Optimal. Leaf size=75 \[ \frac {(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac {(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {B e (a+b x)^5}{5 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 75, 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} \[ \frac {(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac {(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac {B e (a+b x)^5}{5 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x) \, dx &=\int \left (\frac {(A b-a B) (b d-a e) (a+b x)^2}{b^2}+\frac {(b B d+A b e-2 a B e) (a+b x)^3}{b^2}+\frac {B e (a+b x)^4}{b^2}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e) (a+b x)^3}{3 b^3}+\frac {(b B d+A b e-2 a B e) (a+b x)^4}{4 b^3}+\frac {B e (a+b x)^5}{5 b^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 96, normalized size = 1.28 \[ \frac {1}{3} x^3 \left (a^2 B e+2 a A b e+2 a b B d+A b^2 d\right )+a^2 A d x+\frac {1}{4} b x^4 (2 a B e+A b e+b B d)+\frac {1}{2} a x^2 (a A e+a B d+2 A b d)+\frac {1}{5} b^2 B e x^5 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.95, size = 113, normalized size = 1.51 \[ \frac {1}{5} x^{5} e b^{2} B + \frac {1}{4} x^{4} d b^{2} B + \frac {1}{2} x^{4} e b a B + \frac {1}{4} x^{4} e b^{2} A + \frac {2}{3} x^{3} d b a B + \frac {1}{3} x^{3} e a^{2} B + \frac {1}{3} x^{3} d b^{2} A + \frac {2}{3} x^{3} e b a A + \frac {1}{2} x^{2} d a^{2} B + x^{2} d b a A + \frac {1}{2} x^{2} e a^{2} A + x d a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.19, size = 119, normalized size = 1.59 \[ \frac {1}{5} \, B b^{2} x^{5} e + \frac {1}{4} \, B b^{2} d x^{4} + \frac {1}{2} \, B a b x^{4} e + \frac {1}{4} \, A b^{2} x^{4} e + \frac {2}{3} \, B a b d x^{3} + \frac {1}{3} \, A b^{2} d x^{3} + \frac {1}{3} \, B a^{2} x^{3} e + \frac {2}{3} \, A a b x^{3} e + \frac {1}{2} \, B a^{2} d x^{2} + A a b d x^{2} + \frac {1}{2} \, A a^{2} x^{2} e + A a^{2} d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 101, normalized size = 1.35 \[ \frac {B \,b^{2} e \,x^{5}}{5}+A \,a^{2} d x +\frac {\left (B \,b^{2} d +\left (A \,b^{2}+2 B a b \right ) e \right ) x^{4}}{4}+\frac {\left (\left (A \,b^{2}+2 B a b \right ) d +\left (2 A a b +B \,a^{2}\right ) e \right ) x^{3}}{3}+\frac {\left (A \,a^{2} e +\left (2 A a b +B \,a^{2}\right ) d \right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 100, normalized size = 1.33 \[ \frac {1}{5} \, B b^{2} e x^{5} + A a^{2} d x + \frac {1}{4} \, {\left (B b^{2} d + {\left (2 \, B a b + A b^{2}\right )} e\right )} x^{4} + \frac {1}{3} \, {\left ({\left (2 \, B a b + A b^{2}\right )} d + {\left (B a^{2} + 2 \, A a b\right )} e\right )} x^{3} + \frac {1}{2} \, {\left (A a^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 98, normalized size = 1.31 \[ x^3\,\left (\frac {A\,b^2\,d}{3}+\frac {B\,a^2\,e}{3}+\frac {2\,A\,a\,b\,e}{3}+\frac {2\,B\,a\,b\,d}{3}\right )+x^2\,\left (\frac {A\,a^2\,e}{2}+\frac {B\,a^2\,d}{2}+A\,a\,b\,d\right )+x^4\,\left (\frac {A\,b^2\,e}{4}+\frac {B\,b^2\,d}{4}+\frac {B\,a\,b\,e}{2}\right )+A\,a^2\,d\,x+\frac {B\,b^2\,e\,x^5}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.08, size = 116, normalized size = 1.55 \[ A a^{2} d x + \frac {B b^{2} e x^{5}}{5} + x^{4} \left (\frac {A b^{2} e}{4} + \frac {B a b e}{2} + \frac {B b^{2} d}{4}\right ) + x^{3} \left (\frac {2 A a b e}{3} + \frac {A b^{2} d}{3} + \frac {B a^{2} e}{3} + \frac {2 B a b d}{3}\right ) + x^{2} \left (\frac {A a^{2} e}{2} + A a b d + \frac {B a^{2} d}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________