Optimal. Leaf size=80 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right ),-1\right )}{3 d e^{5/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}} \]
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Rubi [A] time = 0.0579532, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081, Rules used = {693, 689, 221} \[ \frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d x e}}{\sqrt{e}}\right )\right |-1\right )}{3 d e^{5/2}}-\frac{2 \sqrt{-c^2-2 c d x-d^2 x^2+1}}{3 d e (c e+d e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 693
Rule 689
Rule 221
Rubi steps
\begin{align*} \int \frac{1}{(c e+d e x)^{5/2} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx &=-\frac{2 \sqrt{1-c^2-2 c d x-d^2 x^2}}{3 d e (c e+d e x)^{3/2}}+\frac{\int \frac{1}{\sqrt{c e+d e x} \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx}{3 e^2}\\ &=-\frac{2 \sqrt{1-c^2-2 c d x-d^2 x^2}}{3 d e (c e+d e x)^{3/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^4}{e^2}}} \, dx,x,\sqrt{c e+d e x}\right )}{3 d e^3}\\ &=-\frac{2 \sqrt{1-c^2-2 c d x-d^2 x^2}}{3 d e (c e+d e x)^{3/2}}+\frac{2 F\left (\left .\sin ^{-1}\left (\frac{\sqrt{c e+d e x}}{\sqrt{e}}\right )\right |-1\right )}{3 d e^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0201096, size = 40, normalized size = 0.5 \[ -\frac{2 (c+d x) \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};(c+d x)^2\right )}{3 d (e (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.286, size = 494, normalized size = 6.2 \begin{align*}{\frac{1}{6\,{e}^{3} \left ( dx+c \right ) ^{2} \left ({d}^{2}{x}^{2}+2\,cdx+{c}^{2}-1 \right ) d} \left ( 5\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) xd+3\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) xd+3\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) xd-3\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) xd+5\,\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}\sqrt{2\,dx+2\,c+2}{\it EllipticF} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) c+3\,\sqrt{2\,dx+2\,c+2}\sqrt{-2\,dx-2\,c+2}\sqrt{dx+c}{\it EllipticE} \left ( 1/2\,\sqrt{-2\,dx-2\,c+2},\sqrt{2} \right ) c+3\,\sqrt{-2\,dx-2\,c+2}\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}{\it EllipticF} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) c-3\,\sqrt{2\,dx+2\,c+2}\sqrt{-dx-c}\sqrt{-2\,dx-2\,c+2}{\it EllipticE} \left ( 1/2\,\sqrt{2\,dx+2\,c+2},\sqrt{2} \right ) c-4\,{d}^{2}{x}^{2}-8\,cdx-4\,{c}^{2}+4 \right ) \sqrt{-{d}^{2}{x}^{2}-2\,cdx-{c}^{2}+1}\sqrt{e \left ( dx+c \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{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{5}{2}}}\,{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^{5} e^{3} x^{5} + 5 \, c d^{4} e^{3} x^{4} +{\left (10 \, c^{2} - 1\right )} d^{3} e^{3} x^{3} +{\left (10 \, c^{3} - 3 \, c\right )} d^{2} e^{3} x^{2} +{\left (5 \, c^{4} - 3 \, c^{2}\right )} d e^{3} x +{\left (c^{5} - c^{3}\right )} e^{3}}, 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}{\left (e \left (c + d x\right )\right )^{\frac{5}{2}} \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{1}{\sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}{\left (d e x + c e\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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