Optimal. Leaf size=62 \[ \frac{3 x}{8 a^2 \left (a+c x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{x}{4 a \left (a+c x^2\right )^2} \]
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Rubi [A] time = 0.0159544, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {199, 205} \[ \frac{3 x}{8 a^2 \left (a+c x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}+\frac{x}{4 a \left (a+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+c x^2\right )^3} \, dx &=\frac{x}{4 a \left (a+c x^2\right )^2}+\frac{3 \int \frac{1}{\left (a+c x^2\right )^2} \, dx}{4 a}\\ &=\frac{x}{4 a \left (a+c x^2\right )^2}+\frac{3 x}{8 a^2 \left (a+c x^2\right )}+\frac{3 \int \frac{1}{a+c x^2} \, dx}{8 a^2}\\ &=\frac{x}{4 a \left (a+c x^2\right )^2}+\frac{3 x}{8 a^2 \left (a+c x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0380062, size = 55, normalized size = 0.89 \[ \frac{5 a x+3 c x^3}{8 a^2 \left (a+c x^2\right )^2}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 51, normalized size = 0.8 \begin{align*}{\frac{x}{4\,a \left ( c{x}^{2}+a \right ) ^{2}}}+{\frac{3\,x}{8\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{3}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 2.07591, size = 401, normalized size = 6.47 \begin{align*} \left [\frac{6 \, a c^{2} x^{3} + 10 \, a^{2} c x - 3 \,{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{16 \,{\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}, \frac{3 \, a c^{2} x^{3} + 5 \, a^{2} c x + 3 \,{\left (c^{2} x^{4} + 2 \, a c x^{2} + a^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{8 \,{\left (a^{3} c^{3} x^{4} + 2 \, a^{4} c^{2} x^{2} + a^{5} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.502658, size = 105, normalized size = 1.69 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x \right )}}{16} + \frac{5 a x + 3 c x^{3}}{8 a^{4} + 16 a^{3} c x^{2} + 8 a^{2} c^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19606, size = 61, normalized size = 0.98 \begin{align*} \frac{3 \, \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2}} + \frac{3 \, c x^{3} + 5 \, a x}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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