Optimal. Leaf size=48 \[ \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 48, 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} \begin {gather*} \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx &=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.77 \begin {gather*} \frac {2 ((d+e x) (a e+c d x))^{3/2}}{3 c d (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 82, normalized size = 1.71 \begin {gather*} \frac {2 \left (a e^2-c d^2+c d (d+e x)\right ) \sqrt {a e (d+e x)-\frac {c d^2 (d+e x)}{e}+\frac {c d (d+e x)^2}{e}}}{3 c d e \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 57, normalized size = 1.19 \begin {gather*} \frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (c d x + a e\right )} \sqrt {e x + d}}{3 \, {\left (c d e x + c d^{2}\right )}} \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*} \int \frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{\sqrt {e x + d}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 50, normalized size = 1.04 \begin {gather*} \frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 \sqrt {e x +d}\, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 18, normalized size = 0.38 \begin {gather*} \frac {2 \, {\left (c d x + a e\right )}^{\frac {3}{2}}}{3 \, c d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.75, size = 49, normalized size = 1.02 \begin {gather*} \frac {\left (\frac {2\,x}{3}+\frac {2\,a\,e}{3\,c\,d}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{\sqrt {d+e\,x}} \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 {\left (d + e x\right ) \left (a e + c d x\right )}}{\sqrt {d + e x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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