Optimal. Leaf size=63 \[ \frac {2 \sqrt {e} E\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{d}-\frac {2 \sqrt {e} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c e+d x e}}{\sqrt {e}}\right )\right |-1\right )}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 63, normalized size of antiderivative = 1.00, 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 221
Rule 307
Rule 424
Rule 690
Rule 1199
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*}
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Mathematica [C] time = 0.02, 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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fricas [F] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1} \sqrt {d e x + c e}}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d e x + c e}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 121, normalized size = 1.92 \[ \frac {\sqrt {\left (d x +c \right ) e}\, \sqrt {-d^{2} x^{2}-2 c d x -c^{2}+1}\, \sqrt {2 d x +2 c +2}\, \sqrt {-d x -c}\, \sqrt {-2 d x -2 c +2}\, \EllipticE \left (\frac {\sqrt {2 d x +2 c +2}}{2}, \sqrt {2}\right )}{\left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}-d x -c \right ) d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d e x + c e}}{\sqrt {-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c\,e+d\,e\,x}}{\sqrt {-c^2-2\,c\,d\,x-d^2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \left (c + d x\right )}}{\sqrt {- \left (c + d x - 1\right ) \left (c + d x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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