*c*d*e^2+30*b*c^2*d^2*e-137*c^3*d^3)/e^6*x-1/60*d^3*(b^3*e^3+6*b^2*c*d*e ^2+30*b*c^2*d^2*e-147*c^3*d^3)/e^7)/(e*x+d)^6+c^3*ln(e*x+d)/e^7
Time = 0.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.79 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x + 60 \, {\left (c^{3} e^{6} x^{6} + 6 \, c^{3} d e^{5} x^{5} + 15 \, c^{3} d^{2} e^{4} x^{4} + 20 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{4} e^{2} x^{2} + 6 \, c^{3} d^{5} e x + c^{3} d^{6}\right )} \log \left (e x + d\right )}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^7,x, algorithm="fricas")
Output:
1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - b^3*d^3*e^3 + 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* e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*b^2*c*d*e^5 - b^3*e^ 6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*b^2*c*d^2*e^4 - b^3*d* e^5)*x^2 + 6*(137*c^3*d^5*e - 30*b*c^2*d^4*e^2 - 6*b^2*c*d^3*e^3 - b^3*d^2 *e^4)*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)
Time = 19.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.50 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log {\left (d + e x \right )}}{e^{7}} + \frac {- b^{3} d^{3} e^{3} - 6 b^{2} c d^{4} e^{2} - 30 b c^{2} d^{5} e + 147 c^{3} d^{6} + x^{5} \left (- 180 b c^{2} e^{6} + 360 c^{3} d e^{5}\right ) + x^{4} \left (- 90 b^{2} c e^{6} - 450 b c^{2} d e^{5} + 1350 c^{3} d^{2} e^{4}\right ) + x^{3} \left (- 20 b^{3} e^{6} - 120 b^{2} c d e^{5} - 600 b c^{2} d^{2} e^{4} + 2200 c^{3} d^{3} e^{3}\right ) + x^{2} \left (- 15 b^{3} d e^{5} - 90 b^{2} c d^{2} e^{4} - 450 b c^{2} d^{3} e^{3} + 1875 c^{3} d^{4} e^{2}\right ) + x \left (- 6 b^{3} d^{2} e^{4} - 36 b^{2} c d^{3} e^{3} - 180 b c^{2} d^{4} e^{2} + 822 c^{3} d^{5} e\right )}{60 d^{6} e^{7} + 360 d^{5} e^{8} x + 900 d^{4} e^{9} x^{2} + 1200 d^{3} e^{10} x^{3} + 900 d^{2} e^{11} x^{4} + 360 d e^{12} x^{5} + 60 e^{13} x^{6}} \] Input:
integrate((c*x**2+b*x)**3/(e*x+d)**7,x)
Output:
c**3*log(d + e*x)/e**7 + (-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 + x**5*(-180*b*c**2*e**6 + 360*c**3*d*e**5) + x**4 *(-90*b**2*c*e**6 - 450*b*c**2*d*e**5 + 1350*c**3*d**2*e**4) + x**3*(-20*b **3*e**6 - 120*b**2*c*d*e**5 - 600*b*c**2*d**2*e**4 + 2200*c**3*d**3*e**3) + x**2*(-15*b**3*d*e**5 - 90*b**2*c*d**2*e**4 - 450*b*c**2*d**3*e**3 + 18 75*c**3*d**4*e**2) + x*(-6*b**3*d**2*e**4 - 36*b**2*c*d**3*e**3 - 180*b*c* *2*d**4*e**2 + 822*c**3*d**5*e))/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d** 4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x** 5 + 60*e**13*x**6)
Time = 0.04 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.45 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3} + 180 \, {\left (2 \, c^{3} d e^{5} - b c^{2} e^{6}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{4} - 5 \, b c^{2} d e^{5} - b^{2} c e^{6}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{3} - 30 \, b c^{2} d^{2} e^{4} - 6 \, b^{2} c d e^{5} - b^{3} e^{6}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e^{2} - 30 \, b c^{2} d^{3} e^{3} - 6 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} e - 30 \, b c^{2} d^{4} e^{2} - 6 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4}\right )} x}{60 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )}} + \frac {c^{3} \log \left (e x + d\right )}{e^{7}} \] Input:
integrate((c*x^2+b*x)^3/(e*x+d)^7,x, algorithm="maxima")
Output:
