Optimal. Leaf size=115 \[ -\frac{6 c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}+\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.0520726, 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} \[ -\frac{6 c^2 d^2 \sqrt{d+e x} \left (c d^2-a e^2\right )}{e^4}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}+\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4} \]
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)^{11/2}} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^{5/2}} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{5/2}}+\frac{3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{3/2}}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 \sqrt{d+e x}}+\frac{c^3 d^3 \sqrt{d+e x}}{e^3}\right ) \, dx\\ &=\frac{2 \left (c d^2-a e^2\right )^3}{3 e^4 (d+e x)^{3/2}}-\frac{6 c d \left (c d^2-a e^2\right )^2}{e^4 \sqrt{d+e x}}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{e^4}+\frac{2 c^3 d^3 (d+e x)^{3/2}}{3 e^4}\\ \end{align*}
Mathematica [A] time = 0.0598344, size = 110, normalized size = 0.96 \[ -\frac{2 \left (3 a^2 c d e^4 (2 d+3 e x)+a^3 e^6-3 a c^2 d^2 e^2 \left (8 d^2+12 d e x+3 e^2 x^2\right )+c^3 d^3 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 130, normalized size = 1.1 \begin{align*} -{\frac{-2\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}-18\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}+12\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+18\,{a}^{2}cd{e}^{5}x-72\,a{c}^{2}{d}^{3}{e}^{3}x+48\,{c}^{3}{d}^{5}ex+2\,{a}^{3}{e}^{6}+12\,{a}^{2}c{d}^{2}{e}^{4}-48\,a{c}^{2}{d}^{4}{e}^{2}+32\,{c}^{3}{d}^{6}}{3\,{e}^{4}} \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 = 1.01749, size = 190, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}} c^{3} d^{3} - 9 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 9 \,{\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 )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.87473, size = 305, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (c^{3} d^{3} e^{3} x^{3} - 16 \, c^{3} d^{6} + 24 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} - 3 \,{\left (2 \, c^{3} d^{4} e^{2} - 3 \, a c^{2} d^{2} e^{4}\right )} x^{2} - 3 \,{\left (8 \, c^{3} d^{5} e - 12 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 31.4873, size = 450, normalized size = 3.91 \begin{align*} \begin{cases} - \frac{2 a^{3} e^{6}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 a^{2} c d^{2} e^{4}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{18 a^{2} c d e^{5} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 a c^{2} d^{4} e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{72 a c^{2} d^{3} e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{18 a c^{2} d^{2} e^{4} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{32 c^{3} d^{6}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{48 c^{3} d^{5} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 c^{3} d^{4} e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{2 c^{3} d^{3} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{c^{3} \sqrt{d} 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 = 1.19244, size = 261, normalized size = 2.27 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{3} e^{8} - 9 \, \sqrt{x e + d} c^{3} d^{4} e^{8} + 9 \, \sqrt{x e + d} a c^{2} d^{2} e^{10}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (9 \,{\left (x e + d\right )}^{4} c^{3} d^{5} -{\left (x e + d\right )}^{3} c^{3} d^{6} - 18 \,{\left (x e + d\right )}^{4} a c^{2} d^{3} e^{2} + 3 \,{\left (x e + d\right )}^{3} a c^{2} d^{4} e^{2} + 9 \,{\left (x e + d\right )}^{4} a^{2} c d e^{4} - 3 \,{\left (x e + d\right )}^{3} a^{2} c d^{2} e^{4} +{\left (x e + d\right )}^{3} a^{3} e^{6}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{9}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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