Integrand size = 20, antiderivative size = 305 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 (2 c d-b e) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \]
(a*e^2-b*d*e+c*d^2)^3*(e*x+d)^(1+m)/e^7/(1+m)-3*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)^2*(e*x+d)^(2+m)/e^7/(2+m)+3*(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*x+d)^(3+m)/e^7/(3+m)-(-b*e+2*c*d)*(10*c^2*d^2+b^2*e^2 -2*c*e*(-3*a*e+5*b*d))*(e*x+d)^(4+m)/e^7/(4+m)+3*c*(5*c^2*d^2+b^2*e^2-c*e* (-a*e+5*b*d))*(e*x+d)^(5+m)/e^7/(5+m)-3*c^2*(-b*e+2*c*d)*(e*x+d)^(6+m)/e^7 /(6+m)+c^3*(e*x+d)^(7+m)/e^7/(7+m)
Time = 1.22 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.56 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left ((a+x (b+c x))^3+\frac {3 (d+e x) \left (2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) (3+m) \left (60 c^2 d^2+b^2 e^2 \left (2+3 m+m^2\right )-4 c e \left (15 b d+a e \left (-13+3 m+m^2\right )\right )\right )-2 (2+m) \left (120 c^4 d^4+b^4 e^4 \left (3+4 m+m^2\right )-2 b^2 c e^3 (1+m) (b d (-3+m)+3 a e (5+m))-8 c^3 d^2 e \left (30 b d+a e \left (-33-2 m+m^2\right )\right )+2 c^2 e^2 \left (4 a b d e \left (-33-2 m+m^2\right )+b^2 d^2 \left (57-2 m+m^2\right )+4 a^2 e^2 \left (24+10 m+m^2\right )\right )\right ) (d+e x)-c e^4 (2+m) (3+m) (4+m) (5+m) (a+x (b+c x))^2 (b e (11+m)+2 c (-5 d+e (6+m) x))-2 e^2 (2+m) (3+m) (a+x (b+c x)) \left (-b^3 e^3 (1+m)+20 c^3 d^2 (-3 d+e (4+m) x)-b c e^2 \left (-2 a e (37+7 m)+b d \left (72+13 m+m^2\right )+b e \left (4+5 m+m^2\right ) x\right )+2 c^2 e \left (5 b d (d (13+m)-2 e (4+m) x)+2 a e \left (d \left (-13+3 m+m^2\right )+e \left (24+10 m+m^2\right ) x\right )\right )\right )\right )}{c e^6 (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)}\right )}{e (1+m)} \]
((d + e*x)^(1 + m)*((a + x*(b + c*x))^3 + (3*(d + e*x)*(2*(2*c*d - b*e)*(c *d^2 + e*(-(b*d) + a*e))*(3 + m)*(60*c^2*d^2 + b^2*e^2*(2 + 3*m + m^2) - 4 *c*e*(15*b*d + a*e*(-13 + 3*m + m^2))) - 2*(2 + m)*(120*c^4*d^4 + b^4*e^4* (3 + 4*m + m^2) - 2*b^2*c*e^3*(1 + m)*(b*d*(-3 + m) + 3*a*e*(5 + m)) - 8*c ^3*d^2*e*(30*b*d + a*e*(-33 - 2*m + m^2)) + 2*c^2*e^2*(4*a*b*d*e*(-33 - 2* m + m^2) + b^2*d^2*(57 - 2*m + m^2) + 4*a^2*e^2*(24 + 10*m + m^2)))*(d + e *x) - c*e^4*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(a + x*(b + c*x))^2*(b*e*(11 + m) + 2*c*(-5*d + e*(6 + m)*x)) - 2*e^2*(2 + m)*(3 + m)*(a + x*(b + c*x))* (-(b^3*e^3*(1 + m)) + 20*c^3*d^2*(-3*d + e*(4 + m)*x) - b*c*e^2*(-2*a*e*(3 7 + 7*m) + b*d*(72 + 13*m + m^2) + b*e*(4 + 5*m + m^2)*x) + 2*c^2*e*(5*b*d *(d*(13 + m) - 2*e*(4 + m)*x) + 2*a*e*(d*(-13 + 3*m + m^2) + e*(24 + 10*m + m^2)*x)))))/(c*e^6*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m))))/(e *(1 + m))
Time = 0.52 (sec) , antiderivative size = 305, 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.
