Integrand size = 23, antiderivative size = 144 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {\left (c d^2-b d e+a e^2\right ) (e f-d g) (d+e x)^{1+m}}{e^4 (1+m)}-\frac {(c d (2 e f-3 d g)-e (b e f-2 b d g+a e g)) (d+e x)^{2+m}}{e^4 (2+m)}+\frac {(c e f-3 c d g+b e g) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {c g (d+e x)^{4+m}}{e^4 (4+m)} \]
(a*e^2-b*d*e+c*d^2)*(-d*g+e*f)*(e*x+d)^(1+m)/e^4/(1+m)-(c*d*(-3*d*g+2*e*f) -e*(a*e*g-2*b*d*g+b*e*f))*(e*x+d)^(2+m)/e^4/(2+m)+(b*e*g-3*c*d*g+c*e*f)*(e *x+d)^(3+m)/e^4/(3+m)+c*g*(e*x+d)^(4+m)/e^4/(4+m)
Time = 0.32 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left (-\frac {\left (c d^2+e (-b d+a e)\right ) (6 c d g+b e g (1+m)-2 c e f (4+m))}{e^2 (1+m)}+\frac {\left (-b^2 e^2 g (2+m)+2 c^2 d (3 d g-e f (4+m))+c e (b d g (-2+m)+2 a e g (3+m)+b e f (4+m))\right ) (d+e x)}{e^2 (2+m)}+(a+x (b+c x)) (b e g+c (-3 d g+e f (4+m)+e g (3+m) x))\right )}{c e^2 (3+m) (4+m)} \]
((d + e*x)^(1 + m)*(-(((c*d^2 + e*(-(b*d) + a*e))*(6*c*d*g + b*e*g*(1 + m) - 2*c*e*f*(4 + m)))/(e^2*(1 + m))) + ((-(b^2*e^2*g*(2 + m)) + 2*c^2*d*(3* d*g - e*f*(4 + m)) + c*e*(b*d*g*(-2 + m) + 2*a*e*g*(3 + m) + b*e*f*(4 + m) ))*(d + e*x))/(e^2*(2 + m)) + (a + x*(b + c*x))*(b*e*g + c*(-3*d*g + e*f*( 4 + m) + e*g*(3 + m)*x))))/(c*e^2*(3 + m)*(4 + m))
Time = 0.32 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1195, 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.
| \(\displaystyle \int (f+g x) \left (a+b x+c x^2\right ) (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(e f-d g) (d+e x)^m \left (a e^2-b d e+c d^2\right )}{e^3}+\frac {(d+e x)^{m+1} (e (a e g-2 b d g+b e f)-c d (2 e f-3 d g))}{e^3}+\frac {(d+e x)^{m+2} (b e g-3 c d g+c e f)}{e^3}+\frac {c g (d+e x)^{m+3}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac {(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac {(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac {c g (d+e x)^{m+4}}{e^4 (m+4)}\) |
((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*(d + e*x)^(1 + m))/(e^4*(1 + m)) - (( c*d*(2*e*f - 3*d*g) - e*(b*e*f - 2*b*d*g + a*e*g))*(d + e*x)^(2 + m))/(e^4 *(2 + m)) + ((c*e*f - 3*c*d*g + b*e*g)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (c*g*(d + e*x)^(4 + m))/(e^4*(4 + m))
3.10.21.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs. \(2(144)=288\).
