Optimal. Leaf size=96 \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right ),-\frac{a d}{(1-e) (b c-a d)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]
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Rubi [A] time = 0.0745662, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {121, 119} \[ \frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}} \]
Antiderivative was successfully verified.
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Rule 121
Rule 119
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx &=\frac{\sqrt{\frac{b (c+d x)}{b c-a d}} \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx}{\sqrt{c+d x}}\\ &=\frac{2 \sqrt{a} \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b \sqrt{1-e} \sqrt{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.405816, size = 126, normalized size = 1.31 \[ -\frac{2 \sqrt{c+d x} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{a}{e-1}}}{\sqrt{a+b x}}\right ),\frac{(e-1) (b c-a d)}{a d}\right )}{d \sqrt{-\frac{a}{e-1}} \sqrt{\frac{b (e-1) x}{a}+e} \sqrt{\frac{b (c+d x)}{d (a+b x)}}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.107, size = 207, normalized size = 2.2 \begin{align*} 2\,{\frac{\sqrt{bx+a}\sqrt{dx+c} \left ( ade-bce+bc \right ) }{ \left ( d{x}^{2}b+adx+bcx+ac \right ) bd \left ( -1+e \right ) }\sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+e \right ) }{a}}}\sqrt{-{\frac{ \left ( dx+c \right ) b \left ( -1+e \right ) }{ade-bce+bc}}}{\it EllipticF} \left ( \sqrt{{\frac{d \left ( bxe+ae-bx \right ) }{ade-bce+bc}}},\sqrt{{\frac{ade-bce+bc}{ad}}} \right ){\frac{1}{\sqrt{{\frac{bxe+ae-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{1}{\sqrt{b x + a} \sqrt{d x + 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} \sqrt{d x + c} a \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}{a^{2} c e +{\left (b^{2} d e - b^{2} d\right )} x^{3} -{\left (b^{2} c + a b d -{\left (b^{2} c + 2 \, a b d\right )} e\right )} x^{2} -{\left (a b c -{\left (2 \, a b c + a^{2} d\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{1}{\sqrt{a + b x} \sqrt{c + d x} \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{1}{\sqrt{b x + a} \sqrt{d x + 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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