Optimal. Leaf size=119 \[ -\frac{6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac{6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac{2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]
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Rubi [A] time = 0.051909, antiderivative size = 119, 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} \[ -\frac{6 c^2 d^2 (d+e x)^{7/2} \left (c d^2-a e^2\right )}{7 e^4}+\frac{6 c d (d+e x)^{5/2} \left (c d^2-a e^2\right )^2}{5 e^4}-\frac{2 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3}{3 e^4}+\frac{2 c^3 d^3 (d+e x)^{9/2}}{9 e^4} \]
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)^{5/2}} \, dx &=\int (a e+c d x)^3 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{\left (-c d^2+a e^2\right )^3 \sqrt{d+e x}}{e^3}+\frac{3 c d \left (c d^2-a e^2\right )^2 (d+e x)^{3/2}}{e^3}-\frac{3 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{5/2}}{e^3}+\frac{c^3 d^3 (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac{2 \left (c d^2-a e^2\right )^3 (d+e x)^{3/2}}{3 e^4}+\frac{6 c d \left (c d^2-a e^2\right )^2 (d+e x)^{5/2}}{5 e^4}-\frac{6 c^2 d^2 \left (c d^2-a e^2\right ) (d+e x)^{7/2}}{7 e^4}+\frac{2 c^3 d^3 (d+e x)^{9/2}}{9 e^4}\\ \end{align*}
Mathematica [A] time = 0.0666193, size = 98, normalized size = 0.82 \[ \frac{2 (d+e x)^{3/2} \left (-135 c^2 d^2 (d+e x)^2 \left (c d^2-a e^2\right )+189 c d (d+e x) \left (c d^2-a e^2\right )^2-105 \left (c d^2-a e^2\right )^3+35 c^3 d^3 (d+e x)^3\right )}{315 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 131, normalized size = 1.1 \begin{align*}{\frac{70\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+270\,a{c}^{2}{d}^{2}{e}^{4}{x}^{2}-60\,{c}^{3}{d}^{4}{e}^{2}{x}^{2}+378\,{a}^{2}cd{e}^{5}x-216\,a{c}^{2}{d}^{3}{e}^{3}x+48\,{c}^{3}{d}^{5}ex+210\,{a}^{3}{e}^{6}-252\,{a}^{2}c{d}^{2}{e}^{4}+144\,a{c}^{2}{d}^{4}{e}^{2}-32\,{c}^{3}{d}^{6}}{315\,{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 = 1.0738, size = 185, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} c^{3} d^{3} - 135 \,{\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\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{5}{2}} - 105 \,{\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 )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8748, size = 383, normalized size = 3.22 \begin{align*} \frac{2 \,{\left (35 \, c^{3} d^{3} e^{4} x^{4} - 16 \, c^{3} d^{7} + 72 \, a c^{2} d^{5} e^{2} - 126 \, a^{2} c d^{3} e^{4} + 105 \, a^{3} d e^{6} + 5 \,{\left (c^{3} d^{4} e^{3} + 27 \, a c^{2} d^{2} e^{5}\right )} x^{3} - 3 \,{\left (2 \, c^{3} d^{5} e^{2} - 9 \, a c^{2} d^{3} e^{4} - 63 \, a^{2} c d e^{6}\right )} x^{2} +{\left (8 \, c^{3} d^{6} e - 36 \, a c^{2} d^{4} e^{3} + 63 \, a^{2} c d^{2} e^{5} + 105 \, a^{3} e^{7}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 102.341, size = 644, normalized size = 5.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20444, size = 250, normalized size = 2.1 \begin{align*} \frac{2}{315} \,{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} c^{3} d^{3} e^{32} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{3} d^{4} e^{32} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{3} d^{5} e^{32} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{3} d^{6} e^{32} + 135 \,{\left (x e + d\right )}^{\frac{7}{2}} a c^{2} d^{2} e^{34} - 378 \,{\left (x e + d\right )}^{\frac{5}{2}} a c^{2} d^{3} e^{34} + 315 \,{\left (x e + d\right )}^{\frac{3}{2}} a c^{2} d^{4} e^{34} + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} a^{2} c d e^{36} - 315 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} c d^{2} e^{36} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} e^{38}\right )} e^{\left (-36\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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