Time = 0.49 (sec) , antiderivative size = 503, normalized size of antiderivative = 3.49
| method | result | size |
| gosper | \(-\frac {\left (e x +d \right )^{1+m} \left (-c \,e^{3} g \,m^{3} x^{3}-b \,e^{3} g \,m^{3} x^{2}-c \,e^{3} f \,m^{3} x^{2}-6 c \,e^{3} g \,m^{2} x^{3}-a \,e^{3} g \,m^{3} x -b \,e^{3} f \,m^{3} x -7 b \,e^{3} g \,m^{2} x^{2}+3 c d \,e^{2} g \,m^{2} x^{2}-7 c \,e^{3} f \,m^{2} x^{2}-11 c \,e^{3} g m \,x^{3}-a \,e^{3} f \,m^{3}-8 a \,e^{3} g \,m^{2} x +2 b d \,e^{2} g \,m^{2} x -8 b \,e^{3} f \,m^{2} x -14 b \,e^{3} g m \,x^{2}+2 c d \,e^{2} f \,m^{2} x +9 c d \,e^{2} g m \,x^{2}-14 c \,e^{3} f m \,x^{2}-6 c \,e^{3} g \,x^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}-19 a \,e^{3} g m x +b d \,e^{2} f \,m^{2}+10 b d \,e^{2} g m x -19 b \,e^{3} f m x -8 b \,e^{3} g \,x^{2}-6 c \,d^{2} e g m x +10 c d \,e^{2} f m x +6 c d \,e^{2} g \,x^{2}-8 c \,e^{3} f \,x^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -12 a \,e^{3} g x -2 b \,d^{2} e g m +7 b d \,e^{2} f m +8 b d \,e^{2} g x -12 b \,e^{3} f x -2 c \,d^{2} e f m -6 c \,d^{2} e g x +8 c d \,e^{2} f x +12 a \,e^{2} g d -24 a \,e^{3} f -8 b \,d^{2} e g +12 b \,e^{2} f d +6 c \,d^{3} g -8 c \,d^{2} e f \right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(503\) |
| norman | \(\frac {c g \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {\left (b e g m +c d g m +c e f m +4 b e g +4 c e f \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}+\frac {\left (a \,e^{2} g \,m^{2}+b d e g \,m^{2}+b \,e^{2} f \,m^{2}+c d e f \,m^{2}+7 a \,e^{2} g m +4 b d e g m +7 b \,e^{2} f m -3 c \,d^{2} g m +4 c d e f m +12 a \,e^{2} g +12 b \,e^{2} f \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (a d \,e^{2} g \,m^{3}+a \,e^{3} f \,m^{3}+b d \,e^{2} f \,m^{3}+7 a d \,e^{2} g \,m^{2}+9 a \,e^{3} f \,m^{2}-2 b \,d^{2} e g \,m^{2}+7 b d \,e^{2} f \,m^{2}-2 c \,d^{2} e f \,m^{2}+12 a d \,e^{2} g m +26 a \,e^{3} f m -8 b \,d^{2} e g m +12 b d \,e^{2} f m +6 c \,d^{3} g m -8 c \,d^{2} e f m +24 a \,e^{3} f \right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}-\frac {d \left (-a \,e^{3} f \,m^{3}+a d \,e^{2} g \,m^{2}-9 a \,e^{3} f \,m^{2}+b d \,e^{2} f \,m^{2}+7 a d \,e^{2} g m -26 a \,e^{3} f m -2 b \,d^{2} e g m +7 b d \,e^{2} f m -2 c \,d^{2} e f m +12 a \,e^{2} g d -24 a \,e^{3} f -8 b \,d^{2} e g +12 b \,e^{2} f d +6 c \,d^{3} g -8 c \,d^{2} e f \right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(535\) |
