Optimal. Leaf size=115 \[ -\frac{6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac{2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.0545124, antiderivative size = 115, 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} \[ -\frac{6 c^2 d^2 (d+e x)^{5/2} \left (c d^2-a e^2\right )}{5 e^4}+\frac{2 c d (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}{e^4}-\frac{2 \sqrt{d+e x} \left (c d^2-a e^2\right )^3}{e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4} \]
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 )^3}{(d+e x)^{7/2}} \, dx &=\int \frac{(a e+c d x)^3}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 \sqrt{d+e x}}+\frac{3 c d \left (c d^2-a e^2\right )^2 \sqrt{d+e x}}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{3/2}}{e^3}+\frac{c^3 d^3 (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^3 \sqrt{d+e x}}{e^4}+\frac{2 c d \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{e^4}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{5 e^4}+\frac{2 c^3 d^3 (d+e x)^{7/2}}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.0695736, size = 110, normalized size = 0.96 \[ \frac{2 \sqrt{d+e x} \left (35 a^2 c d e^4 (e x-2 d)+35 a^3 e^6+7 a c^2 d^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+c^3 d^3 \left (8 d^2 e x-16 d^3-6 d e^2 x^2+5 e^3 x^3\right )\right )}{35 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 131, normalized size = 1.1 \begin{align*}{\frac{10\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+42\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-12\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+70\,{a}^{2}cd{e}^{5}x-56\,a{c}^{2}{d}^{3}{e}^{3}x+16\,{c}^{3}{d}^{5}ex+70\,{a}^{3}{e}^{6}-140\,{a}^{2}c{d}^{2}{e}^{4}+112\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{35\,{e}^{4}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01014, size = 185, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{3} d^{3} - 21 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \sqrt{e x + d}\right )}}{35 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94957, size = 275, normalized size = 2.39 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 56 \, a c^{2} d^{4} e^{2} - 70 \, a^{2} c d^{2} e^{4} + 35 \, a^{3} e^{6} - 3 \,{\left (2 \, c^{3} d^{4} e^{2} - 7 \, a c^{2} d^{2} e^{4}\right )} x^{2} +{\left (8 \, c^{3} d^{5} e - 28 \, a c^{2} d^{3} e^{3} + 35 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 138.656, size = 376, normalized size = 3.27 \begin{align*} \begin{cases} - \frac{\frac{2 a^{3} d e^{3}}{\sqrt{d + e x}} + 2 a^{3} e^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} c d^{2} e \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + 6 a^{2} c d e \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 ) + \frac{6 a c^{2} 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 )}{e} + \frac{6 a c^{2} 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 )}{e} + \frac{2 c^{3} 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^{3}} + \frac{2 c^{3} 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^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{c^{3} d^{\frac{5}{2}} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2541, size = 250, normalized size = 2.17 \begin{align*} \frac{2}{35} \,{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{3} e^{24} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{4} e^{24} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{5} e^{24} - 35 \, \sqrt{x e + d} c^{3} d^{6} e^{24} + 21 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} d^{2} e^{26} - 70 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d^{3} e^{26} + 105 \, \sqrt{x e + d} a c^{2} d^{4} e^{26} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} c d e^{28} - 105 \, \sqrt{x e + d} a^{2} c d^{2} e^{28} + 35 \, \sqrt{x e + d} a^{3} e^{30}\right )} e^{\left (-28\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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