Optimal. Leaf size=43 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )}{\sqrt{a} \sqrt{c}} \]
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Rubi [A] time = 0.0239438, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {63, 217, 203} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a x+a}}{\sqrt{a} \sqrt{c-c x}}\right )}{\sqrt{a} \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+a x} \sqrt{c-c x}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 c-\frac{c x^2}{a}}} \, dx,x,\sqrt{a+a x}\right )}{a}\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+\frac{c x^2}{a}} \, dx,x,\frac{\sqrt{a+a x}}{\sqrt{c-c x}}\right )}{a}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{a+a x}}{\sqrt{a} \sqrt{c-c x}}\right )}{\sqrt{a} \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0163171, size = 47, normalized size = 1.09 \[ \frac{2 \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{x+1}}{\sqrt{c-c x}}\right )}{\sqrt{c} \sqrt{a (x+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 57, normalized size = 1.3 \begin{align*}{\sqrt{ \left ( -cx+c \right ) \left ( ax+a \right ) }\arctan \left ({x\sqrt{ac}{\frac{1}{\sqrt{-ac{x}^{2}+ac}}}} \right ){\frac{1}{\sqrt{ax+a}}}{\frac{1}{\sqrt{-cx+c}}}{\frac{1}{\sqrt{ac}}}} \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 = 1.66179, size = 239, normalized size = 5.56 \begin{align*} \left [-\frac{\sqrt{-a c} \log \left (2 \, a c x^{2} - 2 \, \sqrt{-a c} \sqrt{a x + a} \sqrt{-c x + c} x - a c\right )}{2 \, a c}, -\frac{\sqrt{a c} \arctan \left (\frac{\sqrt{a c} \sqrt{a x + a} \sqrt{-c x + c} x}{a c x^{2} - a c}\right )}{a c}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.1786, size = 85, normalized size = 1.98 \begin{align*} - \frac{i{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{c}} + \frac{{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} \sqrt{a} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14295, size = 66, normalized size = 1.53 \begin{align*} -\frac{2 \, a \log \left ({\left | -\sqrt{-a c} \sqrt{a x + a} + \sqrt{-{\left (a x + a\right )} a c + 2 \, a^{2} c} \right |}\right )}{\sqrt{-a c}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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