Optimal. Leaf size=73 \[ -\frac{d \log \left (a+c x^2\right )}{2 a^2}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{d \log (x)}{a^2}+\frac{d+e x}{2 a \left (a+c x^2\right )} \]
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Rubi [A] time = 0.0619373, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {823, 801, 635, 205, 260} \[ -\frac{d \log \left (a+c x^2\right )}{2 a^2}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{d \log (x)}{a^2}+\frac{d+e x}{2 a \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 823
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{d+e x}{x \left (a+c x^2\right )^2} \, dx &=\frac{d+e x}{2 a \left (a+c x^2\right )}-\frac{\int \frac{-2 a c d-a c e x}{x \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=\frac{d+e x}{2 a \left (a+c x^2\right )}-\frac{\int \left (-\frac{2 c d}{x}+\frac{c (-a e+2 c d x)}{a+c x^2}\right ) \, dx}{2 a^2 c}\\ &=\frac{d+e x}{2 a \left (a+c x^2\right )}+\frac{d \log (x)}{a^2}-\frac{\int \frac{-a e+2 c d x}{a+c x^2} \, dx}{2 a^2}\\ &=\frac{d+e x}{2 a \left (a+c x^2\right )}+\frac{d \log (x)}{a^2}-\frac{(c d) \int \frac{x}{a+c x^2} \, dx}{a^2}+\frac{e \int \frac{1}{a+c x^2} \, dx}{2 a}\\ &=\frac{d+e x}{2 a \left (a+c x^2\right )}+\frac{e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{c}}+\frac{d \log (x)}{a^2}-\frac{d \log \left (a+c x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.059459, size = 65, normalized size = 0.89 \[ \frac{\frac{a (d+e x)}{a+c x^2}-d \log \left (a+c x^2\right )+\frac{\sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c}}+2 d \log (x)}{2 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 74, normalized size = 1. \begin{align*}{\frac{d\ln \left ( x \right ) }{{a}^{2}}}+{\frac{ex}{2\,a \left ( c{x}^{2}+a \right ) }}+{\frac{d}{2\,a \left ( c{x}^{2}+a \right ) }}-{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{e}{2\,a}\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 = 1.66515, size = 487, normalized size = 6.67 \begin{align*} \left [\frac{2 \, a c e x + 2 \, a c d -{\left (c e x^{2} + a e\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (c^{2} d x^{2} + a c d\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (c^{2} d x^{2} + a c d\right )} \log \left (x\right )}{4 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac{a c e x + a c d +{\left (c e x^{2} + a e\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (c^{2} d x^{2} + a c d\right )} \log \left (c x^{2} + a\right ) + 2 \,{\left (c^{2} d x^{2} + a c d\right )} \log \left (x\right )}{2 \,{\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.76729, size = 359, normalized size = 4.92 \begin{align*} \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) \log{\left (x + \frac{- 96 a^{4} c d \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right )^{2} + 4 a^{3} e^{2} \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) + 48 a^{2} c d^{2} \left (- \frac{d}{2 a^{2}} - \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) - 4 a d e^{2} + 48 c d^{3}}{a e^{3} + 36 c d^{2} e} \right )} + \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) \log{\left (x + \frac{- 96 a^{4} c d \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right )^{2} + 4 a^{3} e^{2} \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) + 48 a^{2} c d^{2} \left (- \frac{d}{2 a^{2}} + \frac{e \sqrt{- a^{5} c}}{4 a^{4} c}\right ) - 4 a d e^{2} + 48 c d^{3}}{a e^{3} + 36 c d^{2} e} \right )} + \frac{d + e x}{2 a^{2} + 2 a c x^{2}} + \frac{d \log{\left (x \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13862, size = 90, normalized size = 1.23 \begin{align*} \frac{\arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} a} - \frac{d \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac{d \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{a x e + a d}{2 \,{\left (c x^{2} + a\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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