Optimal. Leaf size=168 \[ \frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{d}} \]
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Rubi [A] time = 0.166877, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {849, 843, 621, 206, 724} \[ \frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 849
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x (d+e x)} \, dx &=\int \frac{a e+c d x}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=(c d) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx+(a e) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=(2 c d) \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )-(2 a e) \operatorname{Subst}\left (\int \frac{1}{4 a d e-x^2} \, dx,x,\frac{2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )\\ &=\frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left (\frac{2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.188225, size = 210, normalized size = 1.25 \[ -\frac{2 \sqrt{a e+c d x} \left (\sqrt{a} \sqrt{c} e \sqrt{d+e x} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )-\sqrt{c d} \sqrt{c d^2-a e^2} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )\right )}{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 439, normalized size = 2.6 \begin{align*}{\frac{1}{d}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{a{e}^{2}}{2\,d}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}+{\frac{cd}{2}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}-{ae\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}-{\frac{1}{d}\sqrt{cde \left ({\frac{d}{e}}+x \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) }}-{\frac{a{e}^{2}}{2\,d}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+ \left ({\frac{d}{e}}+x \right ) cde \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{cde \left ({\frac{d}{e}}+x \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) } \right ){\frac{1}{\sqrt{dec}}}}+{\frac{cd}{2}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+ \left ({\frac{d}{e}}+x \right ) cde \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{cde \left ({\frac{d}{e}}+x \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) } \right ){\frac{1}{\sqrt{dec}}}} \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 = 3.20476, size = 2012, normalized size = 11.98 \begin{align*} \text{result too large to display} \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{\left (d + e x\right ) \left (a e + c d x\right )}}{x \left (d + e x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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