Optimal. Leaf size=41 \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}} \]
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Rubi [A] time = 0.0221359, antiderivative size = 41, normalized size of antiderivative = 1., 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} \[ -\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}} \]
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)^{7/2}} \, dx &=\frac{\int \frac{a e^3+c d e^2 x}{(d+e x)^{5/2}} \, dx}{e^2}\\ &=\frac{\int \left (\frac{-c d^2 e+a e^3}{(d+e x)^{5/2}}+\frac{c d e}{(d+e x)^{3/2}}\right ) \, dx}{e^2}\\ &=-\frac{2 \left (a-\frac{c d^2}{e^2}\right )}{3 (d+e x)^{3/2}}-\frac{2 c d}{e^2 \sqrt{d+e x}}\\ \end{align*}
Mathematica [A] time = 0.0207019, size = 33, normalized size = 0.8 \[ -\frac{2 \left (a e^2+c d (2 d+3 e x)\right )}{3 e^2 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 31, normalized size = 0.8 \begin{align*} -{\frac{6\,cdex+2\,a{e}^{2}+4\,c{d}^{2}}{3\,{e}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03301, size = 45, normalized size = 1.1 \begin{align*} -\frac{2 \,{\left (3 \,{\left (e x + d\right )} c d - c d^{2} + a e^{2}\right )}}{3 \,{\left (e x + d\right )}^{\frac{3}{2}} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1431, size = 111, normalized size = 2.71 \begin{align*} -\frac{2 \,{\left (3 \, c d e x + 2 \, c d^{2} + a e^{2}\right )} \sqrt{e x + d}}{3 \,{\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.62371, size = 126, normalized size = 3.07 \begin{align*} \begin{cases} - \frac{2 a e^{2}}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} - \frac{4 c d^{2}}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} - \frac{6 c d e x}{3 d e^{2} \sqrt{d + e x} + 3 e^{3} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c x^{2}}{2 d^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15023, size = 63, normalized size = 1.54 \begin{align*} -\frac{2 \,{\left (3 \,{\left (x e + d\right )}^{2} c d -{\left (x e + d\right )} c d^{2} +{\left (x e + d\right )} a e^{2}\right )} e^{\left (-2\right )}}{3 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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