Integrand size = 21, antiderivative size = 83 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx=\frac {a A (e x)^{1+m}}{e (1+m)}+\frac {(A b+a B) (e x)^{2+m}}{e^2 (2+m)}+\frac {(b B+A c) (e x)^{3+m}}{e^3 (3+m)}+\frac {B c (e x)^{4+m}}{e^4 (4+m)} \]
a*A*(e*x)^(1+m)/e/(1+m)+(A*b+B*a)*(e*x)^(2+m)/e^2/(2+m)+(A*c+B*b)*(e*x)^(3 +m)/e^3/(3+m)+B*c*(e*x)^(4+m)/e^4/(4+m)
Time = 0.13 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx=\frac {x (e x)^m \left (a \left (12+7 m+m^2\right ) (A (2+m)+B (1+m) x)+(1+m) x (A (4+m) (b (3+m)+c (2+m) x)+B (2+m) x (b (4+m)+c (3+m) x))\right )}{(1+m) (2+m) (3+m) (4+m)} \]
(x*(e*x)^m*(a*(12 + 7*m + m^2)*(A*(2 + m) + B*(1 + m)*x) + (1 + m)*x*(A*(4 + m)*(b*(3 + m) + c*(2 + m)*x) + B*(2 + m)*x*(b*(4 + m) + c*(3 + m)*x)))) /((1 + m)*(2 + m)*(3 + m)*(4 + m))
Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
| \(\displaystyle \int (A+B x) (e x)^m \left (a+b x+c x^2\right ) \, dx\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle \int \left (\frac {(e x)^{m+1} (a B+A b)}{e}+a A (e x)^m+\frac {(e x)^{m+2} (A c+b B)}{e^2}+\frac {B c (e x)^{m+3}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(e x)^{m+2} (a B+A b)}{e^2 (m+2)}+\frac {a A (e x)^{m+1}}{e (m+1)}+\frac {(e x)^{m+3} (A c+b B)}{e^3 (m+3)}+\frac {B c (e x)^{m+4}}{e^4 (m+4)}\) |
(a*A*(e*x)^(1 + m))/(e*(1 + m)) + ((A*b + a*B)*(e*x)^(2 + m))/(e^2*(2 + m) ) + ((b*B + A*c)*(e*x)^(3 + m))/(e^3*(3 + m)) + (B*c*(e*x)^(4 + m))/(e^4*( 4 + m))
3.11.85.3.1 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]
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99
| method | result | size |
| norman | \(\frac {\left (A b +B a \right ) x^{2} {\mathrm e}^{m \ln \left (e x \right )}}{2+m}+\frac {\left (A c +B b \right ) x^{3} {\mathrm e}^{m \ln \left (e x \right )}}{3+m}+\frac {B c \,x^{4} {\mathrm e}^{m \ln \left (e x \right )}}{4+m}+\frac {a A x \,{\mathrm e}^{m \ln \left (e x \right )}}{1+m}\) | \(82\) |
| gosper | \(\frac {x \left (B c \,m^{3} x^{3}+A c \,m^{3} x^{2}+B b \,m^{3} x^{2}+6 B c \,m^{2} x^{3}+A b \,m^{3} x +7 A c \,m^{2} x^{2}+B a \,m^{3} x +7 B b \,m^{2} x^{2}+11 B c m \,x^{3}+A a \,m^{3}+8 A b \,m^{2} x +14 A c m \,x^{2}+8 B a \,m^{2} x +14 B b m \,x^{2}+6 B c \,x^{3}+9 A a \,m^{2}+19 A b m x +8 A c \,x^{2}+19 B a m x +8 B b \,x^{2}+26 A a m +12 A b x +12 a B x +24 a A \right ) \left (e x \right )^{m}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(205\) |
