Optimal. Leaf size=127 \[ \frac{\left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c} e^3}+\frac{d \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^3}-\frac{\sqrt{a+c x^2} (2 d-e x)}{2 e^2} \]
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Rubi [A] time = 0.105262, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {815, 844, 217, 206, 725} \[ \frac{\left (a e^2+2 c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c} e^3}+\frac{d \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{e^3}-\frac{\sqrt{a+c x^2} (2 d-e x)}{2 e^2} \]
Antiderivative was successfully verified.
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Rule 815
Rule 844
Rule 217
Rule 206
Rule 725
Rubi steps
\begin{align*} \int \frac{x \sqrt{a+c x^2}}{d+e x} \, dx &=-\frac{(2 d-e x) \sqrt{a+c x^2}}{2 e^2}+\frac{\int \frac{-a c d e+c \left (2 c d^2+a e^2\right ) x}{(d+e x) \sqrt{a+c x^2}} \, dx}{2 c e^2}\\ &=-\frac{(2 d-e x) \sqrt{a+c x^2}}{2 e^2}-\frac{\left (d \left (c d^2+a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{e^3}+\frac{\left (2 c d^2+a e^2\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 e^3}\\ &=-\frac{(2 d-e x) \sqrt{a+c x^2}}{2 e^2}+\frac{\left (d \left (c d^2+a e^2\right )\right ) \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 )}{e^3}+\frac{\left (2 c d^2+a e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 e^3}\\ &=-\frac{(2 d-e x) \sqrt{a+c x^2}}{2 e^2}+\frac{\left (2 c d^2+a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 \sqrt{c} e^3}+\frac{d \sqrt{c d^2+a e^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{c d^2+a e^2} \sqrt{a+c x^2}}\right )}{e^3}\\ \end{align*}
Mathematica [A] time = 0.303128, size = 175, normalized size = 1.38 \[ \frac{\frac{a^{3/2} e^2 \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{c} \sqrt{a+c x^2}}+2 d \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )+2 \sqrt{c} d^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )-2 d e \sqrt{a+c x^2}+e^2 x \sqrt{a+c x^2}}{2 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.24, size = 423, normalized size = 3.3 \begin{align*}{\frac{x}{2\,e}\sqrt{c{x}^{2}+a}}+{\frac{a}{2\,e}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{d}{{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}}}}}+{\frac{{d}^{2}}{{e}^{3}}\sqrt{c}\ln \left ({ \left ( -{\frac{cd}{e}}+ \left ({\frac{d}{e}}+x \right ) c \right ){\frac{1}{\sqrt{c}}}}+\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 ) }+{\frac{ad}{{e}^{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}}}}}}}+{\frac{c{d}^{3}}{{e}^{4}}\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 [A] time = 3.46924, size = 1507, normalized size = 11.87 \begin{align*} \left [\frac{2 \, \sqrt{c d^{2} + a e^{2}} c d \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 ) +{\left (2 \, c d^{2} + a e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (c e^{2} x - 2 \, c d e\right )} \sqrt{c x^{2} + a}}{4 \, c e^{3}}, \frac{4 \, \sqrt{-c d^{2} - a e^{2}} c d \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 (2 \, c d^{2} + a e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) + 2 \,{\left (c e^{2} x - 2 \, c d e\right )} \sqrt{c x^{2} + a}}{4 \, c e^{3}}, \frac{\sqrt{c d^{2} + a e^{2}} c d \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 ) -{\left (2 \, c d^{2} + a e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (c e^{2} x - 2 \, c d e\right )} \sqrt{c x^{2} + a}}{2 \, c e^{3}}, \frac{2 \, \sqrt{-c d^{2} - a e^{2}} c d \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 (2 \, c d^{2} + a e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) +{\left (c e^{2} x - 2 \, c d e\right )} \sqrt{c x^{2} + a}}{2 \, c e^{3}}\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{x \sqrt{a + c x^{2}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18797, size = 182, normalized size = 1.43 \begin{align*} -\frac{2 \,{\left (c d^{3} + a d e^{2}\right )} \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^{\left (-3\right )}}{\sqrt{-c d^{2} - a e^{2}}} - \frac{{\left (2 \, c^{\frac{3}{2}} d^{2} + a \sqrt{c} e^{2}\right )} e^{\left (-3\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c} + \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (x e^{\left (-1\right )} - 2 \, d e^{\left (-2\right )}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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