\(\displaystyle \int \left (a+b x+c x^2\right )^3 (d+e x)^m \, dx\) |
\(\Big \downarrow \) 1140 |
\(\displaystyle \int \left (\frac {3 (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (a c e^2+b^2 e^2-5 b c d e+5 c^2 d^2\right )}{e^6}+\frac {(2 c d-b e) (d+e x)^{m+3} \left (2 c e (5 b d-3 a e)-b^2 e^2-10 c^2 d^2\right )}{e^6}+\frac {3 c (d+e x)^{m+4} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac {(d+e x)^m \left (a e^2-b d e+c d^2\right )^3}{e^6}+\frac {3 (b e-2 c d) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{m+5}}{e^6}+\frac {c^3 (d+e x)^{m+6}}{e^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 (d+e x)^{m+3} \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+3)}-\frac {(2 c d-b e) (d+e x)^{m+4} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^7 (m+4)}+\frac {3 c (d+e x)^{m+5} \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^7 (m+5)}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^3}{e^7 (m+1)}-\frac {3 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )^2}{e^7 (m+2)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)}\) |
((c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(2 + m))/(e^7*(2 + m)) + (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))*(d + e*x)^(3 + m))/(e^7*(3 + m)) - ((2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (3*c*(5*c^2*d^2 + b^2*e^2 - c *e*(5*b*d - a*e))*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b*e)* (d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)^(7 + m))/(e^7*(7 + m))
3.26.50.3.1 Defintions of rubi rules used
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2337\) vs. \(2(305)=610\).
Time = 0.30 (sec) , antiderivative size = 2338, normalized size of antiderivative = 7.67
method | result | size |
norman | \(\text {Expression too large to display}\) | \(2338\) |
gosper | \(\text {Expression too large to display}\) | \(2927\) |
risch | \(\text {Expression too large to display}\) | \(3658\) |
parallelrisch | \(\text {Expression too large to display}\) | \(5366\) |
c^3/(7+m)*x^7*exp(m*ln(e*x+d))+d*(a^3*e^6*m^6+27*a^3*e^6*m^5-3*a^2*b*d*e^5 *m^5+295*a^3*e^6*m^4-75*a^2*b*d*e^5*m^4+6*a^2*c*d^2*e^4*m^4+6*a*b^2*d^2*e^ 4*m^4+1665*a^3*e^6*m^3-735*a^2*b*d*e^5*m^3+132*a^2*c*d^2*e^4*m^3+132*a*b^2 *d^2*e^4*m^3-36*a*b*c*d^3*e^3*m^3-6*b^3*d^3*e^3*m^3+5104*a^3*e^6*m^2-3525* a^2*b*d*e^5*m^2+1074*a^2*c*d^2*e^4*m^2+1074*a*b^2*d^2*e^4*m^2-648*a*b*c*d^ 3*e^3*m^2+72*a*c^2*d^4*e^2*m^2-108*b^3*d^3*e^3*m^2+72*b^2*c*d^4*e^2*m^2+80 28*a^3*e^6*m-8262*a^2*b*d*e^5*m+3828*a^2*c*d^2*e^4*m+3828*a*b^2*d^2*e^4*m- 3852*a*b*c*d^3*e^3*m+936*a*c^2*d^4*e^2*m-642*b^3*d^3*e^3*m+936*b^2*c*d^4*e ^2*m-360*b*c^2*d^5*e*m+5040*a^3*e^6-7560*a^2*b*d*e^5+5040*a^2*c*d^2*e^4+50 40*a*b^2*d^2*e^4-7560*a*b*c*d^3*e^3+3024*a*c^2*d^4*e^2-1260*b^3*d^3*e^3+30 24*b^2*c*d^4*e^2-2520*b*c^2*d^5*e+720*c^3*d^6)/e^7/(m^7+28*m^6+322*m^5+196 0*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(e*x+d))+(6*a*b*c*e^3*m^3+3 *a*c^2*d*e^2*m^3+b^3*e^3*m^3+3*b^2*c*d*e^2*m^3+108*a*b*c*e^3*m^2+39*a*c^2* d*e^2*m^2+18*b^3*e^3*m^2+39*b^2*c*d*e^2*m^2-15*b*c^2*d^2*e*m^2+642*a*b*c*e ^3*m+126*a*c^2*d*e^2*m+107*b^3*e^3*m+126*b^2*c*d*e^2*m-105*b*c^2*d^2*e*m+3 0*c^3*d^3*m+1260*a*b*c*e^3+210*b^3*e^3)/e^3/(m^4+22*m^3+179*m^2+638*m+840) *x^4*exp(m*ln(e*x+d))+(3*a^2*c*e^4*m^4+3*a*b^2*e^4*m^4+6*a*b*c*d*e^3*m^4+b ^3*d*e^3*m^4+66*a^2*c*e^4*m^3+66*a*b^2*e^4*m^3+108*a*b*c*d*e^3*m^3-12*a*c^ 2*d^2*e^2*m^3+18*b^3*d*e^3*m^3-12*b^2*c*d^2*e^2*m^3+537*a^2*c*e^4*m^2+537* a*b^2*e^4*m^2+642*a*b*c*d*e^3*m^2-156*a*c^2*d^2*e^2*m^2+107*b^3*d*e^3*m...
