Optimal. Leaf size=38 \[ \frac{(a+b x)^4 (b c-a d)}{4 b^2}+\frac{d (a+b x)^5}{5 b^2} \]
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Rubi [A] time = 0.0156864, antiderivative size = 38, 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 = {626, 43} \[ \frac{(a+b x)^4 (b c-a d)}{4 b^2}+\frac{d (a+b x)^5}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (a+b x)^2 \left (a c+(b c+a d) x+b d x^2\right ) \, dx &=\int (a+b x)^3 (c+d x) \, dx\\ &=\int \left (\frac{(b c-a d) (a+b x)^3}{b}+\frac{d (a+b x)^4}{b}\right ) \, dx\\ &=\frac{(b c-a d) (a+b x)^4}{4 b^2}+\frac{d (a+b x)^5}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.010389, size = 67, normalized size = 1.76 \[ \frac{1}{2} a^2 x^2 (a d+3 b c)+a^3 c x+\frac{1}{4} b^2 x^4 (3 a d+b c)+a b x^3 (a d+b c)+\frac{1}{5} b^3 d x^5 \]
Antiderivative was successfully verified.
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Maple [B] time = 0.041, size = 94, normalized size = 2.5 \begin{align*}{\frac{{b}^{3}d{x}^{5}}{5}}+{\frac{ \left ( 2\,a{b}^{2}d+{b}^{2} \left ( ad+bc \right ) \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{2}bd+2\,ab \left ( ad+bc \right ) +ac{b}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ({a}^{2} \left ( ad+bc \right ) +2\,{a}^{2}bc \right ){x}^{2}}{2}}+{a}^{3}cx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16033, size = 93, normalized size = 2.45 \begin{align*} \frac{1}{5} \, b^{3} d x^{5} + a^{3} c x + \frac{1}{4} \,{\left (b^{3} c + 3 \, a b^{2} d\right )} x^{4} +{\left (a b^{2} c + a^{2} b d\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b c + a^{3} d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6618, size = 163, normalized size = 4.29 \begin{align*} \frac{1}{5} x^{5} d b^{3} + \frac{1}{4} x^{4} c b^{3} + \frac{3}{4} x^{4} d b^{2} a + x^{3} c b^{2} a + x^{3} d b a^{2} + \frac{3}{2} x^{2} c b a^{2} + \frac{1}{2} x^{2} d a^{3} + x c a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.084755, size = 73, normalized size = 1.92 \begin{align*} a^{3} c x + \frac{b^{3} d x^{5}}{5} + x^{4} \left (\frac{3 a b^{2} d}{4} + \frac{b^{3} c}{4}\right ) + x^{3} \left (a^{2} b d + a b^{2} c\right ) + x^{2} \left (\frac{a^{3} d}{2} + \frac{3 a^{2} b c}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.17151, size = 97, normalized size = 2.55 \begin{align*} \frac{1}{5} \, b^{3} d x^{5} + \frac{1}{4} \, b^{3} c x^{4} + \frac{3}{4} \, a b^{2} d x^{4} + a b^{2} c x^{3} + a^{2} b d x^{3} + \frac{3}{2} \, a^{2} b c x^{2} + \frac{1}{2} \, a^{3} d x^{2} + a^{3} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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