| risch | \(-\frac {\left (-c \,e^{4} g \,m^{3} x^{4}-b \,e^{4} g \,m^{3} x^{3}-c d \,e^{3} g \,m^{3} x^{3}-c \,e^{4} f \,m^{3} x^{3}-6 c \,e^{4} g \,m^{2} x^{4}-a \,e^{4} g \,m^{3} x^{2}-b d \,e^{3} g \,m^{3} x^{2}-b \,e^{4} f \,m^{3} x^{2}-7 b \,e^{4} g \,m^{2} x^{3}-c d \,e^{3} f \,m^{3} x^{2}-3 c d \,e^{3} g \,m^{2} x^{3}-7 c \,e^{4} f \,m^{2} x^{3}-11 c \,e^{4} g m \,x^{4}-a d \,e^{3} g \,m^{3} x -a \,e^{4} f \,m^{3} x -8 a \,e^{4} g \,m^{2} x^{2}-b d \,e^{3} f \,m^{3} x -5 b d \,e^{3} g \,m^{2} x^{2}-8 b \,e^{4} f \,m^{2} x^{2}-14 b \,e^{4} g m \,x^{3}+3 c \,d^{2} e^{2} g \,m^{2} x^{2}-5 c d \,e^{3} f \,m^{2} x^{2}-2 c d \,e^{3} g m \,x^{3}-14 c \,e^{4} f m \,x^{3}-6 c g \,x^{4} e^{4}-a d \,e^{3} f \,m^{3}-7 a d \,e^{3} g \,m^{2} x -9 a \,e^{4} f \,m^{2} x -19 a \,e^{4} g m \,x^{2}+2 b \,d^{2} e^{2} g \,m^{2} x -7 b d \,e^{3} f \,m^{2} x -4 b d \,e^{3} g m \,x^{2}-19 b \,e^{4} f m \,x^{2}-8 b \,e^{4} g \,x^{3}+2 c \,d^{2} e^{2} f \,m^{2} x +3 c \,d^{2} e^{2} g m \,x^{2}-4 c d \,e^{3} f m \,x^{2}-8 c \,e^{4} f \,x^{3}+a \,d^{2} e^{2} g \,m^{2}-9 a d \,e^{3} f \,m^{2}-12 a d \,e^{3} g m x -26 a \,e^{4} f m x -12 a \,e^{4} g \,x^{2}+b \,d^{2} e^{2} f \,m^{2}+8 b \,d^{2} e^{2} g m x -12 b d \,e^{3} f m x -12 b \,e^{4} f \,x^{2}-6 c \,d^{3} e g m x +8 c \,d^{2} e^{2} f m x +7 a \,d^{2} e^{2} g m -26 a d \,e^{3} f m -24 a \,e^{4} f x -2 b \,d^{3} e g m +7 b \,d^{2} e^{2} f m -2 c \,d^{3} e f m +12 a \,d^{2} e^{2} g -24 a d \,e^{3} f -8 b \,d^{3} e g +12 b \,d^{2} e^{2} f +6 c \,d^{4} g -8 c \,d^{3} e f \right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(734\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1143\) |
-1/e^4*(e*x+d)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(-c*e^3*g*m^3*x^3-b*e^3*g *m^3*x^2-c*e^3*f*m^3*x^2-6*c*e^3*g*m^2*x^3-a*e^3*g*m^3*x-b*e^3*f*m^3*x-7*b *e^3*g*m^2*x^2+3*c*d*e^2*g*m^2*x^2-7*c*e^3*f*m^2*x^2-11*c*e^3*g*m*x^3-a*e^ 3*f*m^3-8*a*e^3*g*m^2*x+2*b*d*e^2*g*m^2*x-8*b*e^3*f*m^2*x-14*b*e^3*g*m*x^2 +2*c*d*e^2*f*m^2*x+9*c*d*e^2*g*m*x^2-14*c*e^3*f*m*x^2-6*c*e^3*g*x^3+a*d*e^ 2*g*m^2-9*a*e^3*f*m^2-19*a*e^3*g*m*x+b*d*e^2*f*m^2+10*b*d*e^2*g*m*x-19*b*e ^3*f*m*x-8*b*e^3*g*x^2-6*c*d^2*e*g*m*x+10*c*d*e^2*f*m*x+6*c*d*e^2*g*x^2-8* c*e^3*f*x^2+7*a*d*e^2*g*m-26*a*e^3*f*m-12*a*e^3*g*x-2*b*d^2*e*g*m+7*b*d*e^ 2*f*m+8*b*d*e^2*g*x-12*b*e^3*f*x-2*c*d^2*e*f*m-6*c*d^2*e*g*x+8*c*d*e^2*f*x +12*a*d*e^2*g-24*a*e^3*f-8*b*d^2*e*g+12*b*d*e^2*f+6*c*d^3*g-8*c*d^2*e*f)
Leaf count of result is larger than twice the leaf count of optimal. 613 vs. \(2 (144) = 288\).
