Optimal. Leaf size=118 \[ \frac{e (a+b x)^5 (-3 a B e+A b e+2 b B d)}{5 b^4}+\frac{(a+b x)^4 (b d-a e) (-3 a B e+2 A b e+b B d)}{4 b^4}+\frac{(a+b x)^3 (A b-a B) (b d-a e)^2}{3 b^4}+\frac{B e^2 (a+b x)^6}{6 b^4} \]
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Rubi [A] time = 0.129242, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {27, 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.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\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 = 0.0536563, 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.
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Maple [A] time = 0., size = 169, normalized size = 1.4 \begin{align*}{\frac{B{e}^{2}{b}^{2}{x}^{6}}{6}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){b}^{2}+2\,B{e}^{2}ab \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){b}^{2}+2\, \left ( A{e}^{2}+2\,Bde \right ) ab+{a}^{2}B{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{2}{b}^{2}+2\, \left ( 2\,Ade+B{d}^{2} \right ) ab+ \left ( A{e}^{2}+2\,Bde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{2}ab+ \left ( 2\,Ade+B{d}^{2} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{2}{a}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19813, size = 227, normalized size = 1.92 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32969, size = 460, normalized size = 3.9 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.090615, size = 202, normalized size = 1.71 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15587, size = 269, normalized size = 2.28 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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