Integrand size = 20, antiderivative size = 178 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {\left (c d^2-b d e+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {2 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)} \] Output:
(a*e^2-b*d*e+c*d^2)^2*(e*x+d)^(1+m)/e^5/(1+m)-2*(-b*e+2*c*d)*(a*e^2-b*d*e+ c*d^2)*(e*x+d)^(2+m)/e^5/(2+m)+(6*c^2*d^2+b^2*e^2-2*c*e*(-a*e+3*b*d))*(e*x +d)^(3+m)/e^5/(3+m)-2*c*(-b*e+2*c*d)*(e*x+d)^(4+m)/e^5/(4+m)+c^2*(e*x+d)^( 5+m)/e^5/(5+m)
Time = 0.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (-\frac {2 (d+e x) (-6 c d+b e (7+m)+2 c e (4+m) x) (a+x (b+c x))}{e^2 (4+m) (5+m)}+(a+x (b+c x))^2+\frac {2 (d+e x) \left (\frac {6 (2 c d-b e) \left (c d^2+e (-b d+a e)\right )}{2+m}-\frac {\left (12 c^2 d^2-b^2 e^2 (1+m)+4 c e (-3 b d+a e (4+m))\right ) (d+e x)}{3+m}\right )}{e^4 (4+m) (5+m)}\right )}{e (1+m)} \] Input:
Integrate[(d + e*x)^m*(a + b*x + c*x^2)^2,x]
Output:
((d + e*x)^(1 + m)*((-2*(d + e*x)*(-6*c*d + b*e*(7 + m) + 2*c*e*(4 + m)*x) *(a + x*(b + c*x)))/(e^2*(4 + m)*(5 + m)) + (a + x*(b + c*x))^2 + (2*(d + e*x)*((6*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(2 + m) - ((12*c^2*d^2 - b^2*e^2*(1 + m) + 4*c*e*(-3*b*d + a*e*(4 + m)))*(d + e*x))/(3 + m)))/(e^ 4*(4 + m)*(5 + m))))/(e*(1 + m))
Time = 0.36 (sec) , antiderivative size = 178, 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.
| \(\displaystyle \int \left (a+b x+c x^2\right )^2 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \int \left (\frac {(d+e x)^{m+2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac {(d+e x)^m \left (a e^2-b d e+c d^2\right )^2}{e^4}+\frac {2 (b e-2 c d) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{m+3}}{e^4}+\frac {c^2 (d+e x)^{m+4}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^{m+3} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 (m+3)}+\frac {(d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^5 (m+1)}-\frac {2 (2 c d-b e) (d+e x)^{m+2} \left (a e^2-b d e+c d^2\right )}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)}\) |
Input:
Int[(d + e*x)^m*(a + b*x + c*x^2)^2,x]
Output:
((c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (2*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(2 + m))/(e^5*(2 + m)) + ((6*c^2*d^ 2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (2*c *(2*c*d - b*e)*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c^2*(d + e*x)^(5 + m))/ (e^5*(5 + m))
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. \(798\) vs. \(2(178)=356\).
