Optimal. Leaf size=118 \[ \frac {e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac {(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac {(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac {(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac {(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 77
Rubi steps
\begin {align*} \int (a+b x)^2 (A+B x) (d+e x)^2 \, dx &=\int \left (\frac {(A b-a B) (b d-a e)^2 (a+b x)^2}{b^3}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^3}{b^3}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^4}{b^3}+\frac {B e^2 (a+b x)^5}{b^3}\right ) \, dx\\ &=\frac {(A b-a B) (b d-a e)^2 (a+b x)^3}{3 b^4}+\frac {(b d-a e) (b B d+2 A b e-3 a B e) (a+b x)^4}{4 b^4}+\frac {e (2 b B d+A b e-3 a B e) (a+b x)^5}{5 b^4}+\frac {B e^2 (a+b x)^6}{6 b^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 157, normalized size = 1.33 \[ \frac {1}{3} x^3 \left (A \left (a^2 e^2+4 a b d e+b^2 d^2\right )+2 a B d (a e+b d)\right )+\frac {1}{4} x^4 \left (a^2 B e^2+2 a b e (A e+2 B d)+b^2 d (2 A e+B d)\right )+a^2 A d^2 x+\frac {1}{5} b e x^5 (2 a B e+A b e+2 b B d)+\frac {1}{2} a d x^2 (2 A (a e+b d)+a B d)+\frac {1}{6} b^2 B e^2 x^6 \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.80, size = 199, normalized size = 1.69 \[ \frac {1}{6} x^{6} e^{2} b^{2} B + \frac {2}{5} x^{5} e d b^{2} B + \frac {2}{5} x^{5} e^{2} b a B + \frac {1}{5} x^{5} e^{2} b^{2} A + \frac {1}{4} x^{4} d^{2} b^{2} B + x^{4} e d b a B + \frac {1}{4} x^{4} e^{2} a^{2} B + \frac {1}{2} x^{4} e d b^{2} A + \frac {1}{2} x^{4} e^{2} b a A + \frac {2}{3} x^{3} d^{2} b a B + \frac {2}{3} x^{3} e d a^{2} B + \frac {1}{3} x^{3} d^{2} b^{2} A + \frac {4}{3} x^{3} e d b a A + \frac {1}{3} x^{3} e^{2} a^{2} A + \frac {1}{2} x^{2} d^{2} a^{2} B + x^{2} d^{2} b a A + x^{2} e d a^{2} A + x d^{2} a^{2} A \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.15, size = 199, normalized size = 1.69 \[ \frac {1}{6} \, B b^{2} x^{6} e^{2} + \frac {2}{5} \, B b^{2} d x^{5} e + \frac {1}{4} \, B b^{2} d^{2} x^{4} + \frac {2}{5} \, B a b x^{5} e^{2} + \frac {1}{5} \, A b^{2} x^{5} e^{2} + B a b d x^{4} e + \frac {1}{2} \, A b^{2} d x^{4} e + \frac {2}{3} \, B a b d^{2} x^{3} + \frac {1}{3} \, A b^{2} d^{2} x^{3} + \frac {1}{4} \, B a^{2} x^{4} e^{2} + \frac {1}{2} \, A a b x^{4} e^{2} + \frac {2}{3} \, B a^{2} d x^{3} e + \frac {4}{3} \, A a b d x^{3} e + \frac {1}{2} \, B a^{2} d^{2} x^{2} + A a b d^{2} x^{2} + \frac {1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} d^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 169, normalized size = 1.43 \[ \frac {B \,b^{2} e^{2} x^{6}}{6}+A \,a^{2} d^{2} x +\frac {\left (2 B \,b^{2} d e +\left (A \,b^{2}+2 B a b \right ) e^{2}\right ) x^{5}}{5}+\frac {\left (B \,b^{2} d^{2}+2 \left (A \,b^{2}+2 B a b \right ) d e +\left (2 A a b +B \,a^{2}\right ) e^{2}\right ) x^{4}}{4}+\frac {\left (A \,a^{2} e^{2}+\left (A \,b^{2}+2 B a b \right ) d^{2}+2 \left (2 A a b +B \,a^{2}\right ) d e \right ) x^{3}}{3}+\frac {\left (2 A \,a^{2} d e +\left (2 A a b +B \,a^{2}\right ) d^{2}\right ) x^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.61, size = 168, normalized size = 1.42 \[ \frac {1}{6} \, B b^{2} e^{2} x^{6} + A a^{2} d^{2} x + \frac {1}{5} \, {\left (2 \, B b^{2} d e + {\left (2 \, B a b + A b^{2}\right )} e^{2}\right )} x^{5} + \frac {1}{4} \, {\left (B b^{2} d^{2} + 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} x^{4} + \frac {1}{3} \, {\left (A a^{2} e^{2} + {\left (2 \, B a b + A b^{2}\right )} d^{2} + 2 \, {\left (B a^{2} + 2 \, A a b\right )} d e\right )} x^{3} + \frac {1}{2} \, {\left (2 \, A a^{2} d e + {\left (B a^{2} + 2 \, A a b\right )} d^{2}\right )} x^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.07, size = 157, normalized size = 1.33 \[ x^3\,\left (\frac {2\,B\,a^2\,d\,e}{3}+\frac {A\,a^2\,e^2}{3}+\frac {2\,B\,a\,b\,d^2}{3}+\frac {4\,A\,a\,b\,d\,e}{3}+\frac {A\,b^2\,d^2}{3}\right )+x^4\,\left (\frac {B\,a^2\,e^2}{4}+B\,a\,b\,d\,e+\frac {A\,a\,b\,e^2}{2}+\frac {B\,b^2\,d^2}{4}+\frac {A\,b^2\,d\,e}{2}\right )+\frac {a\,d\,x^2\,\left (2\,A\,a\,e+2\,A\,b\,d+B\,a\,d\right )}{2}+\frac {b\,e\,x^5\,\left (A\,b\,e+2\,B\,a\,e+2\,B\,b\,d\right )}{5}+A\,a^2\,d^2\,x+\frac {B\,b^2\,e^2\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.10, size = 202, normalized size = 1.71 \[ A a^{2} d^{2} x + \frac {B b^{2} e^{2} x^{6}}{6} + x^{5} \left (\frac {A b^{2} e^{2}}{5} + \frac {2 B a b e^{2}}{5} + \frac {2 B b^{2} d e}{5}\right ) + x^{4} \left (\frac {A a b e^{2}}{2} + \frac {A b^{2} d e}{2} + \frac {B a^{2} e^{2}}{4} + B a b d e + \frac {B b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac {A a^{2} e^{2}}{3} + \frac {4 A a b d e}{3} + \frac {A b^{2} d^{2}}{3} + \frac {2 B a^{2} d e}{3} + \frac {2 B a b d^{2}}{3}\right ) + x^{2} \left (A a^{2} d e + A a b d^{2} + \frac {B a^{2} d^{2}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________