Optimal. Leaf size=75 \[ \frac{(a+b x)^4 (-2 a B e+A b e+b B d)}{4 b^3}+\frac{(a+b x)^3 (A b-a B) (b d-a e)}{3 b^3}+\frac{B e (a+b x)^5}{5 b^3} \]
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Rubi [A] time = 0.0759608, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 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.
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Rule 27
Rule 77
Rubi steps
\begin{align*} \int (A+B x) (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\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 = 0.0282037, 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.
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Maple [A] time = 0.001, size = 94, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}Be{x}^{5}}{5}}+{\frac{ \left ({b}^{2} \left ( Ae+Bd \right ) +2\,Beab \right ){x}^{4}}{4}}+{\frac{ \left ( Ad{b}^{2}+2\, \left ( Ae+Bd \right ) ab+Be{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Adab+ \left ( Ae+Bd \right ){a}^{2} \right ){x}^{2}}{2}}+{a}^{2}Adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.35214, size = 135, normalized size = 1.8 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33083, size = 277, normalized size = 3.69 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082914, size = 116, normalized size = 1.55 \begin{align*} 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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18676, size = 161, normalized size = 2.15 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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