Time = 1.05 (sec) , antiderivative size = 799, normalized size of antiderivative = 4.49
| method | result | size |
| norman | \(\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d \left (a^{2} e^{4} m^{4}+14 a^{2} e^{4} m^{3}-2 a b d \,e^{3} m^{3}+71 a^{2} e^{4} m^{2}-24 a b d \,e^{3} m^{2}+4 a c \,d^{2} e^{2} m^{2}+2 b^{2} d^{2} e^{2} m^{2}+154 a^{2} e^{4} m -94 a b d \,e^{3} m +36 a c \,d^{2} e^{2} m +18 b^{2} d^{2} e^{2} m -12 b c \,d^{3} e m +120 a^{2} e^{4}-120 d \,e^{3} a b +80 a c \,d^{2} e^{2}+40 d^{2} e^{2} b^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {\left (2 a c \,e^{2} m^{2}+b^{2} e^{2} m^{2}+2 b c d e \,m^{2}+18 a c \,e^{2} m +9 b^{2} e^{2} m +10 b c d e m -4 c^{2} d^{2} m +40 a c \,e^{2}+20 b^{2} e^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {\left (2 a b \,e^{3} m^{3}+2 a c d \,e^{2} m^{3}+b^{2} d \,e^{2} m^{3}+24 a b \,e^{3} m^{2}+18 a c d \,e^{2} m^{2}+9 b^{2} d \,e^{2} m^{2}-6 b c \,d^{2} e \,m^{2}+94 a b \,e^{3} m +40 a c d \,e^{2} m +20 b^{2} d \,e^{2} m -30 b c \,d^{2} e m +12 c^{2} d^{3} m +120 a \,e^{3} b \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}+\frac {\left (a^{2} e^{4} m^{4}+2 a b d \,e^{3} m^{4}+14 a^{2} e^{4} m^{3}+24 a b d \,e^{3} m^{3}-4 a c \,d^{2} e^{2} m^{3}-2 b^{2} d^{2} e^{2} m^{3}+71 a^{2} e^{4} m^{2}+94 a b d \,e^{3} m^{2}-36 a c \,d^{2} e^{2} m^{2}-18 b^{2} d^{2} e^{2} m^{2}+12 b c \,d^{3} e \,m^{2}+154 a^{2} e^{4} m +120 a b d \,e^{3} m -80 a c \,d^{2} e^{2} m -40 b^{2} d^{2} e^{2} m +60 b c \,d^{3} e m -24 c^{2} d^{4} m +120 a^{2} e^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {\left (2 b e m +c d m +10 b e \right ) c \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}\) | \(799\) |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+2 a c \,e^{4} m^{4} x^{2}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+2 a b \,e^{4} m^{4} x +24 a c \,e^{4} m^{3} x^{2}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}+a^{2} e^{4} m^{4}+26 a b \,e^{4} m^{3} x -4 a c d \,e^{3} m^{3} x +98 a c \,e^{4} m^{2} x^{2}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}+14 a^{2} e^{4} m^{3}-2 a b d \,e^{3} m^{3}+118 a b \,e^{4} m^{2} x -40 a c d \,e^{3} m^{2} x +156 a c \,e^{4} m \,x^{2}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 x^{3} b c \,e^{4}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 d \,c^{2} x^{3} e^{3}+71 a^{2} e^{4} m^{2}-24 a b d \,e^{3} m^{2}+214 a b \,e^{4} m x +4 a c \,d^{2} e^{2} m^{2}-116 a c d \,e^{3} m x +80 x^{2} a c \,e^{4}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 x^{2} b^{2} e^{4}+72 b c \,d^{2} e^{2} m x -60 x^{2} b c d \,e^{3}-24 c^{2} d^{3} e m x +24 x^{2} c^{2} d^{2} e^{2}+154 a^{2} e^{4} m -94 a b d \,e^{3} m +120 x a b \,e^{4}+36 a c \,d^{2} e^{2} m -80 x a c d \,e^{3}+18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 x b c \,d^{2} e^{2}-24 x \,c^{2} d^{3} e +120 a^{2} e^{4}-120 d \,e^{3} a b +80 a c \,d^{2} e^{2}+40 d^{2} e^{2} b^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(822\) |
