Optimal. Leaf size=43 \[ \frac{2}{5} (d+e x)^{5/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{7/2}}{7 e^2} \]
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Rubi [A] time = 0.0193963, 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} \[ \frac{2}{5} (d+e x)^{5/2} \left (a-\frac{c d^2}{e^2}\right )+\frac{2 c d (d+e x)^{7/2}}{7 e^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+e x} \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)^{3/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right ) (d+e x)^{3/2}}{e}+\frac{c d (d+e x)^{5/2}}{e}\right ) \, dx\\ &=\frac{2}{5} \left (a-\frac{c d^2}{e^2}\right ) (d+e x)^{5/2}+\frac{2 c d (d+e x)^{7/2}}{7 e^2}\\ \end{align*}
Mathematica [A] time = 0.0249812, size = 34, normalized size = 0.79 \[ \frac{2 (d+e x)^{5/2} \left (7 a e^2+c d (5 e x-2 d)\right )}{35 e^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 32, normalized size = 0.7 \begin{align*}{\frac{10\,cdex+14\,a{e}^{2}-4\,c{d}^{2}}{35\,{e}^{2}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976588, size = 51, normalized size = 1.19 \begin{align*} \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} c d - 7 \,{\left (c d^{2} - a e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{35 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.27846, size = 163, normalized size = 3.79 \begin{align*} \frac{2 \,{\left (5 \, c d e^{3} x^{3} - 2 \, c d^{4} + 7 \, a d^{2} e^{2} +{\left (8 \, c d^{2} e^{2} + 7 \, a e^{4}\right )} x^{2} +{\left (c d^{3} e + 14 \, a d e^{3}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.84052, size = 41, normalized size = 0.95 \begin{align*} \frac{2 \left (\frac{c d \left (d + e x\right )^{\frac{7}{2}}}{7 e} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )}{5 e}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13804, size = 157, normalized size = 3.65 \begin{align*} \frac{2}{105} \,{\left (7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} c d^{2} e^{\left (-1\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a d e +{\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 e^{\left (-1\right )} + 7 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a e\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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