Integrand size = 17, antiderivative size = 140 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {\left (c d^2+a e^2\right )^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {4 c d \left (c d^2+a e^2\right ) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {2 c \left (3 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {4 c^2 d (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)} \]
(a*e^2+c*d^2)^2*(e*x+d)^(1+m)/e^5/(1+m)-4*c*d*(a*e^2+c*d^2)*(e*x+d)^(2+m)/ e^5/(2+m)+2*c*(a*e^2+3*c*d^2)*(e*x+d)^(3+m)/e^5/(3+m)-4*c^2*d*(e*x+d)^(4+m )/e^5/(4+m)+c^2*(e*x+d)^(5+m)/e^5/(5+m)
Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.26 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {(d+e x)^{1+m} \left (\left (a+c x^2\right )^2+\frac {4 \left (c d^2+a e^2\right ) \left (a e^2 \left (6+5 m+m^2\right )+c \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )\right )}{e^4 (1+m) (2+m) (3+m)}-\frac {4 c d (d+e x) \left (a e^2 \left (12+7 m+m^2\right )+c \left (2 d^2-2 d e (2+m) x+e^2 \left (6+5 m+m^2\right ) x^2\right )\right )}{e^4 (2+m) (3+m) (4+m)}\right )}{e (5+m)} \]
((d + e*x)^(1 + m)*((a + c*x^2)^2 + (4*(c*d^2 + a*e^2)*(a*e^2*(6 + 5*m + m ^2) + c*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)))/(e^4*(1 + m) *(2 + m)*(3 + m)) - (4*c*d*(d + e*x)*(a*e^2*(12 + 7*m + m^2) + c*(2*d^2 - 2*d*e*(2 + m)*x + e^2*(6 + 5*m + m^2)*x^2)))/(e^4*(2 + m)*(3 + m)*(4 + m)) ))/(e*(5 + m))
Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {476, 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.
| \(\displaystyle \int \left (a+c x^2\right )^2 (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \int \left (\frac {\left (a e^2+c d^2\right )^2 (d+e x)^m}{e^4}-\frac {4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+1}}{e^4}+\frac {2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+2}}{e^4}-\frac {4 c^2 d (d+e x)^{m+3}}{e^4}+\frac {c^2 (d+e x)^{m+4}}{e^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (a e^2+c d^2\right )^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {4 c d \left (a e^2+c d^2\right ) (d+e x)^{m+2}}{e^5 (m+2)}+\frac {2 c \left (a e^2+3 c d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}-\frac {4 c^2 d (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)}\) |
((c*d^2 + a*e^2)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (4*c*d*(c*d^2 + a*e^ 2)*(d + e*x)^(2 + m))/(e^5*(2 + m)) + (2*c*(3*c*d^2 + a*e^2)*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (4*c^2*d*(d + e*x)^(4 + m))/(e^5*(4 + m)) + (c^2*(d + e*x)^(5 + m))/(e^5*(5 + m))
3.8.22.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(140)=280\).
Time = 2.30 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.00