| orering | \(\frac {\left (e x +d \right ) \left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+2 a c \,e^{4} m^{4} x^{2}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+2 a b \,e^{4} m^{4} x +24 a c \,e^{4} m^{3} x^{2}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}+a^{2} e^{4} m^{4}+26 a b \,e^{4} m^{3} x -4 a c d \,e^{3} m^{3} x +98 a c \,e^{4} m^{2} x^{2}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}+14 a^{2} e^{4} m^{3}-2 a b d \,e^{3} m^{3}+118 a b \,e^{4} m^{2} x -40 a c d \,e^{3} m^{2} x +156 a c \,e^{4} m \,x^{2}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 x^{3} b c \,e^{4}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 d \,c^{2} x^{3} e^{3}+71 a^{2} e^{4} m^{2}-24 a b d \,e^{3} m^{2}+214 a b \,e^{4} m x +4 a c \,d^{2} e^{2} m^{2}-116 a c d \,e^{3} m x +80 x^{2} a c \,e^{4}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 x^{2} b^{2} e^{4}+72 b c \,d^{2} e^{2} m x -60 x^{2} b c d \,e^{3}-24 c^{2} d^{3} e m x +24 x^{2} c^{2} d^{2} e^{2}+154 a^{2} e^{4} m -94 a b d \,e^{3} m +120 x a b \,e^{4}+36 a c \,d^{2} e^{2} m -80 x a c d \,e^{3}+18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 x b c \,d^{2} e^{2}-24 x \,c^{2} d^{3} e +120 a^{2} e^{4}-120 d \,e^{3} a b +80 a c \,d^{2} e^{2}+40 d^{2} e^{2} b^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right ) \left (e x +d \right )^{m}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(825\) |
| risch | \(\text {Expression too large to display}\) | \(1112\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1783\) |
Input:
int((e*x+d)^m*(c*x^2+b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
c^2/(5+m)*x^5*exp(m*ln(e*x+d))+d*(a^2*e^4*m^4+14*a^2*e^4*m^3-2*a*b*d*e^3*m ^3+71*a^2*e^4*m^2-24*a*b*d*e^3*m^2+4*a*c*d^2*e^2*m^2+2*b^2*d^2*e^2*m^2+154 *a^2*e^4*m-94*a*b*d*e^3*m+36*a*c*d^2*e^2*m+18*b^2*d^2*e^2*m-12*b*c*d^3*e*m +120*a^2*e^4-120*a*b*d*e^3+80*a*c*d^2*e^2+40*b^2*d^2*e^2-60*b*c*d^3*e+24*c ^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*exp(m*ln(e*x+d))+(2*a*c* e^2*m^2+b^2*e^2*m^2+2*b*c*d*e*m^2+18*a*c*e^2*m+9*b^2*e^2*m+10*b*c*d*e*m-4* c^2*d^2*m+40*a*c*e^2+20*b^2*e^2)/e^2/(m^3+12*m^2+47*m+60)*x^3*exp(m*ln(e*x +d))+(2*a*b*e^3*m^3+2*a*c*d*e^2*m^3+b^2*d*e^2*m^3+24*a*b*e^3*m^2+18*a*c*d* e^2*m^2+9*b^2*d*e^2*m^2-6*b*c*d^2*e*m^2+94*a*b*e^3*m+40*a*c*d*e^2*m+20*b^2 *d*e^2*m-30*b*c*d^2*e*m+12*c^2*d^3*m+120*a*b*e^3)/e^3/(m^4+14*m^3+71*m^2+1 54*m+120)*x^2*exp(m*ln(e*x+d))+(a^2*e^4*m^4+2*a*b*d*e^3*m^4+14*a^2*e^4*m^3 +24*a*b*d*e^3*m^3-4*a*c*d^2*e^2*m^3-2*b^2*d^2*e^2*m^3+71*a^2*e^4*m^2+94*a* b*d*e^3*m^2-36*a*c*d^2*e^2*m^2-18*b^2*d^2*e^2*m^2+12*b*c*d^3*e*m^2+154*a^2 *e^4*m+120*a*b*d*e^3*m-80*a*c*d^2*e^2*m-40*b^2*d^2*e^2*m+60*b*c*d^3*e*m-24 *c^2*d^4*m+120*a^2*e^4)/e^4/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*x*exp(m* ln(e*x+d))+(2*b*e*m+c*d*m+10*b*e)/e*c/(m^2+9*m+20)*x^4*exp(m*ln(e*x+d))
Leaf count of result is larger than twice the leaf count of optimal. 869 vs. \(2 (178) = 356\).