| risch | \(\frac {x \left (B c \,m^{3} x^{3}+A c \,m^{3} x^{2}+B b \,m^{3} x^{2}+6 B c \,m^{2} x^{3}+A b \,m^{3} x +7 A c \,m^{2} x^{2}+B a \,m^{3} x +7 B b \,m^{2} x^{2}+11 B c m \,x^{3}+A a \,m^{3}+8 A b \,m^{2} x +14 A c m \,x^{2}+8 B a \,m^{2} x +14 B b m \,x^{2}+6 B c \,x^{3}+9 A a \,m^{2}+19 A b m x +8 A c \,x^{2}+19 B a m x +8 B b \,x^{2}+26 A a m +12 A b x +12 a B x +24 a A \right ) \left (e x \right )^{m}}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(205\) |
| parallelrisch | \(\frac {8 B \,x^{3} \left (e x \right )^{m} b +12 A \,x^{2} \left (e x \right )^{m} b +6 B \,x^{4} \left (e x \right )^{m} c +8 A \,x^{3} \left (e x \right )^{m} c +12 B \,x^{2} \left (e x \right )^{m} a +24 A x \left (e x \right )^{m} a +19 A \,x^{2} \left (e x \right )^{m} b m +B \,x^{3} \left (e x \right )^{m} b \,m^{3}+A \,x^{2} \left (e x \right )^{m} b \,m^{3}+7 B \,x^{3} \left (e x \right )^{m} b \,m^{2}+8 A \,x^{2} \left (e x \right )^{m} b \,m^{2}+14 B \,x^{3} \left (e x \right )^{m} b m +B \,x^{4} \left (e x \right )^{m} c \,m^{3}+A \,x^{3} \left (e x \right )^{m} c \,m^{3}+6 B \,x^{4} \left (e x \right )^{m} c \,m^{2}+7 A \,x^{3} \left (e x \right )^{m} c \,m^{2}+11 B \,x^{4} \left (e x \right )^{m} c m +B \,x^{2} \left (e x \right )^{m} a \,m^{3}+14 A \,x^{3} \left (e x \right )^{m} c m +A x \left (e x \right )^{m} a \,m^{3}+8 B \,x^{2} \left (e x \right )^{m} a \,m^{2}+9 A x \left (e x \right )^{m} a \,m^{2}+19 B \,x^{2} \left (e x \right )^{m} a m +26 A x \left (e x \right )^{m} a m}{\left (4+m \right ) \left (3+m \right ) \left (2+m \right ) \left (1+m \right )}\) | \(339\) |
(A*b+B*a)/(2+m)*x^2*exp(m*ln(e*x))+(A*c+B*b)/(3+m)*x^3*exp(m*ln(e*x))+B*c/ (4+m)*x^4*exp(m*ln(e*x))+a*A/(1+m)*x*exp(m*ln(e*x))
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.06 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx=\frac {{\left ({\left (B c m^{3} + 6 \, B c m^{2} + 11 \, B c m + 6 \, B c\right )} x^{4} + {\left ({\left (B b + A c\right )} m^{3} + 7 \, {\left (B b + A c\right )} m^{2} + 8 \, B b + 8 \, A c + 14 \, {\left (B b + A c\right )} m\right )} x^{3} + {\left ({\left (B a + A b\right )} m^{3} + 8 \, {\left (B a + A b\right )} m^{2} + 12 \, B a + 12 \, A b + 19 \, {\left (B a + A b\right )} m\right )} x^{2} + {\left (A a m^{3} + 9 \, A a m^{2} + 26 \, A a m + 24 \, A a\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
((B*c*m^3 + 6*B*c*m^2 + 11*B*c*m + 6*B*c)*x^4 + ((B*b + A*c)*m^3 + 7*(B*b + A*c)*m^2 + 8*B*b + 8*A*c + 14*(B*b + A*c)*m)*x^3 + ((B*a + A*b)*m^3 + 8* (B*a + A*b)*m^2 + 12*B*a + 12*A*b + 19*(B*a + A*b)*m)*x^2 + (A*a*m^3 + 9*A *a*m^2 + 26*A*a*m + 24*A*a)*x)*(e*x)^m/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)
Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (71) = 142\).
