Optimal. Leaf size=35 \[ \frac{(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0152227, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 37} \[ \frac{(a e+c d x)^4}{4 (d+e x)^4 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 626
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)^8} \, dx &=\int \frac{(a e+c d x)^3}{(d+e x)^5} \, dx\\ &=\frac{(a e+c d x)^4}{4 \left (c d^2-a e^2\right ) (d+e x)^4}\\ \end{align*}
Mathematica [B] time = 0.0397116, size = 100, normalized size = 2.86 \[ -\frac{a^2 c d e^4 (d+4 e x)+a^3 e^6+a c^2 d^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+c^3 d^3 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )}{4 e^4 (d+e x)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 141, normalized size = 4. \begin{align*} -{\frac{3\,{c}^{2}{d}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) }{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}-{\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}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{3}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{cd \left ({a}^{2}{e}^{4}-2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4} \right ) }{{e}^{4} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13931, size = 213, normalized size = 6.09 \begin{align*} -\frac{4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.52173, size = 308, normalized size = 8.8 \begin{align*} -\frac{4 \, c^{3} d^{3} e^{3} x^{3} + c^{3} d^{6} + a c^{2} d^{4} e^{2} + a^{2} c d^{2} e^{4} + a^{3} e^{6} + 6 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 4 \,{\left (c^{3} d^{5} e + a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x}{4 \,{\left (e^{8} x^{4} + 4 \, d e^{7} x^{3} + 6 \, d^{2} e^{6} x^{2} + 4 \, d^{3} e^{5} x + d^{4} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.7249, size = 168, normalized size = 4.8 \begin{align*} - \frac{a^{3} e^{6} + a^{2} c d^{2} e^{4} + a c^{2} d^{4} e^{2} + c^{3} d^{6} + 4 c^{3} d^{3} e^{3} x^{3} + x^{2} \left (6 a c^{2} d^{2} e^{4} + 6 c^{3} d^{4} e^{2}\right ) + x \left (4 a^{2} c d e^{5} + 4 a c^{2} d^{3} e^{3} + 4 c^{3} d^{5} e\right )}{4 d^{4} e^{4} + 16 d^{3} e^{5} x + 24 d^{2} e^{6} x^{2} + 16 d e^{7} x^{3} + 4 e^{8} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18985, size = 373, normalized size = 10.66 \begin{align*} -\frac{{\left (4 \, c^{3} d^{3} x^{6} e^{6} + 18 \, c^{3} d^{4} x^{5} e^{5} + 34 \, c^{3} d^{5} x^{4} e^{4} + 35 \, c^{3} d^{6} x^{3} e^{3} + 21 \, c^{3} d^{7} x^{2} e^{2} + 7 \, c^{3} d^{8} x e + c^{3} d^{9} + 6 \, a c^{2} d^{2} x^{5} e^{7} + 22 \, a c^{2} d^{3} x^{4} e^{6} + 31 \, a c^{2} d^{4} x^{3} e^{5} + 21 \, a c^{2} d^{5} x^{2} e^{4} + 7 \, a c^{2} d^{6} x e^{3} + a c^{2} d^{7} e^{2} + 4 \, a^{2} c d x^{4} e^{8} + 13 \, a^{2} c d^{2} x^{3} e^{7} + 15 \, a^{2} c d^{3} x^{2} e^{6} + 7 \, a^{2} c d^{4} x e^{5} + a^{2} c d^{5} e^{4} + a^{3} x^{3} e^{9} + 3 \, a^{3} d x^{2} e^{8} + 3 \, a^{3} d^{2} x e^{7} + a^{3} d^{3} e^{6}\right )} e^{\left (-4\right )}}{4 \,{\left (x e + d\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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