Optimal. Leaf size=83 \[ -\frac{4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \]
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Rubi [A] time = 0.0389183, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {626, 43} \[ -\frac{4 c d (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^3}+\frac{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{\sqrt{d+e x}} \, dx &=\int (a e+c d x)^2 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2 (d+e x)^{3/2}}{e^2}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{e^2}+\frac{c^2 d^2 (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^3}-\frac{4 c d \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^3}+\frac{2 c^2 d^2 (d+e x)^{9/2}}{9 e^3}\\ \end{align*}
Mathematica [A] time = 0.043057, size = 67, normalized size = 0.81 \[ \frac{2 (d+e x)^{5/2} \left (63 a^2 e^4+18 a c d e^2 (5 e x-2 d)+c^2 d^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 73, normalized size = 0.9 \begin{align*}{\frac{70\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+180\,acd{e}^{3}x-40\,{c}^{2}{d}^{3}ex+126\,{a}^{2}{e}^{4}-72\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{315\,{e}^{3}} \left ( ex+d \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.00856, size = 378, normalized size = 4.55 \begin{align*} \frac{2 \,{\left (315 \, \sqrt{e x + d} a^{2} d^{2} e^{2} + 42 \,{\left (\frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} c d}{e} + \frac{5 \,{\left (c d^{2} + a e^{2}\right )}{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )}}{e}\right )} a d e + \frac{{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} - 180 \,{\left (e x + d\right )}^{\frac{7}{2}} d + 378 \,{\left (e x + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{e x + d} d^{4}\right )} c^{2} d^{2}}{e^{2}} + \frac{18 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} - 21 \,{\left (e x + d\right )}^{\frac{5}{2}} d + 35 \,{\left (e x + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{e x + d} d^{3}\right )}{\left (c d^{2} + a e^{2}\right )} c d}{e^{2}} + \frac{21 \,{\left (c d^{2} + a e^{2}\right )}^{2}{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )}}{e^{2}}\right )}}{315 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86642, size = 316, normalized size = 3.81 \begin{align*} \frac{2 \,{\left (35 \, c^{2} d^{2} e^{4} x^{4} + 8 \, c^{2} d^{6} - 36 \, a c d^{4} e^{2} + 63 \, a^{2} d^{2} e^{4} + 10 \,{\left (5 \, c^{2} d^{3} e^{3} + 9 \, a c d e^{5}\right )} x^{3} + 3 \,{\left (c^{2} d^{4} e^{2} + 48 \, a c d^{2} e^{4} + 21 \, a^{2} e^{6}\right )} x^{2} - 2 \,{\left (2 \, c^{2} d^{5} e - 9 \, a c d^{3} e^{3} - 63 \, a^{2} d e^{5}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 85.4955, size = 631, normalized size = 7.6 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d^{3} e^{2}}{\sqrt{d + e x}} + 6 a^{2} d^{2} e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} d e^{2} \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 ) + 2 a^{2} e^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + 4 a c d^{4} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 12 a c d^{3} \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 ) + 12 a c d^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right ) + 4 a c d \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right ) + \frac{2 c^{2} d^{5} \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^{2}} + \frac{6 c^{2} d^{4} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{6 c^{2} d^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 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^{2} d^{2} \left (- \frac{d^{5}}{\sqrt{d + e x}} - 5 d^{4} \sqrt{d + e x} + \frac{10 d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} - 2 d^{2} \left (d + e x\right )^{\frac{5}{2}} + \frac{5 d \left (d + e x\right )^{\frac{7}{2}}}{7} - \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{7}{2}} x^{3}}{3} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15654, size = 524, normalized size = 6.31 \begin{align*} \frac{2}{315} \,{\left (21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} c^{2} d^{4} e^{\left (-2\right )} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} c^{2} d^{3} e^{\left (-2\right )} + 210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a c d^{3} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 180 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 378 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 420 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3} + 315 \, \sqrt{x e + d} d^{4}\right )} c^{2} d^{2} e^{\left (-2\right )} + 315 \, \sqrt{x e + d} a^{2} d^{2} e^{2} + 84 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a c d^{2} + 210 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} d e^{2} + 18 \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} a c d + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} a^{2} e^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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