| method | result | size |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (c^{2} e^{4} m^{4} x^{4}+10 c^{2} e^{4} m^{3} x^{4}+2 a c \,e^{4} m^{4} x^{2}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+24 a c \,e^{4} m^{3} x^{2}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}+a^{2} e^{4} m^{4}-4 a c d \,e^{3} m^{3} x +98 a c \,e^{4} m^{2} x^{2}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}+14 a^{2} e^{4} m^{3}-40 a c d \,e^{3} m^{2} x +156 a c \,e^{4} m \,x^{2}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 x^{3} c^{2} d \,e^{3}+71 a^{2} e^{4} m^{2}+4 a c \,d^{2} e^{2} m^{2}-116 a c d \,e^{3} m x +80 x^{2} a c \,e^{4}-24 c^{2} d^{3} e m x +24 x^{2} c^{2} d^{2} e^{2}+154 a^{2} e^{4} m +36 a c \,d^{2} e^{2} m -80 x a c d \,e^{3}-24 x \,c^{2} d^{3} e +120 a^{2} e^{4}+80 a c \,d^{2} e^{2}+24 c^{2} d^{4}\right )}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}\) | \(420\) |
| norman | \(\frac {c^{2} x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{5+m}+\frac {d \left (a^{2} e^{4} m^{4}+14 a^{2} e^{4} m^{3}+71 a^{2} e^{4} m^{2}+4 a c \,d^{2} e^{2} m^{2}+154 a^{2} e^{4} m +36 a c \,d^{2} e^{2} m +120 a^{2} e^{4}+80 a c \,d^{2} e^{2}+24 c^{2} d^{4}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {\left (a^{2} e^{4} m^{4}+14 a^{2} e^{4} m^{3}-4 a c \,d^{2} e^{2} m^{3}+71 a^{2} e^{4} m^{2}-36 a c \,d^{2} e^{2} m^{2}+154 a^{2} e^{4} m -80 a c \,d^{2} e^{2} m -24 c^{2} d^{4} m +120 a^{2} e^{4}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right )}+\frac {c^{2} d m \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+9 m +20\right )}+\frac {2 \left (a \,e^{2} m^{2}+9 a \,e^{2} m -2 c \,d^{2} m +20 e^{2} a \right ) c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+12 m^{2}+47 m +60\right )}+\frac {2 \left (a \,e^{2} m^{2}+9 a \,e^{2} m +20 e^{2} a +6 c \,d^{2}\right ) c d m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right )}\) | \(450\) |
| risch | \(\frac {\left (c^{2} e^{5} m^{4} x^{5}+c^{2} d \,e^{4} m^{4} x^{4}+10 c^{2} e^{5} m^{3} x^{5}+2 a c \,e^{5} m^{4} x^{3}+6 c^{2} d \,e^{4} m^{3} x^{4}+35 c^{2} e^{5} m^{2} x^{5}+2 a c d \,e^{4} m^{4} x^{2}+24 a c \,e^{5} m^{3} x^{3}-4 c^{2} d^{2} e^{3} m^{3} x^{3}+11 c^{2} d \,e^{4} m^{2} x^{4}+50 c^{2} e^{5} m \,x^{5}+a^{2} e^{5} m^{4} x +20 a c d \,e^{4} m^{3} x^{2}+98 a c \,e^{5} m^{2} x^{3}-12 c^{2} d^{2} e^{3} m^{2} x^{3}+6 c^{2} d m \,x^{4} e^{4}+24 c^{2} x^{5} e^{5}+a^{2} d \,e^{4} m^{4}+14 a^{2} e^{5} m^{3} x -4 a c \,d^{2} e^{3} m^{3} x +58 a c d \,e^{4} m^{2} x^{2}+156 a c \,e^{5} m \,x^{3}+12 c^{2} d^{3} e^{2} m^{2} x^{2}-8 c^{2} d^{2} e^{3} m \,x^{3}+14 a^{2} d \,e^{4} m^{3}+71 a^{2} e^{5} m^{2} x -36 a c \,d^{2} e^{3} m^{2} x +40 a c d \,e^{4} m \,x^{2}+80 a c \,e^{5} x^{3}+12 c^{2} d^{3} e^{2} m \,x^{2}+71 a^{2} d \,e^{4} m^{2}+154 a^{2} e^{5} m x +4 a c \,d^{3} e^{2} m^{2}-80 a c \,d^{2} e^{3} m x -24 c^{2} d^{4} e m x +154 a^{2} d \,e^{4} m +120 a^{2} e^{5} x +36 a c \,d^{3} e^{2} m +120 a^{2} d \,e^{4}+80 a \,d^{3} e^{2} c +24 d^{5} c^{2}\right ) \left (e x +d \right )^{m}}{\left (4+m \right ) \left (5+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{5}}\) | \(555\) |
