Optimal. Leaf size=51 \[ \frac {2 \sqrt {\frac {e x}{d}+1} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right ),-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]
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Rubi [A] time = 0.01, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {117, 115} \[ \frac {2 \sqrt {\frac {e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 115
Rule 117
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-3 x} \sqrt {x} \sqrt {d+e x}} \, dx &=\frac {\left (\sqrt {1-\frac {3 x}{2}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {2-3 x} \sqrt {d+e x}}\\ &=\frac {2 \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 72, normalized size = 1.41 \[ -\frac {\sqrt {x} \sqrt {\frac {d+e x}{e (3 x-2)}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {1}{\sqrt {1-\frac {3 x}{2}}}\right ),\frac {3 d}{2 e}+1\right )}{\sqrt {\frac {x}{6 x-4}} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {x} \sqrt {-3 \, x + 2}}{3 \, e x^{3} + {\left (3 \, d - 2 \, e\right )} x^{2} - 2 \, d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 112, normalized size = 2.20 \[ -\frac {2 \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (3 x -2\right ) e}{3 d +2 e}}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {-3 x +2}\, \sqrt {e x +d}\, d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )}{\left (3 e \,x^{2}+3 d x -2 e x -2 d \right ) e \sqrt {x}} \]
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 \[ \int \frac {1}{\sqrt {e x + d} \sqrt {x} \sqrt {-3 \, x + 2}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {x}\,\sqrt {2-3\,x}\,\sqrt {d+e\,x}} \,d x \]
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 \[ \int \frac {1}{\sqrt {x} \sqrt {2 - 3 x} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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