Optimal. Leaf size=17 \[ \frac{\left (a+b x^2\right )^3}{c+d x} \]
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Rubi [A] time = 0.0459761, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {1590} \[ \frac{\left (a+b x^2\right )^3}{c+d x} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (-a d+6 b c x+5 b d x^2\right )}{(c+d x)^2} \, dx &=\frac{\left (a+b x^2\right )^3}{c+d x}\\ \end{align*}
Mathematica [B] time = 0.0385234, size = 90, normalized size = 5.29 \[ \frac{3 a^2 b d^4 \left (c^2+c d x+d^2 x^2\right )+a^3 d^6+3 a b^2 d^2 \left (c^3 d x+c^4+d^4 x^4\right )+b^3 \left (c^5 d x+c^6+d^6 x^6\right )}{d^6 (c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 157, normalized size = 9.2 \begin{align*}{\frac{b \left ({b}^{2}{d}^{4}{x}^{5}-{b}^{2}c{d}^{3}{x}^{4}+3\,ab{d}^{4}{x}^{3}+{b}^{2}{c}^{2}{d}^{2}{x}^{3}-3\,abc{d}^{3}{x}^{2}-{b}^{2}{c}^{3}d{x}^{2}+3\,{a}^{2}{d}^{4}x+3\,ab{c}^{2}{d}^{2}x+{b}^{2}{c}^{4}x \right ) }{{d}^{5}}}-{\frac{-{a}^{3}{d}^{6}-3\,{a}^{2}b{c}^{2}{d}^{4}-3\,a{b}^{2}{c}^{4}{d}^{2}-{b}^{3}{c}^{6}}{{d}^{6} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00438, size = 216, normalized size = 12.71 \begin{align*} \frac{b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6}}{d^{7} x + c d^{6}} + \frac{b^{3} d^{4} x^{5} - b^{3} c d^{3} x^{4} +{\left (b^{3} c^{2} d^{2} + 3 \, a b^{2} d^{4}\right )} x^{3} -{\left (b^{3} c^{3} d + 3 \, a b^{2} c d^{3}\right )} x^{2} +{\left (b^{3} c^{4} + 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b d^{4}\right )} x}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68655, size = 234, normalized size = 13.76 \begin{align*} \frac{b^{3} d^{6} x^{6} + 3 \, a b^{2} d^{6} x^{4} + 3 \, a^{2} b d^{6} x^{2} + b^{3} c^{6} + 3 \, a b^{2} c^{4} d^{2} + 3 \, a^{2} b c^{2} d^{4} + a^{3} d^{6} +{\left (b^{3} c^{5} d + 3 \, a b^{2} c^{3} d^{3} + 3 \, a^{2} b c d^{5}\right )} x}{d^{7} x + c d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.777898, size = 155, normalized size = 9.12 \begin{align*} - \frac{b^{3} c x^{4}}{d^{2}} + \frac{b^{3} x^{5}}{d} + \frac{a^{3} d^{6} + 3 a^{2} b c^{2} d^{4} + 3 a b^{2} c^{4} d^{2} + b^{3} c^{6}}{c d^{6} + d^{7} x} + \frac{x^{3} \left (3 a b^{2} d^{2} + b^{3} c^{2}\right )}{d^{3}} - \frac{x^{2} \left (3 a b^{2} c d^{2} + b^{3} c^{3}\right )}{d^{4}} + \frac{x \left (3 a^{2} b d^{4} + 3 a b^{2} c^{2} d^{2} + b^{3} c^{4}\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16157, size = 292, normalized size = 17.18 \begin{align*} \frac{{\left (b^{3} - \frac{6 \, b^{3} c}{d x + c} + \frac{15 \, b^{3} c^{2}}{{\left (d x + c\right )}^{2}} - \frac{20 \, b^{3} c^{3}}{{\left (d x + c\right )}^{3}} + \frac{15 \, b^{3} c^{4}}{{\left (d x + c\right )}^{4}} + \frac{3 \, a b^{2} d^{2}}{{\left (d x + c\right )}^{2}} - \frac{12 \, a b^{2} c d^{2}}{{\left (d x + c\right )}^{3}} + \frac{18 \, a b^{2} c^{2} d^{2}}{{\left (d x + c\right )}^{4}} + \frac{3 \, a^{2} b d^{4}}{{\left (d x + c\right )}^{4}}\right )}{\left (d x + c\right )}^{5}}{d^{6}} + \frac{\frac{b^{3} c^{6} d^{5}}{d x + c} + \frac{3 \, a b^{2} c^{4} d^{7}}{d x + c} + \frac{3 \, a^{2} b c^{2} d^{9}}{d x + c} + \frac{a^{3} d^{11}}{d x + c}}{d^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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