| parallelrisch | \(\frac {2 x^{2} \left (e x +d \right )^{m} a c \,d^{2} e^{4} m^{4}+98 x^{3} \left (e x +d \right )^{m} a c d \,e^{5} m^{2}+20 x^{2} \left (e x +d \right )^{m} a c \,d^{2} e^{4} m^{3}+156 x^{3} \left (e x +d \right )^{m} a c d \,e^{5} m +58 x^{2} \left (e x +d \right )^{m} a c \,d^{2} e^{4} m^{2}-4 x \left (e x +d \right )^{m} a c \,d^{3} e^{3} m^{3}+40 x^{2} \left (e x +d \right )^{m} a c \,d^{2} e^{4} m -36 x \left (e x +d \right )^{m} a c \,d^{3} e^{3} m^{2}-80 x \left (e x +d \right )^{m} a c \,d^{3} e^{3} m +2 x^{3} \left (e x +d \right )^{m} a c d \,e^{5} m^{4}+71 x \left (e x +d \right )^{m} a^{2} d \,e^{5} m^{2}+154 x \left (e x +d \right )^{m} a^{2} d \,e^{5} m -24 x \left (e x +d \right )^{m} c^{2} d^{5} e m +4 \left (e x +d \right )^{m} a c \,d^{4} e^{2} m^{2}+36 \left (e x +d \right )^{m} a c \,d^{4} e^{2} m +24 x^{3} \left (e x +d \right )^{m} a c d \,e^{5} m^{3}+x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5} m^{4}+10 x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5} m^{3}+x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4} m^{4}+35 x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5} m^{2}+6 x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4} m^{3}+50 x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5} m +11 x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4} m^{2}-4 x^{3} \left (e x +d \right )^{m} c^{2} d^{3} e^{3} m^{3}+6 x^{4} \left (e x +d \right )^{m} c^{2} d^{2} e^{4} m -12 x^{3} \left (e x +d \right )^{m} c^{2} d^{3} e^{3} m^{2}+x \left (e x +d \right )^{m} a^{2} d \,e^{5} m^{4}-8 x^{3} \left (e x +d \right )^{m} c^{2} d^{3} e^{3} m +12 x^{2} \left (e x +d \right )^{m} c^{2} d^{4} e^{2} m^{2}+14 x \left (e x +d \right )^{m} a^{2} d \,e^{5} m^{3}+80 x^{3} \left (e x +d \right )^{m} a c d \,e^{5}+12 x^{2} \left (e x +d \right )^{m} c^{2} d^{4} e^{2} m +71 \left (e x +d \right )^{m} a^{2} d^{2} e^{4} m^{2}+120 x \left (e x +d \right )^{m} a^{2} d \,e^{5}+154 \left (e x +d \right )^{m} a^{2} d^{2} e^{4} m +80 \left (e x +d \right )^{m} a c \,d^{4} e^{2}+120 \left (e x +d \right )^{m} a^{2} d^{2} e^{4}+24 \left (e x +d \right )^{m} c^{2} d^{6}+24 x^{5} \left (e x +d \right )^{m} c^{2} d \,e^{5}+\left (e x +d \right )^{m} a^{2} d^{2} e^{4} m^{4}+14 \left (e x +d \right )^{m} a^{2} d^{2} e^{4} m^{3}}{\left (m^{4}+14 m^{3}+71 m^{2}+154 m +120\right ) d \left (1+m \right ) e^{5}}\) | \(879\) |
1/e^5*(e*x+d)^(1+m)/(m^5+15*m^4+85*m^3+225*m^2+274*m+120)*(c^2*e^4*m^4*x^4 +10*c^2*e^4*m^3*x^4+2*a*c*e^4*m^4*x^2-4*c^2*d*e^3*m^3*x^3+35*c^2*e^4*m^2*x ^4+24*a*c*e^4*m^3*x^2-24*c^2*d*e^3*m^2*x^3+50*c^2*e^4*m*x^4+a^2*e^4*m^4-4* a*c*d*e^3*m^3*x+98*a*c*e^4*m^2*x^2+12*c^2*d^2*e^2*m^2*x^2-44*c^2*d*e^3*m*x ^3+24*c^2*e^4*x^4+14*a^2*e^4*m^3-40*a*c*d*e^3*m^2*x+156*a*c*e^4*m*x^2+36*c ^2*d^2*e^2*m*x^2-24*c^2*d*e^3*x^3+71*a^2*e^4*m^2+4*a*c*d^2*e^2*m^2-116*a*c *d*e^3*m*x+80*a*c*e^4*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+154*a^2*e^4* m+36*a*c*d^2*e^2*m-80*a*c*d*e^3*x-24*c^2*d^3*e*x+120*a^2*e^4+80*a*c*d^2*e^ 2+24*c^2*d^4)
Leaf count of result is larger than twice the leaf count of optimal. 520 vs. \(2 (140) = 280\).
