Optimal. Leaf size=39 \[ -\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}}+\frac {2 c d \sqrt {d+e x}}{e^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {24, 45}
\begin {gather*} \frac {2 c d \sqrt {d+e x}}{e^2}-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 24
Rule 45
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^{5/2}} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^{3/2}} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-c d^2 e+a e^3}{(d+e x)^{3/2}}+\frac {c d e}{\sqrt {d+e x}}\right ) \, dx}{e^2}\\ &=-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}}+\frac {2 c d \sqrt {d+e x}}{e^2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 31, normalized size = 0.79 \begin {gather*} \frac {-2 a e^2+2 c d (2 d+e x)}{e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 38, normalized size = 0.97
method | result | size |
gosper | \(-\frac {2 \left (-c d e x +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\) | \(31\) |
trager | \(-\frac {2 \left (-c d e x +e^{2} a -2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}}\) | \(31\) |
derivativedivides | \(\frac {2 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{2}}\) | \(38\) |
default | \(\frac {2 c d \sqrt {e x +d}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}}{e^{2}}\) | \(38\) |
risch | \(\frac {2 c d \sqrt {e x +d}}{e^{2}}-\frac {2 \left (e^{2} a -c \,d^{2}\right )}{e^{2} \sqrt {e x +d}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 40, normalized size = 1.03 \begin {gather*} 2 \, {\left (\sqrt {x e + d} c d e^{\left (-1\right )} + \frac {{\left (c d^{2} - a e^{2}\right )} e^{\left (-1\right )}}{\sqrt {x e + d}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.78, size = 39, normalized size = 1.00 \begin {gather*} \frac {2 \, {\left (c d x e + 2 \, c d^{2} - a e^{2}\right )} \sqrt {x e + d}}{x e^{3} + d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.42, size = 58, normalized size = 1.49 \begin {gather*} \begin {cases} - \frac {2 a}{\sqrt {d + e x}} + \frac {4 c d^{2}}{e^{2} \sqrt {d + e x}} + \frac {2 c d x}{e \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c x^{2}}{2 \sqrt {d}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.52, size = 38, normalized size = 0.97 \begin {gather*} 2 \, \sqrt {x e + d} c d e^{\left (-2\right )} + \frac {2 \, {\left (c d^{2} - a e^{2}\right )} e^{\left (-2\right )}}{\sqrt {x e + d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 30, normalized size = 0.77 \begin {gather*} \frac {4\,c\,d^2+2\,c\,x\,d\,e-2\,a\,e^2}{e^2\,\sqrt {d+e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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