Time = 0.10 (sec) , antiderivative size = 869, normalized size of antiderivative = 4.88 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:
integrate((e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="fricas")
Output:
(a^2*d*e^4*m^4 + 24*c^2*d^5 - 60*b*c*d^4*e - 120*a*b*d^2*e^3 + 120*a^2*d*e ^4 + 40*(b^2 + 2*a*c)*d^3*e^2 + (c^2*e^5*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5 *m^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^5 + (60*b*c*e^5 + (c^2*d*e^4 + 2*b*c*e ^5)*m^4 + 2*(3*c^2*d*e^4 + 11*b*c*e^5)*m^3 + (11*c^2*d*e^4 + 82*b*c*e^5)*m ^2 + 2*(3*c^2*d*e^4 + 61*b*c*e^5)*m)*x^4 - 2*(a*b*d^2*e^3 - 7*a^2*d*e^4)*m ^3 + (40*(b^2 + 2*a*c)*e^5 + (2*b*c*d*e^4 + (b^2 + 2*a*c)*e^5)*m^4 - 4*(c^ 2*d^2*e^3 - 4*b*c*d*e^4 - 3*(b^2 + 2*a*c)*e^5)*m^3 - (12*c^2*d^2*e^3 - 34* b*c*d*e^4 - 49*(b^2 + 2*a*c)*e^5)*m^2 - 2*(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*(b^2 + 2*a*c)*e^5)*m)*x^3 - (24*a*b*d^2*e^3 - 71*a^2*d*e^4 - 2*(b^2 + 2 *a*c)*d^3*e^2)*m^2 + (120*a*b*e^5 + (2*a*b*e^5 + (b^2 + 2*a*c)*d*e^4)*m^4 - 2*(3*b*c*d^2*e^3 - 13*a*b*e^5 - 5*(b^2 + 2*a*c)*d*e^4)*m^3 + (12*c^2*d^3 *e^2 - 36*b*c*d^2*e^3 + 118*a*b*e^5 + 29*(b^2 + 2*a*c)*d*e^4)*m^2 + 2*(6*c ^2*d^3*e^2 - 15*b*c*d^2*e^3 + 107*a*b*e^5 + 10*(b^2 + 2*a*c)*d*e^4)*m)*x^2 - 2*(6*b*c*d^4*e + 47*a*b*d^2*e^3 - 77*a^2*d*e^4 - 9*(b^2 + 2*a*c)*d^3*e^ 2)*m + (120*a^2*e^5 + (2*a*b*d*e^4 + a^2*e^5)*m^4 + 2*(12*a*b*d*e^4 + 7*a^ 2*e^5 - (b^2 + 2*a*c)*d^2*e^3)*m^3 + (12*b*c*d^3*e^2 + 94*a*b*d*e^4 + 71*a ^2*e^5 - 18*(b^2 + 2*a*c)*d^2*e^3)*m^2 - 2*(12*c^2*d^4*e - 30*b*c*d^3*e^2 - 60*a*b*d*e^4 - 77*a^2*e^5 + 20*(b^2 + 2*a*c)*d^2*e^3)*m)*x)*(e*x + d)^m/ (e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2 + 274*e^5*m + 120*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 10171 vs. \(2 (162) = 324\).
Time = 2.27 (sec) , antiderivative size = 10171, normalized size of antiderivative = 57.14 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)**m*(c*x**2+b*x+a)**2,x)
Output:
Piecewise((d**m*(a**2*x + a*b*x**2 + 2*a*c*x**3/3 + b**2*x**3/3 + b*c*x**4 /2 + c**2*x**5/5), Eq(e, 0)), (-3*a**2*e**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 2*a*b*d*e**3/(12*d **4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x **4) - 8*a*b*e**4*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 4 8*d*e**8*x**3 + 12*e**9*x**4) - 2*a*c*d**2*e**2/(12*d**4*e**5 + 48*d**3*e* *6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 8*a*c*d*e**3*x /(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12* e**9*x**4) - 12*a*c*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e** 7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - b**2*d**2*e**2/(12*d**4*e**5 + 4 8*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*b** 2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8* x**3 + 12*e**9*x**4) - 6*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 7 2*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b*c*d**3*e/(12*d**4* e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 36*b*c*d*e**3*x**2/(12*d**4*e**5 + 48*d* *3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*e* *4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x** 3 + 12*e**9*x**4) + 12*c**2*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e...
Leaf count of result is larger than twice the leaf count of optimal. 455 vs. \(2 (178) = 356\).
Time = 0.05 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.56 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {2 \, {\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a^{2}}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \] Input:
integrate((e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="maxima")
Output:
2*(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a*b/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a^2/(e*(m + 1)) + ((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*b^2/((m^3 + 6*m^2 + 11*m + 6 )*e^3) + 2*((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*c/((m^3 + 6*m^2 + 11*m + 6)*e^3) + 2*((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*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^5 + (m^4 + 6*m^3 + 1 1*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*c^2/((m^5 + 15*m^4 + 85*m ^3 + 225*m^2 + 274*m + 120)*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 1716 vs. \(2 (178) = 356\).