Leaf count of result is larger than twice the leaf count of optimal. 2550 vs. \(2 (305) = 610\).
Time = 0.32 (sec) , antiderivative size = 2550, normalized size of antiderivative = 8.36 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
(a^3*d*e^6*m^6 + 720*c^3*d^7 - 2520*b*c^2*d^6*e - 7560*a^2*b*d^2*e^5 + 504 0*a^3*d*e^6 + 3024*(b^2*c + a*c^2)*d^5*e^2 - 1260*(b^3 + 6*a*b*c)*d^4*e^3 + 5040*(a*b^2 + a^2*c)*d^3*e^4 + (c^3*e^7*m^6 + 21*c^3*e^7*m^5 + 175*c^3*e ^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7 )*x^7 + (2520*b*c^2*e^7 + (c^3*d*e^6 + 3*b*c^2*e^7)*m^6 + 3*(5*c^3*d*e^6 + 22*b*c^2*e^7)*m^5 + 5*(17*c^3*d*e^6 + 114*b*c^2*e^7)*m^4 + 15*(15*c^3*d*e ^6 + 164*b*c^2*e^7)*m^3 + (274*c^3*d*e^6 + 5547*b*c^2*e^7)*m^2 + 6*(20*c^3 *d*e^6 + 1019*b*c^2*e^7)*m)*x^6 - 3*(a^2*b*d^2*e^5 - 9*a^3*d*e^6)*m^5 + 3* (1008*(b^2*c + a*c^2)*e^7 + (b*c^2*d*e^6 + (b^2*c + a*c^2)*e^7)*m^6 - (2*c ^3*d^2*e^5 - 17*b*c^2*d*e^6 - 23*(b^2*c + a*c^2)*e^7)*m^5 - (20*c^3*d^2*e^ 5 - 105*b*c^2*d*e^6 - 207*(b^2*c + a*c^2)*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 5 9*b*c^2*d*e^6 - 185*(b^2*c + a*c^2)*e^7)*m^3 - 2*(50*c^3*d^2*e^5 - 187*b*c ^2*d*e^6 - 1072*(b^2*c + a*c^2)*e^7)*m^2 - 12*(4*c^3*d^2*e^5 - 14*b*c^2*d* e^6 - 201*(b^2*c + a*c^2)*e^7)*m)*x^5 - (75*a^2*b*d^2*e^5 - 295*a^3*d*e^6 - 6*(a*b^2 + a^2*c)*d^3*e^4)*m^4 + (1260*(b^3 + 6*a*b*c)*e^7 + (3*(b^2*c + a*c^2)*d*e^6 + (b^3 + 6*a*b*c)*e^7)*m^6 - 3*(5*b*c^2*d^2*e^5 - 19*(b^2*c + a*c^2)*d*e^6 - 8*(b^3 + 6*a*b*c)*e^7)*m^5 + (30*c^3*d^3*e^4 - 195*b*c^2* d^2*e^5 + 393*(b^2*c + a*c^2)*d*e^6 + 226*(b^3 + 6*a*b*c)*e^7)*m^4 + 3*(60 *c^3*d^3*e^4 - 265*b*c^2*d^2*e^5 + 401*(b^2*c + a*c^2)*d*e^6 + 352*(b^3 + 6*a*b*c)*e^7)*m^3 + 5*(66*c^3*d^3*e^4 - 249*b*c^2*d^2*e^5 + 324*(b^2*c ...
Leaf count of result is larger than twice the leaf count of optimal. 40049 vs. \(2 (284) = 568\).
