Optimal. Leaf size=134 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}} \]
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Rubi [A] time = 0.113946, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {708, 1093, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}} \]
Antiderivative was successfully verified.
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Rule 708
Rule 1093
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a-c x^2\right )} \, dx &=(2 e) \operatorname{Subst}\left (\int \frac{1}{-c d^2+a e^2+2 c d x^2-c x^4} \, dx,x,\sqrt{d+e x}\right )\\ &=-\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{c d-\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{a}}+\frac{\sqrt{c} \operatorname{Subst}\left (\int \frac{1}{c d+\sqrt{a} \sqrt{c} e-c x^2} \, dx,x,\sqrt{d+e x}\right )}{\sqrt{a}}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d-\sqrt{a} e}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d+\sqrt{a} e}}\right )}{\sqrt{a} \sqrt [4]{c} \sqrt{\sqrt{c} d+\sqrt{a} e}}\\ \end{align*}
Mathematica [A] time = 0.0868462, size = 125, normalized size = 0.93 \[ \frac{\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{a} e+\sqrt{c} d}}\right )}{\sqrt{\sqrt{a} e+\sqrt{c} d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{d+e x}}{\sqrt{\sqrt{c} d-\sqrt{a} e}}\right )}{\sqrt{\sqrt{c} d-\sqrt{a} e}}}{\sqrt{a} \sqrt [4]{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.205, size = 110, normalized size = 0.8 \begin{align*}{ce\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( -cd+\sqrt{ac{e}^{2}} \right ) c}}}}+{ce{\it Artanh} \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ac{e}^{2}}}}{\frac{1}{\sqrt{ \left ( cd+\sqrt{ac{e}^{2}} \right ) c}}}} \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{1}{{\left (c x^{2} - a\right )} \sqrt{e x + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.94096, size = 1791, normalized size = 13.37 \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{1}{- a \sqrt{d + e x} + c x^{2} \sqrt{d + e x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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