Time = 0.33 (sec) , antiderivative size = 981, normalized size of antiderivative = 11.82 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx=\begin {cases} \frac {- \frac {A a}{3 x^{3}} - \frac {A b}{2 x^{2}} - \frac {A c}{x} - \frac {B a}{2 x^{2}} - \frac {B b}{x} + B c \log {\left (x \right )}}{e^{4}} & \text {for}\: m = -4 \\\frac {- \frac {A a}{2 x^{2}} - \frac {A b}{x} + A c \log {\left (x \right )} - \frac {B a}{x} + B b \log {\left (x \right )} + B c x}{e^{3}} & \text {for}\: m = -3 \\\frac {- \frac {A a}{x} + A b \log {\left (x \right )} + A c x + B a \log {\left (x \right )} + B b x + \frac {B c x^{2}}{2}}{e^{2}} & \text {for}\: m = -2 \\\frac {A a \log {\left (x \right )} + A b x + \frac {A c x^{2}}{2} + B a x + \frac {B b x^{2}}{2} + \frac {B c x^{3}}{3}}{e} & \text {for}\: m = -1 \\\frac {A a m^{3} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {9 A a m^{2} x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {26 A a m x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {24 A a x \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {A b m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 A b m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 A b m x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 A b x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {A c m^{3} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {7 A c m^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 A c m x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 A c x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B a m^{3} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 B a m^{2} x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {19 B a m x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {12 B a x^{2} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B b m^{3} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {7 B b m^{2} x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {14 B b m x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {8 B b x^{3} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {B c m^{3} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 B c m^{2} x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {11 B c m x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} + \frac {6 B c x^{4} \left (e x\right )^{m}}{m^{4} + 10 m^{3} + 35 m^{2} + 50 m + 24} & \text {otherwise} \end {cases} \]
Piecewise(((-A*a/(3*x**3) - A*b/(2*x**2) - A*c/x - B*a/(2*x**2) - B*b/x + B*c*log(x))/e**4, Eq(m, -4)), ((-A*a/(2*x**2) - A*b/x + A*c*log(x) - B*a/x + B*b*log(x) + B*c*x)/e**3, Eq(m, -3)), ((-A*a/x + A*b*log(x) + A*c*x + B *a*log(x) + B*b*x + B*c*x**2/2)/e**2, Eq(m, -2)), ((A*a*log(x) + A*b*x + A *c*x**2/2 + B*a*x + B*b*x**2/2 + B*c*x**3/3)/e, Eq(m, -1)), (A*a*m**3*x*(e *x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 9*A*a*m**2*x*(e*x)**m/(m** 4 + 10*m**3 + 35*m**2 + 50*m + 24) + 26*A*a*m*x*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 24*A*a*x*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + A*b*m**3*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8 *A*b*m**2*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 19*A*b*m* x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 12*A*b*x**2*(e*x)** m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + A*c*m**3*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 7*A*c*m**2*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 14*A*c*m*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*A*c*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + B*a*m**3*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 8*B*a*m* *2*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 19*B*a*m*x**2*(e *x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 12*B*a*x**2*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + B*b*m**3*x**3*(e*x)**m/(m**4 + 10*m**3 + 35*m**2 + 50*m + 24) + 7*B*b*m**2*x**3*(e*x)**m/(m**4 + 10*m**3 + 35...
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.25 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx=\frac {B c e^{m} x^{4} x^{m}}{m + 4} + \frac {B b e^{m} x^{3} x^{m}}{m + 3} + \frac {A c e^{m} x^{3} x^{m}}{m + 3} + \frac {B a e^{m} x^{2} x^{m}}{m + 2} + \frac {A b e^{m} x^{2} x^{m}}{m + 2} + \frac {\left (e x\right )^{m + 1} A a}{e {\left (m + 1\right )}} \]
B*c*e^m*x^4*x^m/(m + 4) + B*b*e^m*x^3*x^m/(m + 3) + A*c*e^m*x^3*x^m/(m + 3 ) + B*a*e^m*x^2*x^m/(m + 2) + A*b*e^m*x^2*x^m/(m + 2) + (e*x)^(m + 1)*A*a/ (e*(m + 1))
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (83) = 166\).
