Optimal. Leaf size=53 \[ -\frac {2 \sqrt {\frac {e x}{d}+1} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} \sqrt {-x}\right )|\frac {2 e}{3 d}\right )}{\sqrt {3} \sqrt {d+e x}} \]
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Rubi [A] time = 0.03, antiderivative size = 53, 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 = {714, 12, 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 12
Rule 115
Rule 117
Rule 714
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {d+e x} \sqrt {-2 x-3 x^2}} \, dx &=\int \frac {1}{\sqrt {2} \sqrt {-x} \sqrt {1+\frac {3 x}{2}} \sqrt {d+e x}} \, dx\\ &=\frac {\int \frac {1}{\sqrt {-x} \sqrt {1+\frac {3 x}{2}} \sqrt {d+e x}} \, dx}{\sqrt {2}}\\ &=\frac {\sqrt {1+\frac {e x}{d}} \int \frac {1}{\sqrt {-x} \sqrt {1+\frac {3 x}{2}} \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 [C] time = 0.10, size = 82, normalized size = 1.55 \[ \frac {i \sqrt {\frac {4}{x}+6} x^{3/2} \sqrt {\frac {d}{e x}+1} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3 d}{2 e}\right )}{\sqrt {-x (3 x+2)} \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {e x + d} \sqrt {-3 \, x^{2} - 2 \, x}}{3 \, e x^{3} + {\left (3 \, d + 2 \, e\right )} x^{2} + 2 \, d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} - 2 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 115, normalized size = 2.17 \[ -\frac {2 \sqrt {-\frac {e x}{d}}\, \sqrt {-\frac {\left (3 x +2\right ) e}{3 d -2 e}}\, \sqrt {\frac {e x +d}{d}}\, \sqrt {e x +d}\, \sqrt {-\left (3 x +2\right ) x}\, d \EllipticF \left (\sqrt {\frac {e x +d}{d}}, \sqrt {3}\, \sqrt {\frac {d}{3 d -2 e}}\right )}{\left (3 e \,x^{2}+3 d x +2 e x +2 d \right ) e x} \]
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 \[ \int \frac {1}{\sqrt {e x + d} \sqrt {-3 \, x^{2} - 2 \, x}}\,{d x} \]
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 \[ \int \frac {1}{\sqrt {-3\,x^2-2\,x}\,\sqrt {d+e\,x}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- x \left (3 x + 2\right )} \sqrt {d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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