Time = 0.12 (sec) , antiderivative size = 1716, normalized size of antiderivative = 9.64 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^m*(c*x^2+b*x+a)^2,x, algorithm="giac")
Output:
((e*x + d)^m*c^2*e^5*m^4*x^5 + (e*x + d)^m*c^2*d*e^4*m^4*x^4 + 2*(e*x + d) ^m*b*c*e^5*m^4*x^4 + 10*(e*x + d)^m*c^2*e^5*m^3*x^5 + 2*(e*x + d)^m*b*c*d* e^4*m^4*x^3 + (e*x + d)^m*b^2*e^5*m^4*x^3 + 2*(e*x + d)^m*a*c*e^5*m^4*x^3 + 6*(e*x + d)^m*c^2*d*e^4*m^3*x^4 + 22*(e*x + d)^m*b*c*e^5*m^3*x^4 + 35*(e *x + d)^m*c^2*e^5*m^2*x^5 + (e*x + d)^m*b^2*d*e^4*m^4*x^2 + 2*(e*x + d)^m* a*c*d*e^4*m^4*x^2 + 2*(e*x + d)^m*a*b*e^5*m^4*x^2 - 4*(e*x + d)^m*c^2*d^2* e^3*m^3*x^3 + 16*(e*x + d)^m*b*c*d*e^4*m^3*x^3 + 12*(e*x + d)^m*b^2*e^5*m^ 3*x^3 + 24*(e*x + d)^m*a*c*e^5*m^3*x^3 + 11*(e*x + d)^m*c^2*d*e^4*m^2*x^4 + 82*(e*x + d)^m*b*c*e^5*m^2*x^4 + 50*(e*x + d)^m*c^2*e^5*m*x^5 + 2*(e*x + d)^m*a*b*d*e^4*m^4*x + (e*x + d)^m*a^2*e^5*m^4*x - 6*(e*x + d)^m*b*c*d^2* e^3*m^3*x^2 + 10*(e*x + d)^m*b^2*d*e^4*m^3*x^2 + 20*(e*x + d)^m*a*c*d*e^4* m^3*x^2 + 26*(e*x + d)^m*a*b*e^5*m^3*x^2 - 12*(e*x + d)^m*c^2*d^2*e^3*m^2* x^3 + 34*(e*x + d)^m*b*c*d*e^4*m^2*x^3 + 49*(e*x + d)^m*b^2*e^5*m^2*x^3 + 98*(e*x + d)^m*a*c*e^5*m^2*x^3 + 6*(e*x + d)^m*c^2*d*e^4*m*x^4 + 122*(e*x + d)^m*b*c*e^5*m*x^4 + 24*(e*x + d)^m*c^2*e^5*x^5 + (e*x + d)^m*a^2*d*e^4* m^4 - 2*(e*x + d)^m*b^2*d^2*e^3*m^3*x - 4*(e*x + d)^m*a*c*d^2*e^3*m^3*x + 24*(e*x + d)^m*a*b*d*e^4*m^3*x + 14*(e*x + d)^m*a^2*e^5*m^3*x + 12*(e*x + d)^m*c^2*d^3*e^2*m^2*x^2 - 36*(e*x + d)^m*b*c*d^2*e^3*m^2*x^2 + 29*(e*x + d)^m*b^2*d*e^4*m^2*x^2 + 58*(e*x + d)^m*a*c*d*e^4*m^2*x^2 + 118*(e*x + d)^ m*a*b*e^5*m^2*x^2 - 8*(e*x + d)^m*c^2*d^2*e^3*m*x^3 + 20*(e*x + d)^m*b*...
