Optimal. Leaf size=28 \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]
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Rubi [A] time = 0.0093834, antiderivative size = 28, 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, 37} \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^4}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac{a+b x}{(c+d x)^3} \, dx\\ &=\frac{(a+b x)^2}{2 (b c-a d) (c+d x)^2}\\ \end{align*}
Mathematica [A] time = 0.0087586, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 d^2 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 35, normalized size = 1.3 \begin{align*} -{\frac{ad-bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{b}{{d}^{2} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08441, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62033, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.567457, size = 39, normalized size = 1.39 \begin{align*} - \frac{a d + b c + 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17273, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d x + c\right )}^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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