Optimal. Leaf size=38 \[ \frac{(a+b x)^3 (b d-a e)}{3 b^2}+\frac{e (a+b x)^4}{4 b^2} \]
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Rubi [A] time = 0.0297702, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {27, 43} \[ \frac{(a+b x)^3 (b d-a e)}{3 b^2}+\frac{e (a+b x)^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x) \, dx\\ &=\int \left (\frac{(b d-a e) (a+b x)^2}{b}+\frac{e (a+b x)^3}{b}\right ) \, dx\\ &=\frac{(b d-a e) (a+b x)^3}{3 b^2}+\frac{e (a+b x)^4}{4 b^2}\\ \end{align*}
Mathematica [A] time = 0.0098354, size = 46, normalized size = 1.21 \[ \frac{1}{12} x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 49, normalized size = 1.3 \begin{align*}{\frac{{b}^{2}e{x}^{4}}{4}}+{\frac{ \left ( 2\,aeb+{b}^{2}d \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2}e+2\,abd \right ){x}^{2}}{2}}+{a}^{2}dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18585, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac{1}{3} \,{\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69772, size = 115, normalized size = 3.03 \begin{align*} \frac{1}{4} x^{4} e b^{2} + \frac{1}{3} x^{3} d b^{2} + \frac{2}{3} x^{3} e b a + x^{2} d b a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.074636, size = 49, normalized size = 1.29 \begin{align*} a^{2} d x + \frac{b^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a b e}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16536, size = 70, normalized size = 1.84 \begin{align*} \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{3} \, b^{2} d x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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