Optimal. Leaf size=48 \[ \frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3 c d (d+e x)^{3/2}} \]
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Rubi [A]
time = 0.01, 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 = {662}
\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 662
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 = 0.02, size = 37, normalized size = 0.77 \begin {gather*} \frac {2 ((a e+c d x) (d+e x))^{3/2}}{3 c d (d+e x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.79, size = 40, normalized size = 0.83
method | result | size |
default | \(\frac {2 \left (c d x +a e \right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3 d c \sqrt {e x +d}}\) | \(40\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3 d c \sqrt {e x +d}}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 0.40 \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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Fricas [A]
time = 1.94, size = 60, normalized size = 1.25 \begin {gather*} \frac {2 \, \sqrt {c d^{2} x + a x e^{2} + {\left (c d x^{2} + a d\right )} e} {\left (c d x + a e\right )} \sqrt {x e + d}}{3 \, {\left (c d x e + c d^{2}\right )}} \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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Giac [A]
time = 1.76, size = 88, normalized size = 1.83 \begin {gather*} \frac {2}{3} \, {\left (\frac {{\left ({\left (x e + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} e^{\left (-1\right )}}{c d} + \frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d}\right )} e^{\left (-2\right )} \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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