Time = 7.18 (sec) , antiderivative size = 40049, normalized size of antiderivative = 131.31 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
Piecewise((d**m*(a**3*x + 3*a**2*b*x**2/2 + a**2*c*x**3 + a*b**2*x**3 + 3* a*b*c*x**4/2 + 3*a*c**2*x**5/5 + b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x* *6/2 + c**3*x**7/7), Eq(e, 0)), (-10*a**3*e**6/(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 + 3 60*d*e**12*x**5 + 60*e**13*x**6) - 6*a**2*b*d*e**5/(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) - 36*a**2*b*e**6*x/(60*d**6*e**7 + 36 0*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) - 3*a**2*c*d**2*e**4/(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) - 18*a**2*c*d*e**5*x/(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) - 45*a**2*c*e**6*x **2/(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) - 3*a*b**2 *d**2*e**4/(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) - 1 8*a*b**2*d*e**5*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1 200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x* *6) - 45*a*b**2*e**6*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e*...
Leaf count of result is larger than twice the leaf count of optimal. 1132 vs. \(2 (305) = 610\).
Time = 0.24 (sec) , antiderivative size = 1132, normalized size of antiderivative = 3.71 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
3*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a^2*b/((m^2 + 3*m + 2)*e^2 ) + (e*x + d)^(m + 1)*a^3/(e*(m + 1)) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*a*b^2/((m^3 + 6*m^2 + 11 *m + 6)*e^3) + 3*((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e* m*x + 2*d^3)*(e*x + d)^m*a^2*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + ((m^3 + 6* m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2* e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*b^3/((m^4 + 10*m^3 + 35*m^2 + 5 0*m + 24)*e^4) + 6*((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m) *d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*a* b*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3 *m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24*d^4*e*m*x + 24*d^5 )*(e*x + d)^m*b^2*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2 *x^2 - 24*d^4*e*m*x + 24*d^5)*(e*x + d)^m*a*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 3*((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^ 4 + 6*m^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x - 120*d^6)*(e*x + d)^m*b*c^...
Leaf count of result is larger than twice the leaf count of optimal. 5388 vs. \(2 (305) = 610\).
Time = 0.31 (sec) , antiderivative size = 5388, normalized size of antiderivative = 17.67 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
((e*x + d)^m*c^3*e^7*m^6*x^7 + (e*x + d)^m*c^3*d*e^6*m^6*x^6 + 3*(e*x + d) ^m*b*c^2*e^7*m^6*x^6 + 21*(e*x + d)^m*c^3*e^7*m^5*x^7 + 3*(e*x + d)^m*b*c^ 2*d*e^6*m^6*x^5 + 3*(e*x + d)^m*b^2*c*e^7*m^6*x^5 + 3*(e*x + d)^m*a*c^2*e^ 7*m^6*x^5 + 15*(e*x + d)^m*c^3*d*e^6*m^5*x^6 + 66*(e*x + d)^m*b*c^2*e^7*m^ 5*x^6 + 175*(e*x + d)^m*c^3*e^7*m^4*x^7 + 3*(e*x + d)^m*b^2*c*d*e^6*m^6*x^ 4 + 3*(e*x + d)^m*a*c^2*d*e^6*m^6*x^4 + (e*x + d)^m*b^3*e^7*m^6*x^4 + 6*(e *x + d)^m*a*b*c*e^7*m^6*x^4 - 6*(e*x + d)^m*c^3*d^2*e^5*m^5*x^5 + 51*(e*x + d)^m*b*c^2*d*e^6*m^5*x^5 + 69*(e*x + d)^m*b^2*c*e^7*m^5*x^5 + 69*(e*x + d)^m*a*c^2*e^7*m^5*x^5 + 85*(e*x + d)^m*c^3*d*e^6*m^4*x^6 + 570*(e*x + d)^ m*b*c^2*e^7*m^4*x^6 + 735*(e*x + d)^m*c^3*e^7*m^3*x^7 + (e*x + d)^m*b^3*d* e^6*m^6*x^3 + 6*(e*x + d)^m*a*b*c*d*e^6*m^6*x^3 + 3*(e*x + d)^m*a*b^2*e^7* m^6*x^3 + 3*(e*x + d)^m*a^2*c*e^7*m^6*x^3 - 15*(e*x + d)^m*b*c^2*d^2*e^5*m ^5*x^4 + 57*(e*x + d)^m*b^2*c*d*e^6*m^5*x^4 + 57*(e*x + d)^m*a*c^2*d*e^6*m ^5*x^4 + 24*(e*x + d)^m*b^3*e^7*m^5*x^4 + 144*(e*x + d)^m*a*b*c*e^7*m^5*x^ 4 - 60*(e*x + d)^m*c^3*d^2*e^5*m^4*x^5 + 315*(e*x + d)^m*b*c^2*d*e^6*m^4*x ^5 + 621*(e*x + d)^m*b^2*c*e^7*m^4*x^5 + 621*(e*x + d)^m*a*c^2*e^7*m^4*x^5 + 225*(e*x + d)^m*c^3*d*e^6*m^3*x^6 + 2460*(e*x + d)^m*b*c^2*e^7*m^3*x^6 + 1624*(e*x + d)^m*c^3*e^7*m^2*x^7 + 3*(e*x + d)^m*a*b^2*d*e^6*m^6*x^2 + 3 *(e*x + d)^m*a^2*c*d*e^6*m^6*x^2 + 3*(e*x + d)^m*a^2*b*e^7*m^6*x^2 - 12*(e *x + d)^m*b^2*c*d^2*e^5*m^5*x^3 - 12*(e*x + d)^m*a*c^2*d^2*e^5*m^5*x^3 ...
