Optimal. Leaf size=51 \[ \frac {2 \sqrt {d+e x} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {1+\frac {e x}{d}}} \]
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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 = {113, 111}
\begin {gather*} \frac {2 \sqrt {d+e x} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {\frac {e x}{d}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 111
Rule 113
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] Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(51)=102\).
time = 2.57, size = 125, normalized size = 2.45 \begin {gather*} \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 (-2+3 x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {2+\frac {3 d}{e}}}{\sqrt {2-3 x}}\right )|\frac {2 e}{3 d+2 e}\right )}{\sqrt {2+\frac {3 d}{e}} \sqrt {\frac {x}{-2+3 x}}}\right )}{3 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(211\) vs.
\(2(41)=82\).
time = 0.08, size = 212, normalized size = 4.16
method | result | size |
default | \(-\frac {2 \sqrt {e x +d}\, \sqrt {2-3 x}\, d \sqrt {\frac {e x +d}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {-\frac {e x}{d}}\, \left (3 d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right )+2 \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e -3 \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) d -2 \EllipticE \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) e \right )}{3 \sqrt {x}\, e \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\) | \(212\) |
elliptic | \(\frac {\sqrt {-\left (-2+3 x \right ) x \left (e x +d \right )}\, \left (\frac {2 d^{2} \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {2}{3}}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}+\frac {2 d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {2}{3}}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {2}{3}\right ) \EllipticE \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )+\frac {2 \EllipticF \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{3}\right )}{\sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\right )}{\sqrt {2-3 x}\, \sqrt {x}\, \sqrt {e x +d}}\) | \(285\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d + e x}}{\sqrt {x} \sqrt {2 - 3 x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {d+e\,x}}{\sqrt {x}\,\sqrt {2-3\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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