Time = 0.28 (sec) , antiderivative size = 338, normalized size of antiderivative = 4.07 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx=\frac {\left (e x\right )^{m} B c m^{3} x^{4} + \left (e x\right )^{m} B b m^{3} x^{3} + \left (e x\right )^{m} A c m^{3} x^{3} + 6 \, \left (e x\right )^{m} B c m^{2} x^{4} + \left (e x\right )^{m} B a m^{3} x^{2} + \left (e x\right )^{m} A b m^{3} x^{2} + 7 \, \left (e x\right )^{m} B b m^{2} x^{3} + 7 \, \left (e x\right )^{m} A c m^{2} x^{3} + 11 \, \left (e x\right )^{m} B c m x^{4} + \left (e x\right )^{m} A a m^{3} x + 8 \, \left (e x\right )^{m} B a m^{2} x^{2} + 8 \, \left (e x\right )^{m} A b m^{2} x^{2} + 14 \, \left (e x\right )^{m} B b m x^{3} + 14 \, \left (e x\right )^{m} A c m x^{3} + 6 \, \left (e x\right )^{m} B c x^{4} + 9 \, \left (e x\right )^{m} A a m^{2} x + 19 \, \left (e x\right )^{m} B a m x^{2} + 19 \, \left (e x\right )^{m} A b m x^{2} + 8 \, \left (e x\right )^{m} B b x^{3} + 8 \, \left (e x\right )^{m} A c x^{3} + 26 \, \left (e x\right )^{m} A a m x + 12 \, \left (e x\right )^{m} B a x^{2} + 12 \, \left (e x\right )^{m} A b x^{2} + 24 \, \left (e x\right )^{m} A a x}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \]
((e*x)^m*B*c*m^3*x^4 + (e*x)^m*B*b*m^3*x^3 + (e*x)^m*A*c*m^3*x^3 + 6*(e*x) ^m*B*c*m^2*x^4 + (e*x)^m*B*a*m^3*x^2 + (e*x)^m*A*b*m^3*x^2 + 7*(e*x)^m*B*b *m^2*x^3 + 7*(e*x)^m*A*c*m^2*x^3 + 11*(e*x)^m*B*c*m*x^4 + (e*x)^m*A*a*m^3* x + 8*(e*x)^m*B*a*m^2*x^2 + 8*(e*x)^m*A*b*m^2*x^2 + 14*(e*x)^m*B*b*m*x^3 + 14*(e*x)^m*A*c*m*x^3 + 6*(e*x)^m*B*c*x^4 + 9*(e*x)^m*A*a*m^2*x + 19*(e*x) ^m*B*a*m*x^2 + 19*(e*x)^m*A*b*m*x^2 + 8*(e*x)^m*B*b*x^3 + 8*(e*x)^m*A*c*x^ 3 + 26*(e*x)^m*A*a*m*x + 12*(e*x)^m*B*a*x^2 + 12*(e*x)^m*A*b*x^2 + 24*(e*x )^m*A*a*x)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)
Time = 10.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.06 \[ \int (e x)^m (A+B x) \left (a+b x+c x^2\right ) \, dx={\left (e\,x\right )}^m\,\left (\frac {x^2\,\left (A\,b+B\,a\right )\,\left (m^3+8\,m^2+19\,m+12\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {x^3\,\left (A\,c+B\,b\right )\,\left (m^3+7\,m^2+14\,m+8\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {A\,a\,x\,\left (m^3+9\,m^2+26\,m+24\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {B\,c\,x^4\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}\right ) \]
(e*x)^m*((x^2*(A*b + B*a)*(19*m + 8*m^2 + m^3 + 12))/(50*m + 35*m^2 + 10*m ^3 + m^4 + 24) + (x^3*(A*c + B*b)*(14*m + 7*m^2 + m^3 + 8))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (A*a*x*(26*m + 9*m^2 + m^3 + 24))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (B*c*x^4*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24))