Optimal. Leaf size=94 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
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Rubi [A] time = 0.0374351, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {715, 112, 110} \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 715
Rule 112
Rule 110
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{b x+c x^2}} \, dx &=\frac{\left (\sqrt{x} \sqrt{b+c x}\right ) \int \frac{\sqrt{d+e x}}{\sqrt{x} \sqrt{b+c x}} \, dx}{\sqrt{b x+c x^2}}\\ &=\frac{\left (\sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x}\right ) \int \frac{\sqrt{1+\frac{e x}{d}}}{\sqrt{x} \sqrt{1+\frac{c x}{b}}} \, dx}{\sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}\\ &=\frac{2 \sqrt{-b} \sqrt{x} \sqrt{1+\frac{c x}{b}} \sqrt{d+e x} E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{\sqrt{c} \sqrt{1+\frac{e x}{d}} \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.563666, size = 121, normalized size = 1.29 \[ -\frac{2 \sqrt{x} \left (\frac{b}{x}+c\right ) \sqrt{d+e x} \left (\frac{d \sqrt{\frac{d}{e x}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{d}{e}}}{\sqrt{x}}\right )|\frac{b e}{c d}\right )}{\sqrt{-\frac{d}{e}} \sqrt{\frac{b}{c x}+1} \left (\frac{d}{x}+e\right )}-\sqrt{x}\right )}{c \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.262, size = 121, normalized size = 1.3 \begin{align*} -2\,{\frac{\sqrt{ex+d}\sqrt{x \left ( cx+b \right ) }b \left ( be-cd \right ) }{{c}^{2}x \left ( ce{x}^{2}+bxe+cdx+bd \right ) }\sqrt{{\frac{cx+b}{b}}}\sqrt{-{\frac{c \left ( ex+d \right ) }{be-cd}}}\sqrt{-{\frac{cx}{b}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+b}{b}}},\sqrt{{\frac{be}{be-cd}}} \right ) } \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{e x + d}}{\sqrt{c x^{2} + b x}}\,{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{e x + d}}{\sqrt{c x^{2} + b 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{d + e x}}{\sqrt{x \left (b + c x\right )}}\, 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{e x + d}}{\sqrt{c x^{2} + b x}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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