Time = 0.31 (sec) , antiderivative size = 520, normalized size of antiderivative = 3.71 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {{\left (a^{2} d e^{4} m^{4} + 14 \, a^{2} d e^{4} m^{3} + 24 \, c^{2} d^{5} + 80 \, a c d^{3} e^{2} + 120 \, a^{2} d e^{4} + {\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} + {\left (c^{2} d e^{4} m^{4} + 6 \, c^{2} d e^{4} m^{3} + 11 \, c^{2} d e^{4} m^{2} + 6 \, c^{2} d e^{4} m\right )} x^{4} + 2 \, {\left (a c e^{5} m^{4} + 40 \, a c e^{5} - 2 \, {\left (c^{2} d^{2} e^{3} - 6 \, a c e^{5}\right )} m^{3} - {\left (6 \, c^{2} d^{2} e^{3} - 49 \, a c e^{5}\right )} m^{2} - 2 \, {\left (2 \, c^{2} d^{2} e^{3} - 39 \, a c e^{5}\right )} m\right )} x^{3} + {\left (4 \, a c d^{3} e^{2} + 71 \, a^{2} d e^{4}\right )} m^{2} + 2 \, {\left (a c d e^{4} m^{4} + 10 \, a c d e^{4} m^{3} + {\left (6 \, c^{2} d^{3} e^{2} + 29 \, a c d e^{4}\right )} m^{2} + 2 \, {\left (3 \, c^{2} d^{3} e^{2} + 10 \, a c d e^{4}\right )} m\right )} x^{2} + 2 \, {\left (18 \, a c d^{3} e^{2} + 77 \, a^{2} d e^{4}\right )} m + {\left (a^{2} e^{5} m^{4} + 120 \, a^{2} e^{5} - 2 \, {\left (2 \, a c d^{2} e^{3} - 7 \, a^{2} e^{5}\right )} m^{3} - {\left (36 \, a c d^{2} e^{3} - 71 \, a^{2} e^{5}\right )} m^{2} - 2 \, {\left (12 \, c^{2} d^{4} e + 40 \, a c d^{2} e^{3} - 77 \, a^{2} e^{5}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \]
(a^2*d*e^4*m^4 + 14*a^2*d*e^4*m^3 + 24*c^2*d^5 + 80*a*c*d^3*e^2 + 120*a^2* d*e^4 + (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 + (c^2*d*e^4*m^4 + 6*c^2*d*e^4*m^3 + 11*c^2*d*e^4*m^2 + 6*c^ 2*d*e^4*m)*x^4 + 2*(a*c*e^5*m^4 + 40*a*c*e^5 - 2*(c^2*d^2*e^3 - 6*a*c*e^5) *m^3 - (6*c^2*d^2*e^3 - 49*a*c*e^5)*m^2 - 2*(2*c^2*d^2*e^3 - 39*a*c*e^5)*m )*x^3 + (4*a*c*d^3*e^2 + 71*a^2*d*e^4)*m^2 + 2*(a*c*d*e^4*m^4 + 10*a*c*d*e ^4*m^3 + (6*c^2*d^3*e^2 + 29*a*c*d*e^4)*m^2 + 2*(3*c^2*d^3*e^2 + 10*a*c*d* e^4)*m)*x^2 + 2*(18*a*c*d^3*e^2 + 77*a^2*d*e^4)*m + (a^2*e^5*m^4 + 120*a^2 *e^5 - 2*(2*a*c*d^2*e^3 - 7*a^2*e^5)*m^3 - (36*a*c*d^2*e^3 - 71*a^2*e^5)*m ^2 - 2*(12*c^2*d^4*e + 40*a*c*d^2*e^3 - 77*a^2*e^5)*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. 5097 vs. \(2 (128) = 256\).
Time = 1.53 (sec) , antiderivative size = 5097, normalized size of antiderivative = 36.41 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx=\text {Too large to display} \]
Piecewise((d**m*(a**2*x + 2*a*c*x**3/3 + 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*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) - 1 2*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) + 12*c**2*d**4*log(d/e + 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) + 25*c**2* d**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) + 48*c**2*d**3*e*x*log(d/e + 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) + 88*c**2*d**3*e* 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) + 72*c**2*d**2*e**2*x**2*log(d/e + 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) + 108*c**2*d** 2*e**2*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) + 48*c**2*d*e**3*x**3*log(d/e + x)/(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) + 48*c* *2*d*e**3*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*e**4*x**4*log(d/e + 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), Eq...
Time = 0.20 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.68 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {{\left (e x + d\right )}^{m + 1} a^{2}}{e {\left (m + 1\right )}} + \frac {2 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} a c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \]
(e*x + d)^(m + 1)*a^2/(e*(m + 1)) + 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) + ((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*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. 851 vs. \(2 (140) = 280\).
