Optimal. Leaf size=162 \[ -\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}} \]
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Rubi [A] time = 0.183466, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {114, 113} \[ -\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}} \]
Antiderivative was successfully verified.
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Rule 114
Rule 113
Rubi steps
\begin{align*} \int \frac{\sqrt{a+b x}}{\sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx &=\frac{\left (\sqrt{a+b x} \sqrt{\frac{b (-1+c) \left (e+\frac{b (-1+e) x}{a}\right )}{a \left (-\frac{b c (-1+e)}{a}+\frac{b (-1+c) e}{a}\right )}}\right ) \int \frac{\sqrt{\frac{b (-1+c)}{b (-1+c)-b c}+\frac{b^2 (-1+c) x}{a (b (-1+c)-b c)}}}{\sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{\frac{b (-1+c) e}{a \left (-\frac{b c (-1+e)}{a}+\frac{b (-1+c) e}{a}\right )}+\frac{b^2 (-1+c) (-1+e) x}{a^2 \left (-\frac{b c (-1+e)}{a}+\frac{b (-1+c) e}{a}\right )}}} \, dx}{\sqrt{\frac{b (-1+c) (a+b x)}{a (b (-1+c)-b c)}} \sqrt{e+\frac{b (-1+e) x}{a}}}\\ &=-\frac{2 a \sqrt{c-e} \sqrt{a+b x} \sqrt{-\frac{(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{c-\frac{b (1-c) x}{a}}}{\sqrt{c-e}}\right )|\frac{c-e}{1-e}\right )}{b (1-c) \sqrt{1-e} \sqrt{\frac{(1-c) (a+b x)}{a}} \sqrt{e-\frac{b (1-e) x}{a}}}\\ \end{align*}
Mathematica [C] time = 0.202524, size = 103, normalized size = 0.64 \[ -\frac{2 i a \sqrt{a+b x} \left (E\left (i \sinh ^{-1}\left (\sqrt{\frac{(c-1) (a+b x)}{a}}\right )|\frac{e-1}{c-1}\right )-\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{(c-1) (a+b x)}{a}}\right ),\frac{e-1}{c-1}\right )\right )}{b (e-1) \sqrt{\frac{(c-1) (a+b x)}{a}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 183, normalized size = 1.1 \begin{align*} 2\,{\frac{ \left ( c-e \right ){a}^{2}}{\sqrt{bx+a} \left ( c-1 \right ) ^{2}b \left ( -1+e \right ) }{\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 ) \sqrt{{\frac{ \left ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \right ) }{a}}}\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}{\frac{1}{\sqrt{{\frac{bxe+ae-bx}{a}}}}}{\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{b x + a}}{\sqrt{\frac{b{\left (c - 1\right )} x}{a} + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{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^{2} \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 e -{\left (b^{2} c - b^{2} -{\left (b^{2} c - b^{2}\right )} e\right )} x^{2} -{\left (a b c -{\left (2 \, a b c - a b\right )} e\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{a + b x}}{\sqrt{c + \frac{b c x}{a} - \frac{b x}{a}} \sqrt{e + \frac{b e 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{b x + a}}{\sqrt{\frac{b{\left (c - 1\right )} x}{a} + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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