Time = 0.30 (sec) , antiderivative size = 613, normalized size of antiderivative = 4.26 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left (a d e^{3} f m^{3} + {\left (c e^{4} g m^{3} + 6 \, c e^{4} g m^{2} + 11 \, c e^{4} g m + 6 \, c e^{4} g\right )} x^{4} + {\left (8 \, c e^{4} f + 8 \, b e^{4} g + {\left (c e^{4} f + {\left (c d e^{3} + b e^{4}\right )} g\right )} m^{3} + {\left (7 \, c e^{4} f + {\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} g\right )} m^{2} + 2 \, {\left (7 \, c e^{4} f + {\left (c d e^{3} + 7 \, b e^{4}\right )} g\right )} m\right )} x^{3} - {\left (a d^{2} e^{2} g + {\left (b d^{2} e^{2} - 9 \, a d e^{3}\right )} f\right )} m^{2} + {\left (12 \, b e^{4} f + 12 \, a e^{4} g + {\left ({\left (c d e^{3} + b e^{4}\right )} f + {\left (b d e^{3} + a e^{4}\right )} g\right )} m^{3} + {\left ({\left (5 \, c d e^{3} + 8 \, b e^{4}\right )} f - {\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} g\right )} m^{2} + {\left ({\left (4 \, c d e^{3} + 19 \, b e^{4}\right )} f - {\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} g\right )} m\right )} x^{2} + 4 \, {\left (2 \, c d^{3} e - 3 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} f - 2 \, {\left (3 \, c d^{4} - 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} g + {\left ({\left (2 \, c d^{3} e - 7 \, b d^{2} e^{2} + 26 \, a d e^{3}\right )} f + {\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} g\right )} m + {\left (24 \, a e^{4} f + {\left (a d e^{3} g + {\left (b d e^{3} + a e^{4}\right )} f\right )} m^{3} - {\left ({\left (2 \, c d^{2} e^{2} - 7 \, b d e^{3} - 9 \, a e^{4}\right )} f + {\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} g\right )} m^{2} - 2 \, {\left ({\left (4 \, c d^{2} e^{2} - 6 \, b d e^{3} - 13 \, a e^{4}\right )} f - {\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} g\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
(a*d*e^3*f*m^3 + (c*e^4*g*m^3 + 6*c*e^4*g*m^2 + 11*c*e^4*g*m + 6*c*e^4*g)* x^4 + (8*c*e^4*f + 8*b*e^4*g + (c*e^4*f + (c*d*e^3 + b*e^4)*g)*m^3 + (7*c* e^4*f + (3*c*d*e^3 + 7*b*e^4)*g)*m^2 + 2*(7*c*e^4*f + (c*d*e^3 + 7*b*e^4)* g)*m)*x^3 - (a*d^2*e^2*g + (b*d^2*e^2 - 9*a*d*e^3)*f)*m^2 + (12*b*e^4*f + 12*a*e^4*g + ((c*d*e^3 + b*e^4)*f + (b*d*e^3 + a*e^4)*g)*m^3 + ((5*c*d*e^3 + 8*b*e^4)*f - (3*c*d^2*e^2 - 5*b*d*e^3 - 8*a*e^4)*g)*m^2 + ((4*c*d*e^3 + 19*b*e^4)*f - (3*c*d^2*e^2 - 4*b*d*e^3 - 19*a*e^4)*g)*m)*x^2 + 4*(2*c*d^3 *e - 3*b*d^2*e^2 + 6*a*d*e^3)*f - 2*(3*c*d^4 - 4*b*d^3*e + 6*a*d^2*e^2)*g + ((2*c*d^3*e - 7*b*d^2*e^2 + 26*a*d*e^3)*f + (2*b*d^3*e - 7*a*d^2*e^2)*g) *m + (24*a*e^4*f + (a*d*e^3*g + (b*d*e^3 + a*e^4)*f)*m^3 - ((2*c*d^2*e^2 - 7*b*d*e^3 - 9*a*e^4)*f + (2*b*d^2*e^2 - 7*a*d*e^3)*g)*m^2 - 2*((4*c*d^2*e ^2 - 6*b*d*e^3 - 13*a*e^4)*f - (3*c*d^3*e - 4*b*d^2*e^2 + 6*a*d*e^3)*g)*m) *x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50*e^4*m + 24*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 5930 vs. \(2 (134) = 268\).
Time = 1.27 (sec) , antiderivative size = 5930, normalized size of antiderivative = 41.18 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]
Piecewise((d**m*(a*f*x + a*g*x**2/2 + b*f*x**2/2 + b*g*x**3/3 + c*f*x**3/3 + c*g*x**4/4), Eq(e, 0)), (-a*d*e**2*g/(6*d**3*e**4 + 18*d**2*e**5*x + 18 *d*e**6*x**2 + 6*e**7*x**3) - 2*a*e**3*f/(6*d**3*e**4 + 18*d**2*e**5*x + 1 8*d*e**6*x**2 + 6*e**7*x**3) - 3*a*e**3*g*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*b*d**2*e*g/(6*d**3*e**4 + 18*d**2*e**5 *x + 18*d*e**6*x**2 + 6*e**7*x**3) - b*d*e**2*f/(6*d**3*e**4 + 18*d**2*e** 5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*b*d*e**2*g*x/(6*d**3*e**4 + 18*d** 2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 3*b*e**3*f*x/(6*d**3*e**4 + 18* d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*b*e**3*g*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*c*d**3*g*log(d/e + x )/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*c*d** 3*g/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*c*d* *2*e*f/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18* c*d**2*e*g*x*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 27*c*d**2*e*g*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6* x**2 + 6*e**7*x**3) - 6*c*d*e**2*f*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d* e**6*x**2 + 6*e**7*x**3) + 18*c*d*e**2*g*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*c*d*e**2*g*x**2/(6*d** 3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*c*e**3*f*x**2/ (6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*c*e**...
Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (144) = 288\).
Time = 0.23 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.44 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} b f}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} a g}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} a f}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} c f}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b g}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} c g}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*b*f/((m^2 + 3*m + 2)*e^2) + (e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*a*g/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*a*f/(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*c*f/((m^3 + 6*m^2 + 11*m + 6)* e^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*b*g/((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*c*g/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^ 4)
Leaf count of result is larger than twice the leaf count of optimal. 1156 vs. \(2 (144) = 288\).
Time = 0.28 (sec) , antiderivative size = 1156, normalized size of antiderivative = 8.03 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\text {Too large to display} \]
((e*x + d)^m*c*e^4*g*m^3*x^4 + (e*x + d)^m*c*e^4*f*m^3*x^3 + (e*x + d)^m*c *d*e^3*g*m^3*x^3 + (e*x + d)^m*b*e^4*g*m^3*x^3 + 6*(e*x + d)^m*c*e^4*g*m^2 *x^4 + (e*x + d)^m*c*d*e^3*f*m^3*x^2 + (e*x + d)^m*b*e^4*f*m^3*x^2 + (e*x + d)^m*b*d*e^3*g*m^3*x^2 + (e*x + d)^m*a*e^4*g*m^3*x^2 + 7*(e*x + d)^m*c*e ^4*f*m^2*x^3 + 3*(e*x + d)^m*c*d*e^3*g*m^2*x^3 + 7*(e*x + d)^m*b*e^4*g*m^2 *x^3 + 11*(e*x + d)^m*c*e^4*g*m*x^4 + (e*x + d)^m*b*d*e^3*f*m^3*x + (e*x + d)^m*a*e^4*f*m^3*x + (e*x + d)^m*a*d*e^3*g*m^3*x + 5*(e*x + d)^m*c*d*e^3* f*m^2*x^2 + 8*(e*x + d)^m*b*e^4*f*m^2*x^2 - 3*(e*x + d)^m*c*d^2*e^2*g*m^2* x^2 + 5*(e*x + d)^m*b*d*e^3*g*m^2*x^2 + 8*(e*x + d)^m*a*e^4*g*m^2*x^2 + 14 *(e*x + d)^m*c*e^4*f*m*x^3 + 2*(e*x + d)^m*c*d*e^3*g*m*x^3 + 14*(e*x + d)^ m*b*e^4*g*m*x^3 + 6*(e*x + d)^m*c*e^4*g*x^4 + (e*x + d)^m*a*d*e^3*f*m^3 - 2*(e*x + d)^m*c*d^2*e^2*f*m^2*x + 7*(e*x + d)^m*b*d*e^3*f*m^2*x + 9*(e*x + d)^m*a*e^4*f*m^2*x - 2*(e*x + d)^m*b*d^2*e^2*g*m^2*x + 7*(e*x + d)^m*a*d* e^3*g*m^2*x + 4*(e*x + d)^m*c*d*e^3*f*m*x^2 + 19*(e*x + d)^m*b*e^4*f*m*x^2 - 3*(e*x + d)^m*c*d^2*e^2*g*m*x^2 + 4*(e*x + d)^m*b*d*e^3*g*m*x^2 + 19*(e *x + d)^m*a*e^4*g*m*x^2 + 8*(e*x + d)^m*c*e^4*f*x^3 + 8*(e*x + d)^m*b*e^4* g*x^3 - (e*x + d)^m*b*d^2*e^2*f*m^2 + 9*(e*x + d)^m*a*d*e^3*f*m^2 - (e*x + d)^m*a*d^2*e^2*g*m^2 - 8*(e*x + d)^m*c*d^2*e^2*f*m*x + 12*(e*x + d)^m*b*d *e^3*f*m*x + 26*(e*x + d)^m*a*e^4*f*m*x + 6*(e*x + d)^m*c*d^3*e*g*m*x - 8* (e*x + d)^m*b*d^2*e^2*g*m*x + 12*(e*x + d)^m*a*d*e^3*g*m*x + 12*(e*x + ...
