Optimal. Leaf size=51 \[ \frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \]
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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 = {112, 110} \[ \frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \]
Antiderivative was successfully verified.
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Rule 110
Rule 112
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {2-3 x} \sqrt {x}} \, dx &=\frac {\left (\sqrt {1-\frac {3 x}{2}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {1-\frac {3 x}{2}} \sqrt {x}} \, dx}{\sqrt {2-3 x} \sqrt {1+\frac {e x}{d}}}\\ &=\frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}}\\ \end {align*}
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Mathematica [B] time = 0.84, size = 125, normalized size = 2.45 \[ \frac {2 \sqrt {x} \left (\frac {3 (d+e x)}{\sqrt {2-3 x}}-\frac {(3 d+2 e) \sqrt {\frac {d+e x}{e (3 x-2)}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {3 d}{e}+2}}{\sqrt {2-3 x}}\right )|\frac {2 e}{3 d+2 e}\right )}{\sqrt {\frac {x}{3 x-2}} \sqrt {\frac {3 d}{e}+2}}\right )}{3 \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {x} \sqrt {-3 \, x + 2}}{3 \, x^{2} - 2 \, x}, x\right ) \]
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 = 0.02, size = 212, normalized size = 4.16 \[ -\frac {2 \sqrt {e x +d}\, \sqrt {-3 x +2}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (3 x -2\right ) e}{3 d +2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (-3 d \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+3 d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )-2 e \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+2 e \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )\right ) d}{3 \left (3 e \,x^{2}+3 d x -2 e x -2 d \right ) e \sqrt {x}} \]
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 \[ \int \frac {\sqrt {e x + d}}{\sqrt {x} \sqrt {-3 \, x + 2}}\,{d x} \]
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 \[ \int \frac {\sqrt {d+e\,x}}{\sqrt {x}\,\sqrt {2-3\,x}} \,d x \]
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 \[ \int \frac {\sqrt {d + e x}}{\sqrt {x} \sqrt {2 - 3 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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