Optimal. Leaf size=160 \[ -\frac{\sqrt{a} e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{e \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]
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Rubi [A] time = 0.21066, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {961, 266, 47, 63, 208, 277, 217, 206, 50, 735, 844, 725} \[ -\frac{\sqrt{a} e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}+\frac{e \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}-\frac{\sqrt{a+c x^2}}{2 d x^2}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 961
Rule 266
Rule 47
Rule 63
Rule 208
Rule 277
Rule 217
Rule 206
Rule 50
Rule 735
Rule 844
Rule 725
Rubi steps
\begin{align*} \int \frac{\sqrt{a+c x^2}}{x^3 (d+e x)} \, dx &=\int \left (\frac{\sqrt{a+c x^2}}{d x^3}-\frac{e \sqrt{a+c x^2}}{d^2 x^2}+\frac{e^2 \sqrt{a+c x^2}}{d^3 x}-\frac{e^3 \sqrt{a+c x^2}}{d^3 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{\sqrt{a+c x^2}}{x^3} \, dx}{d}-\frac{e \int \frac{\sqrt{a+c x^2}}{x^2} \, dx}{d^2}+\frac{e^2 \int \frac{\sqrt{a+c x^2}}{x} \, dx}{d^3}-\frac{e^3 \int \frac{\sqrt{a+c x^2}}{d+e x} \, dx}{d^3}\\ &=-\frac{e^2 \sqrt{a+c x^2}}{d^3}+\frac{e \sqrt{a+c x^2}}{d^2 x}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x^2} \, dx,x,x^2\right )}{2 d}-\frac{(c e) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{d^2}+\frac{e^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+c x}}{x} \, dx,x,x^2\right )}{2 d^3}-\frac{e^2 \int \frac{a e-c d x}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^3}\\ &=-\frac{\sqrt{a+c x^2}}{2 d x^2}+\frac{e \sqrt{a+c x^2}}{d^2 x}+\frac{c \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{4 d}+\frac{(c e) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{d^2}-\frac{(c e) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{d^2}+\frac{\left (a e^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,x^2\right )}{2 d^3}-\frac{\left (e \left (c d^2+a e^2\right )\right ) \int \frac{1}{(d+e x) \sqrt{a+c x^2}} \, dx}{d^3}\\ &=-\frac{\sqrt{a+c x^2}}{2 d x^2}+\frac{e \sqrt{a+c x^2}}{d^2 x}-\frac{\sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{d^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{2 d}+\frac{(c e) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{d^2}+\frac{\left (a e^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+c x^2}\right )}{c d^3}+\frac{\left (e \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 )}{d^3}\\ &=-\frac{\sqrt{a+c x^2}}{2 d x^2}+\frac{e \sqrt{a+c x^2}}{d^2 x}+\frac{e \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 )}{d^3}-\frac{c \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{2 \sqrt{a} d}-\frac{\sqrt{a} e^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.424999, size = 283, normalized size = 1.77 \[ -\frac{-2 e x^2 \sqrt{a+c x^2} \sqrt{a e^2+c d^2} \tanh ^{-1}\left (\frac{a e-c d x}{\sqrt{a+c x^2} \sqrt{a e^2+c d^2}}\right )+c d^2 x^2 \sqrt{\frac{c x^2}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{c x^2}{a}+1}\right )+2 \sqrt{a} \sqrt{c} d e x^2 \sqrt{\frac{c x^2}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )-2 \sqrt{c} d e x^2 \sqrt{a+c x^2} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )+2 \sqrt{a} e^2 x^2 \sqrt{a+c x^2} \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )+a d^2-2 a d e x+c d^2 x^2-2 c d e x^3}{2 d^3 x^2 \sqrt{a+c x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.239, size = 567, normalized size = 3.5 \begin{align*} -{\frac{{e}^{2}}{{d}^{3}}\sqrt{a}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ) }+{\frac{{e}^{2}}{{d}^{3}}\sqrt{c{x}^{2}+a}}-{\frac{{e}^{2}}{{d}^{3}}\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}{{d}^{2}}\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{a{e}^{2}}{{d}^{3}}\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}\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{1}{2\,ad{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{c}{2\,d}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{c}{2\,ad}\sqrt{c{x}^{2}+a}}+{\frac{e}{a{d}^{2}x} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{cex}{a{d}^{2}}\sqrt{c{x}^{2}+a}}-{\frac{e}{{d}^{2}}\sqrt{c}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c x^{2} + a}}{{\left (e x + d\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.25257, size = 1577, normalized size = 9.86 \begin{align*} \left [\frac{2 \, \sqrt{c d^{2} + a e^{2}} a e x^{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 ) +{\left (c d^{2} + 2 \, a e^{2}\right )} \sqrt{a} x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, a d e x - a d^{2}\right )} \sqrt{c x^{2} + a}}{4 \, a d^{3} x^{2}}, \frac{4 \, \sqrt{-c d^{2} - a e^{2}} a e x^{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 (c d^{2} + 2 \, a e^{2}\right )} \sqrt{a} x^{2} \log \left (-\frac{c x^{2} - 2 \, \sqrt{c x^{2} + a} \sqrt{a} + 2 \, a}{x^{2}}\right ) + 2 \,{\left (2 \, a d e x - a d^{2}\right )} \sqrt{c x^{2} + a}}{4 \, a d^{3} x^{2}}, \frac{\sqrt{c d^{2} + a e^{2}} a e x^{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 ) +{\left (c d^{2} + 2 \, a e^{2}\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, a d e x - a d^{2}\right )} \sqrt{c x^{2} + a}}{2 \, a d^{3} x^{2}}, \frac{2 \, \sqrt{-c d^{2} - a e^{2}} a e x^{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 (c d^{2} + 2 \, a e^{2}\right )} \sqrt{-a} x^{2} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (2 \, a d e x - a d^{2}\right )} \sqrt{c x^{2} + a}}{2 \, a d^{3} 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{\sqrt{a + c x^{2}}}{x^{3} \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 = 1.1892, size = 311, normalized size = 1.94 \begin{align*} -\frac{2 \,{\left (c d^{2} e + a e^{3}\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 )}{\sqrt{-c d^{2} - a e^{2}} d^{3}} + \frac{{\left (c d^{2} + 2 \, a e^{2}\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} d^{3}} + \frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} c d - 2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} a \sqrt{c} e +{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} a c d + 2 \, a^{2} \sqrt{c} e}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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