Optimal. Leaf size=58 \[ \frac{2 \sqrt{a} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b \sqrt{1-c}} \]
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Rubi [A] time = 0.0214087, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.025, Rules used = {113} \[ \frac{2 \sqrt{a} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b \sqrt{1-c}} \]
Antiderivative was successfully verified.
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Rule 113
Rubi steps
\begin{align*} \int \frac{\sqrt{e+\frac{b (-1+e) x}{a}}}{\sqrt{a+b x} \sqrt{c+\frac{b (-1+c) x}{a}}} \, dx &=\frac{2 \sqrt{a} E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b \sqrt{1-c}}\\ \end{align*}
Mathematica [B] time = 0.669944, size = 191, normalized size = 3.29 \[ -\frac{2 (a+b x)^{3/2} \left (\frac{a \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{e-1}{c-1}\right )}{\sqrt{a+b x}}-\frac{\sqrt{-\frac{a}{e-1}} \left (\frac{a}{a+b x}+c-1\right ) \left (\frac{a}{a+b x}+e-1\right )}{c-1}\right )}{a b \sqrt{-\frac{a}{e-1}} \sqrt{\frac{b (c-1) x}{a}+c} \sqrt{\frac{b (e-1) x}{a}+e}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 548, normalized size = 9.5 \begin{align*} -2\,{\frac{{a}^{2}\sqrt{bx+a}}{ \left ({b}^{2}e{x}^{2}+2\,abex-{b}^{2}{x}^{2}+{a}^{2}e-abx \right ) \left ( c-1 \right ) ^{2}b \left ( -1+e \right ) }\sqrt{{\frac{bxe+ae-bx}{a}}}\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}} \left ({\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ){c}^{2}-{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ce-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ce+{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ){e}^{2}-{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) c+{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) e+{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) c-{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) e \right ){\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}} \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{\frac{b{\left (e - 1\right )} x}{a} + e}}{\sqrt{b x + a} \sqrt{\frac{b{\left (c - 1\right )} x}{a} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x + a} a \sqrt{\frac{a c +{\left (b c - b\right )} x}{a}} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}{a^{2} c +{\left (b^{2} c - b^{2}\right )} x^{2} +{\left (2 \, a b c - a b\right )} x}, x\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{e + \frac{b e x}{a} - \frac{b x}{a}}}{\sqrt{a + b x} \sqrt{c + \frac{b c x}{a} - \frac{b x}{a}}}\, 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*} \int \frac{\sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}{\sqrt{b x + a} \sqrt{\frac{b{\left (c - 1\right )} x}{a} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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