1/60*(147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - b^3*d^3*e^3 + 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* e^6)*x^4 + 20*(110*c^3*d^3*e^3 - 30*b*c^2*d^2*e^4 - 6*b^2*c*d*e^5 - b^3*e^ 6)*x^3 + 15*(125*c^3*d^4*e^2 - 30*b*c^2*d^3*e^3 - 6*b^2*c*d^2*e^4 - b^3*d* e^5)*x^2 + 6*(137*c^3*d^5*e - 30*b*c^2*d^4*e^2 - 6*b^2*c*d^3*e^3 - b^3*d^2 *e^4)*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 = 0.13 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.20 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^{3} \log \left ({\left | e x + d \right |}\right )}{e^{7}} + \frac {180 \, {\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{5} + 90 \, {\left (15 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} - b^{2} c e^{5}\right )} x^{4} + 20 \, {\left (110 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} - 6 \, b^{2} c d e^{4} - b^{3} e^{5}\right )} x^{3} + 15 \, {\left (125 \, c^{3} d^{4} e - 30 \, b c^{2} d^{3} e^{2} - 6 \, b^{2} c d^{2} e^{3} - b^{3} d e^{4}\right )} x^{2} + 6 \, {\left (137 \, c^{3} d^{5} - 30 \, b c^{2} d^{4} e - 6 \, b^{2} c d^{3} e^{2} - b^{3} d^{2} e^{3}\right )} x + \frac {147 \, c^{3} d^{6} - 30 \, b c^{2} d^{5} e - 6 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}}{e}}{60 \, {\left (e x + d\right )}^{6} e^{6}} \] Input:
integrate((c*x^2+b*x)^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)*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 - b^3*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 - b^3*d*e^4)*x^2 + 6*(137*c^3*d^5 - 30*b*c^2* d^4*e - 6*b^2*c*d^3*e^2 - b^3*d^2*e^3)*x + (147*c^3*d^6 - 30*b*c^2*d^5*e - 6*b^2*c*d^4*e^2 - b^3*d^3*e^3)/e)/((e*x + d)^6*e^6)
Time = 8.98 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.18 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {c^3\,\ln \left (d+e\,x\right )}{e^7}-\frac {x^5\,\left (3\,b\,c^2\,e^6-6\,c^3\,d\,e^5\right )+x^4\,\left (\frac {3\,b^2\,c\,e^6}{2}+\frac {15\,b\,c^2\,d\,e^5}{2}-\frac {45\,c^3\,d^2\,e^4}{2}\right )+x\,\left (\frac {b^3\,d^2\,e^4}{10}+\frac {3\,b^2\,c\,d^3\,e^3}{5}+3\,b\,c^2\,d^4\,e^2-\frac {137\,c^3\,d^5\,e}{10}\right )+x^2\,\left (\frac {b^3\,d\,e^5}{4}+\frac {3\,b^2\,c\,d^2\,e^4}{2}+\frac {15\,b\,c^2\,d^3\,e^3}{2}-\frac {125\,c^3\,d^4\,e^2}{4}\right )+x^3\,\left (\frac {b^3\,e^6}{3}+2\,b^2\,c\,d\,e^5+10\,b\,c^2\,d^2\,e^4-\frac {110\,c^3\,d^3\,e^3}{3}\right )-\frac {49\,c^3\,d^6}{20}+\frac {b^3\,d^3\,e^3}{60}+\frac {b^2\,c\,d^4\,e^2}{10}+\frac {b\,c^2\,d^5\,e}{2}}{e^7\,{\left (d+e\,x\right )}^6} \] Input:
int((b*x + c*x^2)^3/(d + e*x)^7,x)
Output:
(c^3*log(d + e*x))/e^7 - (x^5*(3*b*c^2*e^6 - 6*c^3*d*e^5) + x^4*((3*b^2*c* e^6)/2 - (45*c^3*d^2*e^4)/2 + (15*b*c^2*d*e^5)/2) + x*((b^3*d^2*e^4)/10 - (137*c^3*d^5*e)/10 + 3*b*c^2*d^4*e^2 + (3*b^2*c*d^3*e^3)/5) + x^2*((b^3*d* e^5)/4 - (125*c^3*d^4*e^2)/4 + (15*b*c^2*d^3*e^3)/2 + (3*b^2*c*d^2*e^4)/2) + x^3*((b^3*e^6)/3 - (110*c^3*d^3*e^3)/3 + 10*b*c^2*d^2*e^4 + 2*b^2*c*d*e ^5) - (49*c^3*d^6)/20 + (b^3*d^3*e^3)/60 + (b^2*c*d^4*e^2)/10 + (b*c^2*d^5 *e)/2)/(e^7*(d + e*x)^6)
Time = 0.22 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.75 \[ \int \frac {\left (b x+c x^2\right )^3}{(d+e x)^7} \, dx=\frac {60 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{7}+360 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{6} e x +900 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{5} e^{2} x^{2}+1200 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{4} e^{3} x^{3}+900 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{3} e^{4} x^{4}+360 \,\mathrm {log}\left (e x +d \right ) c^{3} d^{2} e^{5} x^{5}+60 \,\mathrm {log}\left (e x +d \right ) c^{3} d \,e^{6} x^{6}-b^{3} d^{4} e^{3}-6 b^{3} d^{3} e^{4} x -15 b^{3} d^{2} e^{5} x^{2}-20 b^{3} d \,e^{6} x^{3}-6 b^{2} c \,d^{5} e^{2}-36 b^{2} c \,d^{4} e^{3} x -90 b^{2} c \,d^{3} e^{4} x^{2}-120 b^{2} c \,d^{2} e^{5} x^{3}-90 b^{2} c d \,e^{6} x^{4}+30 b \,c^{2} e^{7} x^{6}+87 c^{3} d^{7}+462 c^{3} d^{6} e x +975 c^{3} d^{5} e^{2} x^{2}+1000 c^{3} d^{4} e^{3} x^{3}+450 c^{3} d^{3} e^{4} x^{4}-60 c^{3} d \,e^{6} x^{6}}{60 d \,e^{7} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int((c*x^2+b*x)^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 - 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 + 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)/(60*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))