Optimal. Leaf size=46 \[ \frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}} \]
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Rubi [A] time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {648} \[ \frac {2 \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \int \frac {\sqrt {d+e x}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac {2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d \sqrt {d+e x}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 35, normalized size = 0.76 \[ \frac {2 \sqrt {(d+e x) (a e+c d x)}}{c d \sqrt {d+e x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 49, normalized size = 1.07 \[ \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{c d e x + c d^{2}} \]
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 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 1.09 \[ \frac {2 \left (c d x +a e \right ) \sqrt {e x +d}}{\sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}\, c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 18, normalized size = 0.39 \[ \frac {2 \, \sqrt {c d x + a e}}{c d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.20, size = 54, normalized size = 1.17 \[ \frac {2\,\sqrt {d+e\,x}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{c\,d\,e\,\left (x+\frac {d}{e}\right )} \]
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 \[ \int \frac {\sqrt {d + e x}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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