Optimal. Leaf size=94 \[ \frac{a e+c d x}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]
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Rubi [A] time = 0.0461382, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {741, 12, 725, 206} \[ \frac{a e+c d x}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 741
Rule 12
Rule 725
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(d+e x) \left (a+c x^2\right )^{3/2}} \, dx &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{\int \frac{a e^2}{(d+e x) \sqrt{a+c x^2}} \, dx}{a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}+\frac{e^2 \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{c d^2+a e^2}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}-\frac{e^2 \operatorname{Subst}\left (\int \frac{1}{c d^2+a e^2-x^2} \, dx,x,\frac{a e-c d x}{\sqrt{a+c x^2}}\right )}{c d^2+a e^2}\\ &=\frac{a e+c d x}{a \left (c d^2+a e^2\right ) \sqrt{a+c x^2}}-\frac{e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{\left (c d^2+a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0501925, size = 94, normalized size = 1. \[ \frac{a e+c d x}{a \sqrt{a+c x^2} \left (a e^2+c d^2\right )}-\frac{e^2 \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{\left (a e^2+c d^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.225, size = 260, normalized size = 2.8 \begin{align*}{\frac{e}{a{e}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}+{\frac{cdx}{ \left ( a{e}^{2}+c{d}^{2} \right ) a}{\frac{1}{\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{e}{a{e}^{2}+c{d}^{2}}\ln \left ({ \left ( 2\,{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ({\frac{d}{e}}+x \right ) ^{2}c-2\,{\frac{cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}+c{d}^{2}}{{e}^{2}}}}}}} \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 [B] time = 2.67039, size = 913, normalized size = 9.71 \begin{align*} \left [\frac{{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt{c d^{2} + a e^{2}} \log \left (\frac{2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} -{\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} - 2 \, \sqrt{c d^{2} + a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{2 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )}}, -\frac{{\left (a c e^{2} x^{2} + a^{2} e^{2}\right )} \sqrt{-c d^{2} - a e^{2}} \arctan \left (\frac{\sqrt{-c d^{2} - a e^{2}}{\left (c d x - a e\right )} \sqrt{c x^{2} + a}}{a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) -{\left (a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + c x^{2}\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17332, size = 232, normalized size = 2.47 \begin{align*} \frac{\frac{{\left (c^{2} d^{3} + a c d e^{2}\right )} x}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}} + \frac{a c d^{2} e + a^{2} e^{3}}{a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}}}{\sqrt{c x^{2} + a}} - \frac{2 \, \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} - a e^{2}}}\right ) e^{2}}{{\left (c d^{2} + a e^{2}\right )} \sqrt{-c d^{2} - a e^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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