Optimal. Leaf size=73 \[ \frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0262825, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {626, 45, 37} \[ \frac{c d (a e+c d x)^4}{20 (d+e x)^4 \left (c d^2-a e^2\right )^2}+\frac{(a e+c d x)^4}{5 (d+e x)^5 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 626
Rule 45
Rule 37
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)^9} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^6} \, dx\\ &=\frac{(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{(c d) \int \frac{(a e+c d x)^3}{(d+e x)^5} \, dx}{5 \left (c d^2-a e^2\right )}\\ &=\frac{(a e+c d x)^4}{5 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac{c d (a e+c d x)^4}{20 \left (c d^2-a e^2\right )^2 (d+e x)^4}\\ \end{align*}
Mathematica [A] time = 0.0406616, size = 103, normalized size = 1.41 \[ -\frac{3 a^2 c d e^4 (d+5 e x)+4 a^3 e^6+2 a c^2 d^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+c^3 d^3 \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )}{20 e^4 (d+e x)^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 141, normalized size = 1.9 \begin{align*} -{\frac{{c}^{3}{d}^{3}}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{3\,cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{3}{e}^{6}-3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{c}^{2}{d}^{4}{e}^{2}-{c}^{3}{d}^{6}}{5\,{e}^{4} \left ( ex+d \right ) ^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.12977, size = 236, normalized size = 3.23 \begin{align*} -\frac{10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.50647, size = 352, normalized size = 4.82 \begin{align*} -\frac{10 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + 2 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + 4 \, a^{3} e^{6} + 10 \,{\left (c^{3} d^{4} e^{2} + 2 \, a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \,{\left (c^{3} d^{5} e + 2 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x}{20 \,{\left (e^{9} x^{5} + 5 \, d e^{8} x^{4} + 10 \, d^{2} e^{7} x^{3} + 10 \, d^{3} e^{6} x^{2} + 5 \, d^{4} e^{5} x + d^{5} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.4928, size = 185, normalized size = 2.53 \begin{align*} - \frac{4 a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 2 a c^{2} d^{4} e^{2} + c^{3} d^{6} + 10 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (20 a c^{2} d^{2} e^{4} + 10 c^{3} d^{4} e^{2}\right ) + x \left (15 a^{2} c d e^{5} + 10 a c^{2} d^{3} e^{3} + 5 c^{3} d^{5} e\right )}{20 d^{5} e^{4} + 100 d^{4} e^{5} x + 200 d^{3} e^{6} x^{2} + 200 d^{2} e^{7} x^{3} + 100 d e^{8} x^{4} + 20 e^{9} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21506, size = 378, normalized size = 5.18 \begin{align*} -\frac{{\left (10 \, c^{3} d^{3} x^{6} e^{6} + 40 \, c^{3} d^{4} x^{5} e^{5} + 65 \, c^{3} d^{5} x^{4} e^{4} + 56 \, c^{3} d^{6} x^{3} e^{3} + 28 \, c^{3} d^{7} x^{2} e^{2} + 8 \, c^{3} d^{8} x e + c^{3} d^{9} + 20 \, a c^{2} d^{2} x^{5} e^{7} + 70 \, a c^{2} d^{3} x^{4} e^{6} + 92 \, a c^{2} d^{4} x^{3} e^{5} + 56 \, a c^{2} d^{5} x^{2} e^{4} + 16 \, a c^{2} d^{6} x e^{3} + 2 \, a c^{2} d^{7} e^{2} + 15 \, a^{2} c d x^{4} e^{8} + 48 \, a^{2} c d^{2} x^{3} e^{7} + 54 \, a^{2} c d^{3} x^{2} e^{6} + 24 \, a^{2} c d^{4} x e^{5} + 3 \, a^{2} c d^{5} e^{4} + 4 \, a^{3} x^{3} e^{9} + 12 \, a^{3} d x^{2} e^{8} + 12 \, a^{3} d^{2} x e^{7} + 4 \, a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{20 \,{\left (x e + d\right )}^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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