Optimal. Leaf size=35 \[ -\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]
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Rubi [A] time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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} \[ -\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 626
Rubi steps
\begin {align*} \int \frac {(d+e x)^4}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac {d+e x}{(a e+c d x)^3} \, dx\\ &=-\frac {(d+e x)^2}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 1.00 \[ -\frac {a e^2+c d (d+2 e x)}{2 c^2 d^2 (a e+c d x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 56, normalized size = 1.60 \[ -\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.70, size = 377, normalized size = 10.77 \[ -\frac {2 \, c^{5} d^{9} x^{3} e^{3} + 5 \, c^{5} d^{10} x^{2} e^{2} + 4 \, c^{5} d^{11} x e + c^{5} d^{12} - 8 \, a c^{4} d^{7} x^{3} e^{5} - 19 \, a c^{4} d^{8} x^{2} e^{4} - 14 \, a c^{4} d^{9} x e^{3} - 3 \, a c^{4} d^{10} e^{2} + 12 \, a^{2} c^{3} d^{5} x^{3} e^{7} + 26 \, a^{2} c^{3} d^{6} x^{2} e^{6} + 16 \, a^{2} c^{3} d^{7} x e^{5} + 2 \, a^{2} c^{3} d^{8} e^{4} - 8 \, a^{3} c^{2} d^{3} x^{3} e^{9} - 14 \, a^{3} c^{2} d^{4} x^{2} e^{8} - 4 \, a^{3} c^{2} d^{5} x e^{7} + 2 \, a^{3} c^{2} d^{6} e^{6} + 2 \, a^{4} c d x^{3} e^{11} + a^{4} c d^{2} x^{2} e^{10} - 4 \, a^{4} c d^{3} x e^{9} - 3 \, a^{4} c d^{4} e^{8} + a^{5} x^{2} e^{12} + 2 \, a^{5} d x e^{11} + a^{5} d^{2} e^{10}}{2 \, {\left (c^{6} d^{10} - 4 \, a c^{5} d^{8} e^{2} + 6 \, a^{2} c^{4} d^{6} e^{4} - 4 \, a^{3} c^{3} d^{4} e^{6} + a^{4} c^{2} d^{2} e^{8}\right )} {\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 51, normalized size = 1.46 \[ -\frac {e}{\left (c d x +a e \right ) c^{2} d^{2}}-\frac {-a \,e^{2}+c \,d^{2}}{2 \left (c d x +a e \right )^{2} c^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 56, normalized size = 1.60 \[ -\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 43, normalized size = 1.23 \[ -\frac {\frac {1}{2\,c}-\frac {x^2}{2\,a}}{a^2\,e^2+2\,a\,c\,d\,e\,x+c^2\,d^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.38, size = 60, normalized size = 1.71 \[ \frac {- a e^{2} - c d^{2} - 2 c d e x}{2 a^{2} c^{2} d^{2} e^{2} + 4 a c^{3} d^{3} e x + 2 c^{4} d^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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