Optimal. Leaf size=83 \[ -\frac{4 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^3}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{2 c^2 d^2 (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A] time = 0.0391043, 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} \[ -\frac{4 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^3}+\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{3 e^3}+\frac{2 c^2 d^2 (d+e x)^{7/2}}{7 e^3} \]
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}{(d+e x)^{3/2}} \, dx &=\int (a e+c d x)^2 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^2 \sqrt{d+e x}}{e^2}-\frac{2 c d \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^2}+\frac{c^2 d^2 (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{3 e^3}-\frac{4 c d \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 e^3}+\frac{2 c^2 d^2 (d+e x)^{7/2}}{7 e^3}\\ \end{align*}
Mathematica [A] time = 0.0396866, size = 67, normalized size = 0.81 \[ \frac{2 (d+e x)^{3/2} \left (35 a^2 e^4+14 a c d e^2 (3 e x-2 d)+c^2 d^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 73, normalized size = 0.9 \begin{align*}{\frac{30\,{c}^{2}{d}^{2}{x}^{2}{e}^{2}+84\,acd{e}^{3}x-24\,{c}^{2}{d}^{3}ex+70\,{a}^{2}{e}^{4}-56\,ac{d}^{2}{e}^{2}+16\,{c}^{2}{d}^{4}}{105\,{e}^{3}} \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 = 1.03428, size = 108, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} d^{2} - 42 \,{\left (c^{2} d^{3} - a c d e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92341, size = 236, normalized size = 2.84 \begin{align*} \frac{2 \,{\left (15 \, c^{2} d^{2} e^{3} x^{3} + 8 \, c^{2} d^{5} - 28 \, a c d^{3} e^{2} + 35 \, a^{2} d e^{4} + 3 \,{\left (c^{2} d^{3} e^{2} + 14 \, a c d e^{4}\right )} x^{2} -{\left (4 \, c^{2} d^{4} e - 14 \, a c d^{2} e^{3} - 35 \, a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 50.8447, size = 411, normalized size = 4.95 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d^{2} e^{2}}{\sqrt{d + e x}} + 4 a^{2} d e^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 2 a^{2} 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 ) + 4 a c d^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 8 a c d^{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 ) + 4 a c d \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 ) + \frac{2 c^{2} d^{4} \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{4 c^{2} d^{3} \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{2 c^{2} d^{2} \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}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{2} d^{\frac{5}{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 [A] time = 1.20345, size = 143, normalized size = 1.72 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} d^{2} e^{18} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d^{3} e^{18} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{4} e^{18} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} a c d e^{20} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c d^{2} e^{20} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} e^{22}\right )} e^{\left (-21\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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