Time = 11.56 (sec) , antiderivative size = 2542, normalized size of antiderivative = 8.33 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
((d + e*x)^m*(720*c^3*d^7 + 5040*a^3*d*e^6 - 1260*b^3*d^4*e^3 + 5040*a*b^2 *d^3*e^4 - 7560*a^2*b*d^2*e^5 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 3024*b^2*c*d^5*e^2 + 5104*a^3*d*e^6*m^2 + 1665*a^3*d*e^6*m^3 + 295*a^3*d*e ^6*m^4 + 27*a^3*d*e^6*m^5 + a^3*d*e^6*m^6 - 642*b^3*d^4*e^3*m - 108*b^3*d^ 4*e^3*m^2 - 6*b^3*d^4*e^3*m^3 - 2520*b*c^2*d^6*e + 8028*a^3*d*e^6*m - 7560 *a*b*c*d^4*e^3 - 360*b*c^2*d^6*e*m + 3828*a*b^2*d^3*e^4*m - 8262*a^2*b*d^2 *e^5*m + 936*a*c^2*d^5*e^2*m + 3828*a^2*c*d^3*e^4*m + 936*b^2*c*d^5*e^2*m + 1074*a*b^2*d^3*e^4*m^2 - 3525*a^2*b*d^2*e^5*m^2 + 132*a*b^2*d^3*e^4*m^3 - 735*a^2*b*d^2*e^5*m^3 + 6*a*b^2*d^3*e^4*m^4 - 75*a^2*b*d^2*e^5*m^4 - 3*a ^2*b*d^2*e^5*m^5 + 72*a*c^2*d^5*e^2*m^2 + 1074*a^2*c*d^3*e^4*m^2 + 132*a^2 *c*d^3*e^4*m^3 + 6*a^2*c*d^3*e^4*m^4 + 72*b^2*c*d^5*e^2*m^2 - 3852*a*b*c*d ^4*e^3*m - 648*a*b*c*d^4*e^3*m^2 - 36*a*b*c*d^4*e^3*m^3))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x*(d + e*x)^m*(5040*a^3*e^7 + 8028*a^3*e^7*m + 5104*a^3*e^7*m^2 + 1665*a^3*e^7* m^3 + 295*a^3*e^7*m^4 + 27*a^3*e^7*m^5 + a^3*e^7*m^6 + 1260*b^3*d^3*e^4*m + 642*b^3*d^3*e^4*m^2 + 108*b^3*d^3*e^4*m^3 + 6*b^3*d^3*e^4*m^4 - 720*c^3* d^6*e*m + 7560*a^2*b*d*e^6*m - 5040*a*b^2*d^2*e^5*m + 8262*a^2*b*d*e^6*m^2 + 3525*a^2*b*d*e^6*m^3 + 735*a^2*b*d*e^6*m^4 + 75*a^2*b*d*e^6*m^5 + 3*a^2 *b*d*e^6*m^6 - 3024*a*c^2*d^4*e^3*m - 5040*a^2*c*d^2*e^5*m + 2520*b*c^2*d^ 5*e^2*m - 3024*b^2*c*d^4*e^3*m - 3828*a*b^2*d^2*e^5*m^2 - 1074*a*b^2*d^...