Optimal. Leaf size=38 \[ \frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2} \]
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Rubi [A] time = 0.034688, antiderivative size = 38, 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 = {626, 43} \[ \frac{b (c+d x)^4}{4 d^2}-\frac{(c+d x)^3 (b c-a d)}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{a+b x} \, dx &=\int (a+b x) (c+d x)^2 \, dx\\ &=\int \left (\frac{(-b c+a d) (c+d x)^2}{d}+\frac{b (c+d x)^3}{d}\right ) \, dx\\ &=-\frac{(b c-a d) (c+d x)^3}{3 d^2}+\frac{b (c+d x)^4}{4 d^2}\\ \end{align*}
Mathematica [A] time = 0.009309, size = 47, normalized size = 1.24 \[ \frac{1}{12} x \left (4 d x^2 (a d+2 b c)+6 c x (2 a d+b c)+12 a c^2+3 b d^2 x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 55, normalized size = 1.5 \begin{align*}{\frac{b{d}^{2}{x}^{4}}{4}}+{\frac{ \left ( bcd+d \left ( ad+bc \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( c \left ( ad+bc \right ) +acd \right ){x}^{2}}{2}}+xa{c}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05459, size = 65, normalized size = 1.71 \begin{align*} \frac{1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac{1}{3} \,{\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{2} + 2 \, a c d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.79481, size = 109, normalized size = 2.87 \begin{align*} \frac{1}{4} \, b d^{2} x^{4} + a c^{2} x + \frac{1}{3} \,{\left (2 \, b c d + a d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (b c^{2} + 2 \, a c d\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128249, size = 49, normalized size = 1.29 \begin{align*} a c^{2} x + \frac{b d^{2} x^{4}}{4} + x^{3} \left (\frac{a d^{2}}{3} + \frac{2 b c d}{3}\right ) + x^{2} \left (a c d + \frac{b c^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20942, size = 66, normalized size = 1.74 \begin{align*} \frac{1}{4} \, b d^{2} x^{4} + \frac{2}{3} \, b c d x^{3} + \frac{1}{3} \, a d^{2} x^{3} + \frac{1}{2} \, b c^{2} x^{2} + a c d x^{2} + a c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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