Optimal. Leaf size=90 \[ \frac {(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac {(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac {b B (d+e x)^{m+3}}{e^3 (m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ \frac {(b d-a e) (B d-A e) (d+e x)^{m+1}}{e^3 (m+1)}-\frac {(d+e x)^{m+2} (-a B e-A b e+2 b B d)}{e^3 (m+2)}+\frac {b B (d+e x)^{m+3}}{e^3 (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int (a+b x) (A+B x) (d+e x)^m \, dx &=\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*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 79, normalized size = 0.88 \[ \frac {(d+e x)^{m+1} \left (-\frac {(d+e x) (-a B e-A b e+2 b B d)}{m+2}+\frac {(b d-a e) (B d-A e)}{m+1}+\frac {b B (d+e x)^2}{m+3}\right )}{e^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.83, size = 256, normalized size = 2.84 \[ \frac {{\left (A a d e^{2} m^{2} + 2 \, B b d^{3} + 6 \, A a d e^{2} - 3 \, {\left (B a + A b\right )} d^{2} e + {\left (B b e^{3} m^{2} + 3 \, B b e^{3} m + 2 \, B b e^{3}\right )} x^{3} + {\left (3 \, {\left (B a + A b\right )} e^{3} + {\left (B b d e^{2} + {\left (B a + A b\right )} e^{3}\right )} m^{2} + {\left (B b d e^{2} + 4 \, {\left (B a + A b\right )} e^{3}\right )} m\right )} x^{2} + {\left (5 \, A a d e^{2} - {\left (B a + A b\right )} d^{2} e\right )} m + {\left (6 \, A a e^{3} + {\left (A a e^{3} + {\left (B a + A b\right )} d e^{2}\right )} m^{2} - {\left (2 \, B b d^{2} e - 5 \, A a e^{3} - 3 \, {\left (B a + A b\right )} d e^{2}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{3} m^{3} + 6 \, e^{3} m^{2} + 11 \, e^{3} m + 6 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.00, size = 497, normalized size = 5.52 \[ \frac {{\left (x e + d\right )}^{m} B b m^{2} x^{3} e^{3} + {\left (x e + d\right )}^{m} B b d m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} B a m^{2} x^{2} e^{3} + {\left (x e + d\right )}^{m} A b m^{2} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} B b m x^{3} e^{3} + {\left (x e + d\right )}^{m} B a d m^{2} x e^{2} + {\left (x e + d\right )}^{m} A b d m^{2} x e^{2} + {\left (x e + d\right )}^{m} B b d m x^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} B b d^{2} m x e + {\left (x e + d\right )}^{m} A a m^{2} x e^{3} + 4 \, {\left (x e + d\right )}^{m} B a m x^{2} e^{3} + 4 \, {\left (x e + d\right )}^{m} A b m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} B b x^{3} e^{3} + {\left (x e + d\right )}^{m} A a d m^{2} e^{2} + 3 \, {\left (x e + d\right )}^{m} B a d m x e^{2} + 3 \, {\left (x e + d\right )}^{m} A b d m x e^{2} - {\left (x e + d\right )}^{m} B a d^{2} m e - {\left (x e + d\right )}^{m} A b d^{2} m e + 2 \, {\left (x e + d\right )}^{m} B b d^{3} + 5 \, {\left (x e + d\right )}^{m} A a m x e^{3} + 3 \, {\left (x e + d\right )}^{m} B a x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} A b x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} A a d m e^{2} - 3 \, {\left (x e + d\right )}^{m} B a d^{2} e - 3 \, {\left (x e + d\right )}^{m} A b d^{2} e + 6 \, {\left (x e + d\right )}^{m} A a x e^{3} + 6 \, {\left (x e + d\right )}^{m} A a d e^{2}}{m^{3} e^{3} + 6 \, m^{2} e^{3} + 11 \, m e^{3} + 6 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.01, size = 189, normalized size = 2.10 \[ \frac {\left (B b \,e^{2} m^{2} x^{2}+A b \,e^{2} m^{2} x +B a \,e^{2} m^{2} x +3 B b \,e^{2} m \,x^{2}+A a \,e^{2} m^{2}+4 A b \,e^{2} m x +4 B a \,e^{2} m x -2 B b d e m x +2 B b \,x^{2} e^{2}+5 A a \,e^{2} m -A b d e m +3 A b \,e^{2} x -B a d e m +3 B a \,e^{2} x -2 B b d e x +6 A a \,e^{2}-3 A b d e -3 B a d e +2 B b \,d^{2}\right ) \left (e x +d \right )^{m +1}}{\left (m^{3}+6 m^{2}+11 m +6\right ) e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.50, size = 179, normalized size = 1.99 \[ \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} A b}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} B b}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.56, size = 259, normalized size = 2.88 \[ {\left (d+e\,x\right )}^m\,\left (\frac {x\,\left (6\,A\,a\,e^3+5\,A\,a\,e^3\,m+A\,a\,e^3\,m^2+3\,A\,b\,d\,e^2\,m+3\,B\,a\,d\,e^2\,m-2\,B\,b\,d^2\,e\,m+A\,b\,d\,e^2\,m^2+B\,a\,d\,e^2\,m^2\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {d\,\left (6\,A\,a\,e^2+2\,B\,b\,d^2+5\,A\,a\,e^2\,m+A\,a\,e^2\,m^2-3\,A\,b\,d\,e-3\,B\,a\,d\,e-A\,b\,d\,e\,m-B\,a\,d\,e\,m\right )}{e^3\,\left (m^3+6\,m^2+11\,m+6\right )}+\frac {B\,b\,x^3\,\left (m^2+3\,m+2\right )}{m^3+6\,m^2+11\,m+6}+\frac {x^2\,\left (m+1\right )\,\left (3\,A\,b\,e+3\,B\,a\,e+A\,b\,e\,m+B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^3+6\,m^2+11\,m+6\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.79, size = 1982, normalized size = 22.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________