Optimal. Leaf size=54 \[ -\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2} \]
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Rubi [A] time = 0.0390207, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 43} \[ -\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(d+e x)^5}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx &=\int \frac{d+e x}{(a e+c d x)^4} \, dx\\ &=\int \left (\frac{c d^2-a e^2}{c d (a e+c d x)^4}+\frac{e}{c d (a e+c d x)^3}\right ) \, dx\\ &=-\frac{c d^2-a e^2}{3 c^2 d^2 (a e+c d x)^3}-\frac{e}{2 c^2 d^2 (a e+c d x)^2}\\ \end{align*}
Mathematica [A] time = 0.0169026, size = 37, normalized size = 0.69 \[ -\frac{a e^2+c d (2 d+3 e x)}{6 c^2 d^2 (a e+c d x)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 51, normalized size = 0.9 \begin{align*} -{\frac{-a{e}^{2}+c{d}^{2}}{3\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{3}}}-{\frac{e}{2\,{c}^{2}{d}^{2} \left ( cdx+ae \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0737, size = 100, normalized size = 1.85 \begin{align*} -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98335, size = 149, normalized size = 2.76 \begin{align*} -\frac{3 \, c d e x + 2 \, c d^{2} + a e^{2}}{6 \,{\left (c^{5} d^{5} x^{3} + 3 \, a c^{4} d^{4} e x^{2} + 3 \, a^{2} c^{3} d^{3} e^{2} x + a^{3} c^{2} d^{2} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.851355, size = 80, normalized size = 1.48 \begin{align*} - \frac{a e^{2} + 2 c d^{2} + 3 c d e x}{6 a^{3} c^{2} d^{2} e^{3} + 18 a^{2} c^{3} d^{3} e^{2} x + 18 a c^{4} d^{4} e x^{2} + 6 c^{5} d^{5} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.5247, size = 865, normalized size = 16.02 \begin{align*} -\frac{3 \, c^{7} d^{13} x^{4} e^{4} + 11 \, c^{7} d^{14} x^{3} e^{3} + 15 \, c^{7} d^{15} x^{2} e^{2} + 9 \, c^{7} d^{16} x e + 2 \, c^{7} d^{17} - 18 \, a c^{6} d^{11} x^{4} e^{6} - 65 \, a c^{6} d^{12} x^{3} e^{5} - 87 \, a c^{6} d^{13} x^{2} e^{4} - 51 \, a c^{6} d^{14} x e^{3} - 11 \, a c^{6} d^{15} e^{2} + 45 \, a^{2} c^{5} d^{9} x^{4} e^{8} + 159 \, a^{2} c^{5} d^{10} x^{3} e^{7} + 207 \, a^{2} c^{5} d^{11} x^{2} e^{6} + 117 \, a^{2} c^{5} d^{12} x e^{5} + 24 \, a^{2} c^{5} d^{13} e^{4} - 60 \, a^{3} c^{4} d^{7} x^{4} e^{10} - 205 \, a^{3} c^{4} d^{8} x^{3} e^{9} - 255 \, a^{3} c^{4} d^{9} x^{2} e^{8} - 135 \, a^{3} c^{4} d^{10} x e^{7} - 25 \, a^{3} c^{4} d^{11} e^{6} + 45 \, a^{4} c^{3} d^{5} x^{4} e^{12} + 145 \, a^{4} c^{3} d^{6} x^{3} e^{11} + 165 \, a^{4} c^{3} d^{7} x^{2} e^{10} + 75 \, a^{4} c^{3} d^{8} x e^{9} + 10 \, a^{4} c^{3} d^{9} e^{8} - 18 \, a^{5} c^{2} d^{3} x^{4} e^{14} - 51 \, a^{5} c^{2} d^{4} x^{3} e^{13} - 45 \, a^{5} c^{2} d^{5} x^{2} e^{12} - 9 \, a^{5} c^{2} d^{6} x e^{11} + 3 \, a^{5} c^{2} d^{7} e^{10} + 3 \, a^{6} c d x^{4} e^{16} + 5 \, a^{6} c d^{2} x^{3} e^{15} - 3 \, a^{6} c d^{3} x^{2} e^{14} - 9 \, a^{6} c d^{4} x e^{13} - 4 \, a^{6} c d^{5} e^{12} + a^{7} x^{3} e^{17} + 3 \, a^{7} d x^{2} e^{16} + 3 \, a^{7} d^{2} x e^{15} + a^{7} d^{3} e^{14}}{6 \,{\left (c^{8} d^{14} - 6 \, a c^{7} d^{12} e^{2} + 15 \, a^{2} c^{6} d^{10} e^{4} - 20 \, a^{3} c^{5} d^{8} e^{6} + 15 \, a^{4} c^{4} d^{6} e^{8} - 6 \, a^{5} c^{3} d^{4} e^{10} + a^{6} c^{2} d^{2} e^{12}\right )}{\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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