Time = 6.73 (sec) , antiderivative size = 895, normalized size of antiderivative = 5.03 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (a^2\,d\,e^4\,m^4+14\,a^2\,d\,e^4\,m^3+71\,a^2\,d\,e^4\,m^2+154\,a^2\,d\,e^4\,m+120\,a^2\,d\,e^4-2\,a\,b\,d^2\,e^3\,m^3-24\,a\,b\,d^2\,e^3\,m^2-94\,a\,b\,d^2\,e^3\,m-120\,a\,b\,d^2\,e^3+4\,a\,c\,d^3\,e^2\,m^2+36\,a\,c\,d^3\,e^2\,m+80\,a\,c\,d^3\,e^2+2\,b^2\,d^3\,e^2\,m^2+18\,b^2\,d^3\,e^2\,m+40\,b^2\,d^3\,e^2-12\,b\,c\,d^4\,e\,m-60\,b\,c\,d^4\,e+24\,c^2\,d^5\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^5\,m^4+14\,a^2\,e^5\,m^3+71\,a^2\,e^5\,m^2+154\,a^2\,e^5\,m+120\,a^2\,e^5+2\,a\,b\,d\,e^4\,m^4+24\,a\,b\,d\,e^4\,m^3+94\,a\,b\,d\,e^4\,m^2+120\,a\,b\,d\,e^4\,m-4\,a\,c\,d^2\,e^3\,m^3-36\,a\,c\,d^2\,e^3\,m^2-80\,a\,c\,d^2\,e^3\,m-2\,b^2\,d^2\,e^3\,m^3-18\,b^2\,d^2\,e^3\,m^2-40\,b^2\,d^2\,e^3\,m+12\,b\,c\,d^3\,e^2\,m^2+60\,b\,c\,d^3\,e^2\,m-24\,c^2\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2+2\,b\,c\,d\,e\,m^2+10\,b\,c\,d\,e\,m-4\,c^2\,d^2\,m+2\,a\,c\,e^2\,m^2+18\,a\,c\,e^2\,m+40\,a\,c\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (b^2\,d\,e^2\,m^3+9\,b^2\,d\,e^2\,m^2+20\,b^2\,d\,e^2\,m-6\,b\,c\,d^2\,e\,m^2-30\,b\,c\,d^2\,e\,m+2\,a\,b\,e^3\,m^3+24\,a\,b\,e^3\,m^2+94\,a\,b\,e^3\,m+120\,a\,b\,e^3+12\,c^2\,d^3\,m+2\,a\,c\,d\,e^2\,m^3+18\,a\,c\,d\,e^2\,m^2+40\,a\,c\,d\,e^2\,m\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,x^4\,{\left (d+e\,x\right )}^m\,\left (10\,b\,e+2\,b\,e\,m+c\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )} \] Input:
int((d + e*x)^m*(a + b*x + c*x^2)^2,x)
Output:
((d + e*x)^m*(24*c^2*d^5 + 120*a^2*d*e^4 + 40*b^2*d^3*e^2 + 71*a^2*d*e^4*m ^2 + 14*a^2*d*e^4*m^3 + a^2*d*e^4*m^4 + 18*b^2*d^3*e^2*m - 60*b*c*d^4*e + 2*b^2*d^3*e^2*m^2 - 120*a*b*d^2*e^3 + 80*a*c*d^3*e^2 + 154*a^2*d*e^4*m - 9 4*a*b*d^2*e^3*m + 36*a*c*d^3*e^2*m - 24*a*b*d^2*e^3*m^2 - 2*a*b*d^2*e^3*m^ 3 + 4*a*c*d^3*e^2*m^2 - 12*b*c*d^4*e*m))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (x*(d + e*x)^m*(120*a^2*e^5 + 154*a^2*e^5*m + 71*a^ 2*e^5*m^2 + 14*a^2*e^5*m^3 + a^2*e^5*m^4 - 40*b^2*d^2*e^3*m - 18*b^2*d^2*e ^3*m^2 - 2*b^2*d^2*e^3*m^3 - 24*c^2*d^4*e*m + 94*a*b*d*e^4*m^2 + 24*a*b*d* e^4*m^3 + 2*a*b*d*e^4*m^4 - 80*a*c*d^2*e^3*m + 60*b*c*d^3*e^2*m - 36*a*c*d ^2*e^3*m^2 - 4*a*c*d^2*e^3*m^3 + 12*b*c*d^3*e^2*m^2 + 120*a*b*d*e^4*m))/(e ^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c^2*x^5*(d + e*x)^m *(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(20*b^2*e^2 + 9*b^2*e^2*m - 4*c^2*d^2*m + b^2*e^2*m^2 + 40*a*c*e^2 + 18*a*c*e^2*m + 2*a*c*e^2*m^2 + 1 0*b*c*d*e*m + 2*b*c*d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^ 5 + 120)) + (x^2*(m + 1)*(d + e*x)^m*(12*c^2*d^3*m + 120*a*b*e^3 + 9*b^2*d *e^2*m^2 + b^2*d*e^2*m^3 + 94*a*b*e^3*m + 24*a*b*e^3*m^2 + 2*a*b*e^3*m^3 + 20*b^2*d*e^2*m + 18*a*c*d*e^2*m^2 + 2*a*c*d*e^2*m^3 - 6*b*c*d^2*e*m^2 + 4 0*a*c*d*e^2*m - 30*b*c*d^2*e*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c*x^4*(d + e*x)^m*(10*b*e + 2*b*e*m + c*d*m)*(11*m + 6*...
