Integrand size = 18, antiderivative size = 90 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^{1+m}}{e^3 (1+m)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {b B (d+e x)^{3+m}}{e^3 (3+m)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {78} \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\frac {(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac {(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac {b B (d+e x)^{m+3}}{e^3 (m+3)} \]
[In]
[Out]
Rule 78
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b d+a e) (-B d+A e) (d+e x)^m}{e^2}+\frac {(-2 b B d+A b e+a B e) (d+e x)^{1+m}}{e^2}+\frac {b B (d+e x)^{2+m}}{e^2}\right ) \, dx \\ & = \frac {(b d-a e) (B d-A e) (d+e x)^{1+m}}{e^3 (1+m)}-\frac {(2 b B d-A b e-a B e) (d+e x)^{2+m}}{e^3 (2+m)}+\frac {b B (d+e x)^{3+m}}{e^3 (3+m)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.88 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\frac {(d+e x)^{1+m} \left (\frac {(b d-a e) (B d-A e)}{1+m}-\frac {(2 b B d-A b e-a B e) (d+e x)}{2+m}+\frac {b B (d+e x)^2}{3+m}\right )}{e^3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(188\) vs. \(2(90)=180\).
Time = 1.50 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.10
method | result | size |
gosper | \(\frac {\left (e x +d \right )^{1+m} \left (B b \,e^{2} m^{2} x^{2}+A b \,e^{2} m^{2} x +B a \,e^{2} m^{2} x +3 B b \,e^{2} m \,x^{2}+A a \,e^{2} m^{2}+4 A b \,e^{2} m x +4 B a \,e^{2} m x -2 B b d e m x +2 b B \,x^{2} e^{2}+5 A a \,e^{2} m -A b d e m +3 A b \,e^{2} x -B a d e m +3 B a \,e^{2} x -2 B b d e x +6 A a \,e^{2}-3 A b d e -3 B a d e +2 b B \,d^{2}\right )}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(189\) |
norman | \(\frac {b B \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{3+m}+\frac {d \left (A a \,e^{2} m^{2}+5 A a \,e^{2} m -A b d e m -B a d e m +6 A a \,e^{2}-3 A b d e -3 B a d e +2 b B \,d^{2}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{3}+6 m^{2}+11 m +6\right )}+\frac {\left (A b e m +B a e m +B b d m +3 A b e +3 B a e \right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+5 m +6\right )}+\frac {\left (A a \,e^{2} m^{2}+A b d e \,m^{2}+B a d e \,m^{2}+5 A a \,e^{2} m +3 A b d e m +3 B a d e m -2 B b \,d^{2} m +6 A a \,e^{2}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+6 m^{2}+11 m +6\right )}\) | \(253\) |
risch | \(\frac {\left (B b \,e^{3} m^{2} x^{3}+A b \,e^{3} m^{2} x^{2}+B a \,e^{3} m^{2} x^{2}+B b d \,e^{2} m^{2} x^{2}+3 B b \,e^{3} m \,x^{3}+A a \,e^{3} m^{2} x +A b d \,e^{2} m^{2} x +4 A b \,e^{3} m \,x^{2}+B a d \,e^{2} m^{2} x +4 B a \,e^{3} m \,x^{2}+B b d \,e^{2} m \,x^{2}+2 b B \,x^{3} e^{3}+A a d \,e^{2} m^{2}+5 A a \,e^{3} m x +3 A b d \,e^{2} m x +3 A b \,e^{3} x^{2}+3 B a d \,e^{2} m x +3 B a \,e^{3} x^{2}-2 B b \,d^{2} e m x +5 A a d \,e^{2} m +6 A a \,e^{3} x -A b \,d^{2} e m -B a \,d^{2} e m +6 A a d \,e^{2}-3 A b \,d^{2} e -3 B a \,d^{2} e +2 b B \,d^{3}\right ) \left (e x +d \right )^{m}}{\left (2+m \right ) \left (3+m \right ) \left (1+m \right ) e^{3}}\) | \(298\) |
parallelrisch | \(\frac {2 B \,x^{3} \left (e x +d \right )^{m} b d \,e^{3}+3 A \,x^{2} \left (e x +d \right )^{m} b d \,e^{3}+A \left (e x +d \right )^{m} a \,d^{2} e^{2} m^{2}+3 B \,x^{2} \left (e x +d \right )^{m} a d \,e^{3}+6 A x \left (e x +d \right )^{m} a d \,e^{3}+5 A \left (e x +d \right )^{m} a \,d^{2} e^{2} m -A \left (e x +d \right )^{m} b \,d^{3} e m -B \left (e x +d \right )^{m} a \,d^{3} e m +B \,x^{2} \left (e x +d \right )^{m} a d \,e^{3} m^{2}+B \,x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{2} m^{2}+4 A \,x^{2} \left (e x +d \right )^{m} b d \,e^{3} m +A x \left (e x +d \right )^{m} a d \,e^{3} m^{2}+A x \left (e x +d \right )^{m} b \,d^{2} e^{2} m^{2}+4 B \,x^{2} \left (e x +d \right )^{m} a d \,e^{3} m +B \,x^{2} \left (e x +d \right )^{m} b \,d^{2} e^{2} m +B x \left (e x +d \right )^{m} a \,d^{2} e^{2} m^{2}+5 A x \left (e x +d \right )^{m} a d \,e^{3} m +3 A x \left (e x +d \right )^{m} b \,d^{2} e^{2} m +3 B x \left (e x +d \right )^{m} a \,d^{2} e^{2} m -2 B x \left (e x +d \right )^{m} b \,d^{3} e m +2 B \left (e x +d \right )^{m} b \,d^{4}+B \,x^{3} \left (e x +d \right )^{m} b d \,e^{3} m^{2}+A \,x^{2} \left (e x +d \right )^{m} b d \,e^{3} m^{2}+3 B \,x^{3} \left (e x +d \right )^{m} b d \,e^{3} m +6 A \left (e x +d \right )^{m} a \,d^{2} e^{2}-3 A \left (e x +d \right )^{m} b \,d^{3} e -3 B \left (e x +d \right )^{m} a \,d^{3} e}{\left (3+m \right ) \left (2+m \right ) \left (1+m \right ) e^{3} d}\) | \(513\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (90) = 180\).