Time = 0.31 (sec) , antiderivative size = 851, normalized size of antiderivative = 6.08 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx=\frac {{\left (e x + d\right )}^{m} c^{2} e^{5} m^{4} x^{5} + {\left (e x + d\right )}^{m} c^{2} d e^{4} m^{4} x^{4} + 10 \, {\left (e x + d\right )}^{m} c^{2} e^{5} m^{3} x^{5} + 2 \, {\left (e x + d\right )}^{m} a c e^{5} m^{4} x^{3} + 6 \, {\left (e x + d\right )}^{m} c^{2} d e^{4} m^{3} x^{4} + 35 \, {\left (e x + d\right )}^{m} c^{2} e^{5} m^{2} x^{5} + 2 \, {\left (e x + d\right )}^{m} a c d e^{4} m^{4} x^{2} - 4 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{3} m^{3} x^{3} + 24 \, {\left (e x + d\right )}^{m} a c e^{5} m^{3} x^{3} + 11 \, {\left (e x + d\right )}^{m} c^{2} d e^{4} m^{2} x^{4} + 50 \, {\left (e x + d\right )}^{m} c^{2} e^{5} m x^{5} + {\left (e x + d\right )}^{m} a^{2} e^{5} m^{4} x + 20 \, {\left (e x + d\right )}^{m} a c d e^{4} m^{3} x^{2} - 12 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{3} m^{2} x^{3} + 98 \, {\left (e x + d\right )}^{m} a c e^{5} m^{2} x^{3} + 6 \, {\left (e x + d\right )}^{m} c^{2} d e^{4} m x^{4} + 24 \, {\left (e x + d\right )}^{m} c^{2} e^{5} x^{5} + {\left (e x + d\right )}^{m} a^{2} d e^{4} m^{4} - 4 \, {\left (e x + d\right )}^{m} a c d^{2} e^{3} m^{3} x + 14 \, {\left (e x + d\right )}^{m} a^{2} e^{5} m^{3} x + 12 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{2} m^{2} x^{2} + 58 \, {\left (e x + d\right )}^{m} a c d e^{4} m^{2} x^{2} - 8 \, {\left (e x + d\right )}^{m} c^{2} d^{2} e^{3} m x^{3} + 156 \, {\left (e x + d\right )}^{m} a c e^{5} m x^{3} + 14 \, {\left (e x + d\right )}^{m} a^{2} d e^{4} m^{3} - 36 \, {\left (e x + d\right )}^{m} a c d^{2} e^{3} m^{2} x + 71 \, {\left (e x + d\right )}^{m} a^{2} e^{5} m^{2} x + 12 \, {\left (e x + d\right )}^{m} c^{2} d^{3} e^{2} m x^{2} + 40 \, {\left (e x + d\right )}^{m} a c d e^{4} m x^{2} + 80 \, {\left (e x + d\right )}^{m} a c e^{5} x^{3} + 4 \, {\left (e x + d\right )}^{m} a c d^{3} e^{2} m^{2} + 71 \, {\left (e x + d\right )}^{m} a^{2} d e^{4} m^{2} - 24 \, {\left (e x + d\right )}^{m} c^{2} d^{4} e m x - 80 \, {\left (e x + d\right )}^{m} a c d^{2} e^{3} m x + 154 \, {\left (e x + d\right )}^{m} a^{2} e^{5} m x + 36 \, {\left (e x + d\right )}^{m} a c d^{3} e^{2} m + 154 \, {\left (e x + d\right )}^{m} a^{2} d e^{4} m + 120 \, {\left (e x + d\right )}^{m} a^{2} e^{5} x + 24 \, {\left (e x + d\right )}^{m} c^{2} d^{5} + 80 \, {\left (e x + d\right )}^{m} a c d^{3} e^{2} + 120 \, {\left (e x + d\right )}^{m} a^{2} d e^{4}}{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}} \]
