Optimal. Leaf size=39 \[ \frac {2 c d \sqrt {d+e x}}{e^2}-\frac {2 \left (a-\frac {c d^2}{e^2}\right )}{\sqrt {d+e x}} \]
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Rubi [A] time = 0.02, 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, 43} \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 43
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 = 0.02, size = 31, normalized size = 0.79 \begin {gather*} \frac {2 c d (2 d+e x)-2 a e^2}{e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 34, normalized size = 0.87 \begin {gather*} \frac {2 \left (-a e^2+c d^2+c d (d+e x)\right )}{e^2 \sqrt {d+e x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 40, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} \sqrt {e x + d}}{e^{3} x + d e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 50, normalized size = 1.28 \begin {gather*} 2 \, \sqrt {x e + d} c d e^{\left (-2\right )} + \frac {2 \, {\left ({\left (x e + d\right )} c d^{2} - {\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.79 \begin {gather*} -\frac {2 \left (-c d e x +a \,e^{2}-2 c \,d^{2}\right )}{\sqrt {e x +d}\, e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 42, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left (\frac {\sqrt {e x + d} c d}{e} + \frac {c d^{2} - a e^{2}}{\sqrt {e x + d} e}\right )}}{e} \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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sympy [A] time = 1.42, 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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