Time = 0.26 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.84 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\frac {{\left (A a d e^{2} m^{2} + 2 \, B b d^{3} + 6 \, A a d e^{2} - 3 \, {\left (B a + A b\right )} d^{2} e + {\left (B b e^{3} m^{2} + 3 \, B b e^{3} m + 2 \, B b e^{3}\right )} x^{3} + {\left (3 \, {\left (B a + A b\right )} e^{3} + {\left (B b d e^{2} + {\left (B a + A b\right )} e^{3}\right )} m^{2} + {\left (B b d e^{2} + 4 \, {\left (B a + A b\right )} e^{3}\right )} m\right )} x^{2} + {\left (5 \, A a d e^{2} - {\left (B a + A b\right )} d^{2} e\right )} m + {\left (6 \, A a e^{3} + {\left (A a e^{3} + {\left (B a + A b\right )} d e^{2}\right )} m^{2} - {\left (2 \, B b d^{2} e - 5 \, A a e^{3} - 3 \, {\left (B a + A b\right )} d e^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1982 vs. \(2 (78) = 156\).
Time = 0.70 (sec) , antiderivative size = 1982, normalized size of antiderivative = 22.02 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\text {Too large to display} \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.99 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} B b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (90) = 180\).
Time = 0.28 (sec) , antiderivative size = 490, normalized size of antiderivative = 5.44 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx=\frac {{\left (e x + d\right )}^{m} B b e^{3} m^{2} x^{3} + {\left (e x + d\right )}^{m} B b d e^{2} m^{2} x^{2} + {\left (e x + d\right )}^{m} B a e^{3} m^{2} x^{2} + {\left (e x + d\right )}^{m} A b e^{3} m^{2} x^{2} + 3 \, {\left (e x + d\right )}^{m} B b e^{3} m x^{3} + {\left (e x + d\right )}^{m} B a d e^{2} m^{2} x + {\left (e x + d\right )}^{m} A b d e^{2} m^{2} x + {\left (e x + d\right )}^{m} A a e^{3} m^{2} x + {\left (e x + d\right )}^{m} B b d e^{2} m x^{2} + 4 \, {\left (e x + d\right )}^{m} B a e^{3} m x^{2} + 4 \, {\left (e x + d\right )}^{m} A b e^{3} m x^{2} + 2 \, {\left (e x + d\right )}^{m} B b e^{3} x^{3} + {\left (e x + d\right )}^{m} A a d e^{2} m^{2} - 2 \, {\left (e x + d\right )}^{m} B b d^{2} e m x + 3 \, {\left (e x + d\right )}^{m} B a d e^{2} m x + 3 \, {\left (e x + d\right )}^{m} A b d e^{2} m x + 5 \, {\left (e x + d\right )}^{m} A a e^{3} m x + 3 \, {\left (e x + d\right )}^{m} B a e^{3} x^{2} + 3 \, {\left (e x + d\right )}^{m} A b e^{3} x^{2} - {\left (e x + d\right )}^{m} B a d^{2} e m - {\left (e x + d\right )}^{m} A b d^{2} e m + 5 \, {\left (e x + d\right )}^{m} A a d e^{2} m + 6 \, {\left (e x + d\right )}^{m} A a e^{3} x + 2 \, {\left (e x + d\right )}^{m} B b d^{3} - 3 \, {\left (e x + d\right )}^{m} B a d^{2} e - 3 \, {\left (e x + d\right )}^{m} A b d^{2} e + 6 \, {\left (e x + d\right )}^{m} A a d e^{2}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
[In]
[Out]
Time = 3.02 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.88 \[ \int (a+b x) (A+B x) (d+e x)^m \, dx={\left (d+e\,x\right )}^m\,\left (\frac {x\,\left (6\,A\,a\,e^3+5\,A\,a\,e^3\,m+A\,a\,e^3\,m^2+3\,A\,b\,d\,e^2\,m+3\,B\,a\,d\,e^2\,m-2\,B\,b\,d^2\,e\,m+A\,b\,d\,e^2\,m^2+B\,a\,d\,e^2\,m^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,\left (6\,A\,a\,e^2+2\,B\,b\,d^2+5\,A\,a\,e^2\,m+A\,a\,e^2\,m^2-3\,A\,b\,d\,e-3\,B\,a\,d\,e-A\,b\,d\,e\,m-B\,a\,d\,e\,m\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {B\,b\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {x^2\,\left (m+1\right )\,\left (3\,A\,b\,e+3\,B\,a\,e+A\,b\,e\,m+B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \]
[In]
[Out]