Time = 13.54 (sec) , antiderivative size = 602, normalized size of antiderivative = 4.18 \[ \int (d+e x)^m (f+g x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (24\,a\,d\,e^3\,f-6\,c\,d^4\,g+8\,b\,d^3\,e\,g+8\,c\,d^3\,e\,f-12\,a\,d^2\,e^2\,g-12\,b\,d^2\,e^2\,f+9\,a\,d\,e^3\,f\,m^2+a\,d\,e^3\,f\,m^3-7\,a\,d^2\,e^2\,g\,m-7\,b\,d^2\,e^2\,f\,m-a\,d^2\,e^2\,g\,m^2-b\,d^2\,e^2\,f\,m^2+26\,a\,d\,e^3\,f\,m+2\,b\,d^3\,e\,g\,m+2\,c\,d^3\,e\,f\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (24\,a\,e^4\,f+26\,a\,e^4\,f\,m+9\,a\,e^4\,f\,m^2+a\,e^4\,f\,m^3+7\,a\,d\,e^3\,g\,m^2+7\,b\,d\,e^3\,f\,m^2+a\,d\,e^3\,g\,m^3+b\,d\,e^3\,f\,m^3-8\,b\,d^2\,e^2\,g\,m-8\,c\,d^2\,e^2\,f\,m-2\,b\,d^2\,e^2\,g\,m^2-2\,c\,d^2\,e^2\,f\,m^2+12\,a\,d\,e^3\,g\,m+12\,b\,d\,e^3\,f\,m+6\,c\,d^3\,e\,g\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (12\,a\,e^2\,g+12\,b\,e^2\,f+7\,a\,e^2\,g\,m+7\,b\,e^2\,f\,m-3\,c\,d^2\,g\,m+a\,e^2\,g\,m^2+b\,e^2\,f\,m^2+4\,b\,d\,e\,g\,m+4\,c\,d\,e\,f\,m+b\,d\,e\,g\,m^2+c\,d\,e\,f\,m^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {c\,g\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,b\,e\,g+4\,c\,e\,f+b\,e\,g\,m+c\,d\,g\,m+c\,e\,f\,m\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]
((d + e*x)^m*(24*a*d*e^3*f - 6*c*d^4*g + 8*b*d^3*e*g + 8*c*d^3*e*f - 12*a* d^2*e^2*g - 12*b*d^2*e^2*f + 9*a*d*e^3*f*m^2 + a*d*e^3*f*m^3 - 7*a*d^2*e^2 *g*m - 7*b*d^2*e^2*f*m - a*d^2*e^2*g*m^2 - b*d^2*e^2*f*m^2 + 26*a*d*e^3*f* m + 2*b*d^3*e*g*m + 2*c*d^3*e*f*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 2 4)) + (x*(d + e*x)^m*(24*a*e^4*f + 26*a*e^4*f*m + 9*a*e^4*f*m^2 + a*e^4*f* m^3 + 7*a*d*e^3*g*m^2 + 7*b*d*e^3*f*m^2 + a*d*e^3*g*m^3 + b*d*e^3*f*m^3 - 8*b*d^2*e^2*g*m - 8*c*d^2*e^2*f*m - 2*b*d^2*e^2*g*m^2 - 2*c*d^2*e^2*f*m^2 + 12*a*d*e^3*g*m + 12*b*d*e^3*f*m + 6*c*d^3*e*g*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x^2*(m + 1)*(d + e*x)^m*(12*a*e^2*g + 12*b*e^2*f + 7*a*e^2*g*m + 7*b*e^2*f*m - 3*c*d^2*g*m + a*e^2*g*m^2 + b*e^2*f*m^2 + 4*b* d*e*g*m + 4*c*d*e*f*m + b*d*e*g*m^2 + c*d*e*f*m^2))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (c*g*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (x^3*(d + e*x)^m*(3*m + m^2 + 2)*(4*b*e*g + 4*c*e*f + b*e*g*m + c*d*g*m + c*e*f*m))/(e*(50*m + 35*m^2 + 10*m^3 + m^ 4 + 24))