Time = 0.18 (sec) , antiderivative size = 1111, normalized size of antiderivative = 6.24 \[ \int (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx =\text {Too large to display} \] Input:
int((e*x+d)^m*(c*x^2+b*x+a)^2,x)
Output:
((d + e*x)**m*(a**2*d*e**4*m**4 + 14*a**2*d*e**4*m**3 + 71*a**2*d*e**4*m** 2 + 154*a**2*d*e**4*m + 120*a**2*d*e**4 + a**2*e**5*m**4*x + 14*a**2*e**5* m**3*x + 71*a**2*e**5*m**2*x + 154*a**2*e**5*m*x + 120*a**2*e**5*x - 2*a*b *d**2*e**3*m**3 - 24*a*b*d**2*e**3*m**2 - 94*a*b*d**2*e**3*m - 120*a*b*d** 2*e**3 + 2*a*b*d*e**4*m**4*x + 24*a*b*d*e**4*m**3*x + 94*a*b*d*e**4*m**2*x + 120*a*b*d*e**4*m*x + 2*a*b*e**5*m**4*x**2 + 26*a*b*e**5*m**3*x**2 + 118 *a*b*e**5*m**2*x**2 + 214*a*b*e**5*m*x**2 + 120*a*b*e**5*x**2 + 4*a*c*d**3 *e**2*m**2 + 36*a*c*d**3*e**2*m + 80*a*c*d**3*e**2 - 4*a*c*d**2*e**3*m**3* x - 36*a*c*d**2*e**3*m**2*x - 80*a*c*d**2*e**3*m*x + 2*a*c*d*e**4*m**4*x** 2 + 20*a*c*d*e**4*m**3*x**2 + 58*a*c*d*e**4*m**2*x**2 + 40*a*c*d*e**4*m*x* *2 + 2*a*c*e**5*m**4*x**3 + 24*a*c*e**5*m**3*x**3 + 98*a*c*e**5*m**2*x**3 + 156*a*c*e**5*m*x**3 + 80*a*c*e**5*x**3 + 2*b**2*d**3*e**2*m**2 + 18*b**2 *d**3*e**2*m + 40*b**2*d**3*e**2 - 2*b**2*d**2*e**3*m**3*x - 18*b**2*d**2* e**3*m**2*x - 40*b**2*d**2*e**3*m*x + b**2*d*e**4*m**4*x**2 + 10*b**2*d*e* *4*m**3*x**2 + 29*b**2*d*e**4*m**2*x**2 + 20*b**2*d*e**4*m*x**2 + b**2*e** 5*m**4*x**3 + 12*b**2*e**5*m**3*x**3 + 49*b**2*e**5*m**2*x**3 + 78*b**2*e* *5*m*x**3 + 40*b**2*e**5*x**3 - 12*b*c*d**4*e*m - 60*b*c*d**4*e + 12*b*c*d **3*e**2*m**2*x + 60*b*c*d**3*e**2*m*x - 6*b*c*d**2*e**3*m**3*x**2 - 36*b* c*d**2*e**3*m**2*x**2 - 30*b*c*d**2*e**3*m*x**2 + 2*b*c*d*e**4*m**4*x**3 + 16*b*c*d*e**4*m**3*x**3 + 34*b*c*d*e**4*m**2*x**3 + 20*b*c*d*e**4*m*x*...