Optimal. Leaf size=51 \[ \frac {2 \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {728, 12, 118,
116} \begin {gather*} \frac {2 \sqrt {\frac {e x}{d}+1} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{2}} \sqrt {x}\right )|-\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 116
Rule 118
Rule 728
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {2 x-3 x^2}} \, dx &=\int \frac {1}{\sqrt {2} \sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {d+e x}} \, dx\\ &=\frac {\int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {d+e x}} \, dx}{\sqrt {2}}\\ &=\frac {\sqrt {1+\frac {e x}{d}} \int \frac {1}{\sqrt {1-\frac {3 x}{2}} \sqrt {x} \sqrt {1+\frac {e x}{d}}} \, dx}{\sqrt {2} \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 = 1.27, size = 76, normalized size = 1.49 \begin {gather*} -\frac {\sqrt {6-\frac {4}{x}} \sqrt {1+\frac {d}{e x}} x^{3/2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|-\frac {3 d}{2 e}\right )}{\sqrt {-x (-2+3 x)} \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. \(114\) vs.
\(2(41)=82\).
time = 0.45, size = 115, normalized size = 2.25
method | result | size |
default | \(-\frac {2 \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d +2 e}}\right ) \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (-2+3 x \right ) e}{3 d +2 e}}\, \sqrt {\frac {e x +d}{d}}\, d \sqrt {e x +d}\, \sqrt {-x \left (-2+3 x \right )}}{e x \left (3 e \,x^{2}+3 d x -2 e x -2 d \right )}\) | \(115\) |
elliptic | \(\frac {2 \sqrt {-x \left (-2+3 x \right ) \left (e x +d \right )}\, d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {2}{3}}{-\frac {d}{e}-\frac {2}{3}}}\, \sqrt {-\frac {e x}{d}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {2}{3}\right )}}\right )}{\sqrt {-x \left (-2+3 x \right )}\, \sqrt {e x +d}\, e \sqrt {-3 e \,x^{3}-3 d \,x^{2}+2 e \,x^{2}+2 d x}}\) | \(136\) |
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 {1}{\sqrt {- x \left (3 x - 2\right )} \sqrt {d + e x}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {1}{\sqrt {2\,x-3\,x^2}\,\sqrt {d+e\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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