Optimal. Leaf size=43 \[ \frac{2}{7} (d+e x)^{7/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{9/2}}{9 e^2} \]
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Rubi [A] time = 0.0199979, antiderivative size = 43, 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 = {626, 43} \[ \frac{2}{7} (d+e x)^{7/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{9/2}}{9 e^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right ) \, dx &=\int (a e+c d x) (d+e x)^{5/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^{5/2}}{e}+\frac{c d (d+e x)^{7/2}}{e}\right ) \, dx\\ &=\frac{2}{7} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^{7/2}+\frac{2 c d (d+e x)^{9/2}}{9 e^2}\\ \end{align*}
Mathematica [A] time = 0.0337937, size = 34, normalized size = 0.79 \[ \frac{2 (d+e x)^{7/2} \left (9 a e^2+c d (7 e x-2 d)\right )}{63 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 32, normalized size = 0.7 \begin{align*}{\frac{14\,cdex+18\,a{e}^{2}-4\,c{d}^{2}}{63\,{e}^{2}} \left ( ex+d \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.985135, size = 51, normalized size = 1.19 \begin{align*} \frac{2 \,{\left (7 \,{\left (e x + d\right )}^{\frac{9}{2}} c d - 9 \,{\left (c d^{2} - a e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}}\right )}}{63 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.33905, size = 213, normalized size = 4.95 \begin{align*} \frac{2 \,{\left (7 \, c d e^{4} x^{4} - 2 \, c d^{5} + 9 \, a d^{3} e^{2} +{\left (19 \, c d^{2} e^{3} + 9 \, a e^{5}\right )} x^{3} + 3 \,{\left (5 \, c d^{3} e^{2} + 9 \, a d e^{4}\right )} x^{2} +{\left (c d^{4} e + 27 \, a d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{63 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.7528, size = 235, normalized size = 5.47 \begin{align*} a d^{2} e \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + 4 a d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + 2 a \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right ) + \frac{2 c d^{3} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 c d^{2} \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 c d \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13888, size = 290, normalized size = 6.74 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} c d^{3} e^{\left (-1\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a d^{2} e + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} c d^{2} e^{\left (-1\right )} + 42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a d e +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} c d e^{\left (-1\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a e\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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