Optimal. Leaf size=17 \[ \frac{\left (a+b x^2\right )^2}{c+d x} \]
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Rubi [A] time = 0.0188685, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {1590} \[ \frac{\left (a+b x^2\right )^2}{c+d x} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) (-a d+b x (4 c+3 d x))}{(c+d x)^2} \, dx &=\frac{\left (a+b x^2\right )^2}{c+d x}\\ \end{align*}
Mathematica [B] time = 0.013127, size = 62, normalized size = 3.65 \[ \frac{a^2 d^4+2 a b d^2 \left (c^2+c d x+d^2 x^2\right )+b^2 \left (c^3 d x+c^4+d^4 x^4\right )}{d^4 (c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 76, normalized size = 4.5 \begin{align*}{\frac{b \left ( b{d}^{2}{x}^{3}-bcd{x}^{2}+2\,a{d}^{2}x+b{c}^{2}x \right ) }{{d}^{3}}}-{\frac{-{a}^{2}{d}^{4}-2\,ab{c}^{2}{d}^{2}-{b}^{2}{c}^{4}}{{d}^{4} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.958837, size = 111, normalized size = 6.53 \begin{align*} \frac{b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{d^{5} x + c d^{4}} + \frac{b^{2} d^{2} x^{3} - b^{2} c d x^{2} +{\left (b^{2} c^{2} + 2 \, a b d^{2}\right )} x}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.67211, size = 153, normalized size = 9. \begin{align*} \frac{b^{2} d^{4} x^{4} + 2 \, a b d^{4} x^{2} + b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4} +{\left (b^{2} c^{3} d + 2 \, a b c d^{3}\right )} x}{d^{5} x + c d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.545774, size = 75, normalized size = 4.41 \begin{align*} - \frac{b^{2} c x^{2}}{d^{2}} + \frac{b^{2} x^{3}}{d} + \frac{a^{2} d^{4} + 2 a b c^{2} d^{2} + b^{2} c^{4}}{c d^{4} + d^{5} x} + \frac{x \left (2 a b d^{2} + b^{2} c^{2}\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16933, size = 150, normalized size = 8.82 \begin{align*} \frac{{\left (b^{2} - \frac{4 \, b^{2} c}{d x + c} + \frac{6 \, b^{2} c^{2}}{{\left (d x + c\right )}^{2}} + \frac{2 \, a b d^{2}}{{\left (d x + c\right )}^{2}}\right )}{\left (d x + c\right )}^{3}}{d^{4}} + \frac{\frac{b^{2} c^{4} d^{3}}{d x + c} + \frac{2 \, a b c^{2} d^{5}}{d x + c} + \frac{a^{2} d^{7}}{d x + c}}{d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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