Optimal. Leaf size=58 \[ \frac{x (b c-a d)}{b^2}-\frac{\sqrt{a} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{d x^3}{3 b} \]
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Rubi [A] time = 0.034077, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {459, 321, 205} \[ \frac{x (b c-a d)}{b^2}-\frac{\sqrt{a} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{d x^3}{3 b} \]
Antiderivative was successfully verified.
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Rule 459
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x^2\right )}{a+b x^2} \, dx &=\frac{d x^3}{3 b}-\frac{(-3 b c+3 a d) \int \frac{x^2}{a+b x^2} \, dx}{3 b}\\ &=\frac{(b c-a d) x}{b^2}+\frac{d x^3}{3 b}-\frac{(a (b c-a d)) \int \frac{1}{a+b x^2} \, dx}{b^2}\\ &=\frac{(b c-a d) x}{b^2}+\frac{d x^3}{3 b}-\frac{\sqrt{a} (b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0409964, size = 57, normalized size = 0.98 \[ \frac{x (b c-a d)}{b^2}+\frac{\sqrt{a} (a d-b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{d x^3}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 68, normalized size = 1.2 \begin{align*}{\frac{d{x}^{3}}{3\,b}}-{\frac{adx}{{b}^{2}}}+{\frac{cx}{b}}+{\frac{{a}^{2}d}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{ac}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53942, size = 277, normalized size = 4.78 \begin{align*} \left [\frac{2 \, b d x^{3} - 3 \,{\left (b c - a d\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 6 \,{\left (b c - a d\right )} x}{6 \, b^{2}}, \frac{b d x^{3} - 3 \,{\left (b c - a d\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 3 \,{\left (b c - a d\right )} x}{3 \, b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.444769, size = 90, normalized size = 1.55 \begin{align*} - \frac{\sqrt{- \frac{a}{b^{5}}} \left (a d - b c\right ) \log{\left (- b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{a}{b^{5}}} \left (a d - b c\right ) \log{\left (b^{2} \sqrt{- \frac{a}{b^{5}}} + x \right )}}{2} + \frac{d x^{3}}{3 b} - \frac{x \left (a d - b c\right )}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14221, size = 78, normalized size = 1.34 \begin{align*} -\frac{{\left (a b c - a^{2} d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} d x^{3} + 3 \, b^{2} c x - 3 \, a b d x}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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