Optimal. Leaf size=41 \[ 2 \sqrt {d+e x} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{3/2}}{3 e^2} \]
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Rubi [A] time = 0.02, antiderivative size = 41, 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} \[ 2 \sqrt {d+e x} \left (a-\frac {c d^2}{e^2}\right )+\frac {2 c d (d+e x)^{3/2}}{3 e^2} \]
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)^{3/2}} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{\sqrt {d+e x}} \, dx}{e^2}\\ &=\frac {\int \left (\frac {-c d^2 e+a e^3}{\sqrt {d+e x}}+c d e \sqrt {d+e x}\right ) \, dx}{e^2}\\ &=2 \left (a-\frac {c d^2}{e^2}\right ) \sqrt {d+e x}+\frac {2 c d (d+e x)^{3/2}}{3 e^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.80 \[ \frac {2 \sqrt {d+e x} \left (3 a e^2+c d (e x-2 d)\right )}{3 e^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 30, normalized size = 0.73 \[ \frac {2 \, {\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt {e x + d}}{3 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 47, normalized size = 1.15 \[ \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} c d e^{4} - 3 \, \sqrt {x e + d} c d^{2} e^{4} + 3 \, \sqrt {x e + d} a e^{6}\right )} e^{\left (-6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 31, normalized size = 0.76 \[ \frac {2 \sqrt {e x +d}\, \left (c d e x +3 a \,e^{2}-2 c \,d^{2}\right )}{3 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 37, normalized size = 0.90 \[ \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} c d - 3 \, {\left (c d^{2} - a e^{2}\right )} \sqrt {e x + d}\right )}}{3 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 33, normalized size = 0.80 \[ \frac {2\,\sqrt {d+e\,x}\,\left (3\,a\,e^2-3\,c\,d^2+c\,d\,\left (d+e\,x\right )\right )}{3\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.48, size = 124, normalized size = 3.02 \[ \begin {cases} \frac {- \frac {2 a d e}{\sqrt {d + e x}} - 2 a e \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right ) - \frac {2 c d^{2} \left (- \frac {d}{\sqrt {d + e x}} - \sqrt {d + e x}\right )}{e} - \frac {2 c d \left (\frac {d^{2}}{\sqrt {d + e x}} + 2 d \sqrt {d + e x} - \frac {\left (d + e x\right )^{\frac {3}{2}}}{3}\right )}{e}}{e} & \text {for}\: e \neq 0 \\\frac {c \sqrt {d} x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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