((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 + 10*(e*x + d )^m*c^2*e^5*m^3*x^5 + 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 + 35*(e*x + d)^m*c^2*e^5*m^2*x^5 + 2*(e*x + d)^m*a*c*d*e^4*m^4 *x^2 - 4*(e*x + d)^m*c^2*d^2*e^3*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 + 50*(e*x + d)^m*c^2*e^5*m*x^5 + (e*x + d)^m*a^2*e^5*m^4*x + 20*(e*x + d)^m*a*c*d*e^4*m^3*x^2 - 12*(e*x + d)^m*c^ 2*d^2*e^3*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 + 24*(e*x + d)^m*c^2*e^5*x^5 + (e*x + d)^m*a^2*d*e^4*m^4 - 4*(e*x + d)^m*a*c*d^2*e^3*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 + 58*(e*x + d)^m*a*c*d*e^4*m^2*x^2 - 8*(e*x + d)^m*c^2 *d^2*e^3*m*x^3 + 156*(e*x + d)^m*a*c*e^5*m*x^3 + 14*(e*x + d)^m*a^2*d*e^4* m^3 - 36*(e*x + d)^m*a*c*d^2*e^3*m^2*x + 71*(e*x + d)^m*a^2*e^5*m^2*x + 12 *(e*x + d)^m*c^2*d^3*e^2*m*x^2 + 40*(e*x + d)^m*a*c*d*e^4*m*x^2 + 80*(e*x + d)^m*a*c*e^5*x^3 + 4*(e*x + d)^m*a*c*d^3*e^2*m^2 + 71*(e*x + d)^m*a^2*d* e^4*m^2 - 24*(e*x + d)^m*c^2*d^4*e*m*x - 80*(e*x + d)^m*a*c*d^2*e^3*m*x + 154*(e*x + d)^m*a^2*e^5*m*x + 36*(e*x + d)^m*a*c*d^3*e^2*m + 154*(e*x + d) ^m*a^2*d*e^4*m + 120*(e*x + d)^m*a^2*e^5*x + 24*(e*x + d)^m*c^2*d^5 + 80*( e*x + d)^m*a*c*d^3*e^2 + 120*(e*x + d)^m*a^2*d*e^4)/(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)
Time = 9.79 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.54 \[ \int (d+e x)^m \left (a+c x^2\right )^2 \, dx={\left (d+e\,x\right )}^m\,\left (\frac {c^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {d\,\left (a^2\,e^4\,m^4+14\,a^2\,e^4\,m^3+71\,a^2\,e^4\,m^2+154\,a^2\,e^4\,m+120\,a^2\,e^4+4\,a\,c\,d^2\,e^2\,m^2+36\,a\,c\,d^2\,e^2\,m+80\,a\,c\,d^2\,e^2+24\,c^2\,d^4\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x\,\left (a^2\,e^5\,m^4+14\,a^2\,e^5\,m^3+71\,a^2\,e^5\,m^2+154\,a^2\,e^5\,m+120\,a^2\,e^5-4\,a\,c\,d^2\,e^3\,m^3-36\,a\,c\,d^2\,e^3\,m^2-80\,a\,c\,d^2\,e^3\,m-24\,c^2\,d^4\,e\,m\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,c\,x^3\,\left (m^2+3\,m+2\right )\,\left (-2\,c\,d^2\,m+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c^2\,d\,m\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {2\,c\,d\,m\,x^2\,\left (m+1\right )\,\left (6\,c\,d^2+a\,e^2\,m^2+9\,a\,e^2\,m+20\,a\,e^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \]
(d + e*x)^m*((c^2*x^5*(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) + (d*(120*a^2*e^4 + 24*c^2*d^4 + 154*a^2* e^4*m + 71*a^2*e^4*m^2 + 14*a^2*e^4*m^3 + a^2*e^4*m^4 + 80*a*c*d^2*e^2 + 3 6*a*c*d^2*e^2*m + 4*a*c*d^2*e^2*m^2))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15* m^4 + m^5 + 120)) + (x*(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 - 24*c^2*d^4*e*m - 80*a*c*d^2*e^3*m - 36*a*c*d^2 *e^3*m^2 - 4*a*c*d^2*e^3*m^3))/(e^5*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m ^5 + 120)) + (2*c*x^3*(3*m + m^2 + 2)*(20*a*e^2 + a*e^2*m^2 + 9*a*e^2*m - 2*c*d^2*m))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c^2*d *m*x^4*(11*m + 6*m^2 + m^3 + 6))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m ^5 + 120)) + (2*c*d*m*x^2*(m + 1)*(20*a*e^2 + 6*c*d^2 + a*e^2*m^2 + 9*a*e^ 2*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)))