Optimal. Leaf size=14 \[ \frac{(c+d x)^3}{3 d} \]
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Rubi [A] time = 0.0099224, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {626, 32} \[ \frac{(c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 32
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^2} \, dx &=\int (c+d x)^2 \, dx\\ &=\frac{(c+d x)^3}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0011442, size = 14, normalized size = 1. \[ \frac{(c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 13, normalized size = 0.9 \begin{align*}{\frac{ \left ( dx+c \right ) ^{3}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04203, size = 27, normalized size = 1.93 \begin{align*} \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66728, size = 42, normalized size = 3. \begin{align*} \frac{1}{3} \, d^{2} x^{3} + c d x^{2} + c^{2} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.189729, size = 19, normalized size = 1.36 \begin{align*} c^{2} x + c d x^{2} + \frac{d^{2} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22177, size = 113, normalized size = 8.07 \begin{align*} \frac{{\left (\frac{3 \, b^{2} c^{2}}{{\left (b x + a\right )}^{2}} + \frac{3 \, b c d}{b x + a} - \frac{6 \, a b c d}{{\left (b x + a\right )}^{2}} - \frac{3 \, a d^{2}}{b x + a} + \frac{3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{2}} + d^{2}\right )}{\left (b x + a\right )}^{3}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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