Optimal. Leaf size=63 \[ \frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right ),-1\right )}{d} \]
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Rubi [A] time = 0.0553057, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {690, 307, 221, 1199, 424} \[ \frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{d} \]
Antiderivative was successfully verified.
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Rule 690
Rule 307
Rule 221
Rule 1199
Rule 424
Rubi steps
\begin{align*} \int \frac{\sqrt{c e+d e x}}{\sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{d e}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{d}+\frac{2 \operatorname{Subst}\left (\int \frac{1+\frac{x^2}{e}}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{d}\\ &=-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{d}+\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x^2}{e}}}{\sqrt{1-\frac{x^2}{e}}} \, dx,x,\sqrt{c e+d e x}\right )}{d}\\ &=\frac{2 \sqrt{e} E\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{d}-\frac{2 \sqrt{e} F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{d}\\ \end{align*}
Mathematica [C] time = 0.0165099, size = 40, normalized size = 0.63 \[ \frac{2 (c+d x) \sqrt{e (c+d x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};(c+d x)^2\right )}{3 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.17, size = 204, normalized size = 3.2 \begin{align*}{\frac{1}{6\,d \left ({x}^{3}{d}^{3}+3\,{x}^{2}c{d}^{2}+3\,x{c}^{2}d+{c}^{3}-dx-c \right ) }\sqrt{e \left ( dx+c \right ) }\sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2} \left ( \sqrt{-dx-c}{\it EllipticF} \left ({\frac{1}{2}\sqrt{2\,dx+2\,c+2}},\sqrt{2} \right ) +3\,\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) +\sqrt{dx+c}{\it EllipticF} \left ({\frac{1}{2}\sqrt{-2\,dx-2\,c+2}},\sqrt{2} \right ) +3\,\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) \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{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{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{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt{d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, 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 \left (c + d x\right )}}{\sqrt{- \left (c + d x - 1\right ) \left (c + d x + 1\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{d e x + c e}}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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