Optimal. Leaf size=77 \[ -\frac{(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac{(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac{b B (d+e x)^5}{5 e^3} \]
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Rubi [A] time = 0.0710015, antiderivative size = 77, normalized size of antiderivative = 1., 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} \[ -\frac{(d+e x)^4 (-a B e-A b e+2 b B d)}{4 e^3}+\frac{(d+e x)^3 (b d-a e) (B d-A e)}{3 e^3}+\frac{b B (d+e x)^5}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin{align*} \int (a+b x) (A+B x) (d+e x)^2 \, dx &=\int \left (\frac{(-b d+a e) (-B d+A e) (d+e x)^2}{e^2}+\frac{(-2 b B d+A b e+a B e) (d+e x)^3}{e^2}+\frac{b B (d+e x)^4}{e^2}\right ) \, dx\\ &=\frac{(b d-a e) (B d-A e) (d+e x)^3}{3 e^3}-\frac{(2 b B d-A b e-a B e) (d+e x)^4}{4 e^3}+\frac{b B (d+e x)^5}{5 e^3}\\ \end{align*}
Mathematica [A] time = 0.0328729, size = 96, normalized size = 1.25 \[ \frac{1}{3} x^3 \left (a A e^2+2 a B d e+2 A b d e+b B d^2\right )+\frac{1}{4} e x^4 (a B e+A b e+2 b B d)+\frac{1}{2} d x^2 (2 a A e+a B d+A b d)+a A d^2 x+\frac{1}{5} b B e^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 94, normalized size = 1.2 \begin{align*}{\frac{bB{e}^{2}{x}^{5}}{5}}+{\frac{ \left ( \left ( Ab+Ba \right ){e}^{2}+2\,bBde \right ){x}^{4}}{4}}+{\frac{ \left ( aA{e}^{2}+2\, \left ( Ab+Ba \right ) de+bB{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,aAde+ \left ( Ab+Ba \right ){d}^{2} \right ){x}^{2}}{2}}+aA{d}^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08316, size = 126, normalized size = 1.64 \begin{align*} \frac{1}{5} \, B b e^{2} x^{5} + A a d^{2} x + \frac{1}{4} \,{\left (2 \, B b d e +{\left (B a + A b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (B b d^{2} + A a e^{2} + 2 \,{\left (B a + A b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a d e +{\left (B 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 = 1.49302, size = 277, normalized size = 3.6 \begin{align*} \frac{1}{5} x^{5} e^{2} b B + \frac{1}{2} x^{4} e d b B + \frac{1}{4} x^{4} e^{2} a B + \frac{1}{4} x^{4} e^{2} b A + \frac{1}{3} x^{3} d^{2} b B + \frac{2}{3} x^{3} e d a B + \frac{2}{3} x^{3} e d b A + \frac{1}{3} x^{3} e^{2} a A + \frac{1}{2} x^{2} d^{2} a B + \frac{1}{2} x^{2} d^{2} b A + x^{2} e d a A + x d^{2} a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082934, size = 116, normalized size = 1.51 \begin{align*} A a d^{2} x + \frac{B b e^{2} x^{5}}{5} + x^{4} \left (\frac{A b e^{2}}{4} + \frac{B a e^{2}}{4} + \frac{B b d e}{2}\right ) + x^{3} \left (\frac{A a e^{2}}{3} + \frac{2 A b d e}{3} + \frac{2 B a d e}{3} + \frac{B b d^{2}}{3}\right ) + x^{2} \left (A a d e + \frac{A b d^{2}}{2} + \frac{B a d^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.7954, size = 153, normalized size = 1.99 \begin{align*} \frac{1}{5} \, B b x^{5} e^{2} + \frac{1}{2} \, B b d x^{4} e + \frac{1}{3} \, B b d^{2} x^{3} + \frac{1}{4} \, B a x^{4} e^{2} + \frac{1}{4} \, A b x^{4} e^{2} + \frac{2}{3} \, B a d x^{3} e + \frac{2}{3} \, A b d x^{3} e + \frac{1}{2} \, B a d^{2} x^{2} + \frac{1}{2} \, A b d^{2} x^{2} + \frac{1}{3} \, A a x^{3} e^{2} + A a d x^{2} e + A a d^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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