Optimal. Leaf size=35 \[ \frac {(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]
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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 = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {24, 37} \[ \frac {(a e+c d x)^2}{2 (d+e x)^2 \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 24
Rule 37
Rubi steps
\begin {align*} \int \frac {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^4} \, dx &=\frac {\int \frac {a e^3+c d e^2 x}{(d+e x)^3} \, dx}{e^2}\\ &=\frac {(a e+c d x)^2}{2 \left (c d^2-a e^2\right ) (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 0.83 \[ -\frac {a e^2+c d (d+2 e x)}{2 e^2 (d+e x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 43, normalized size = 1.23 \[ -\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 1.31 \[ -\frac {{\left (2 \, c d x^{2} e^{2} + 3 \, c d^{2} x e + c d^{3} + a x e^{3} + a d e^{2}\right )} e^{\left (-2\right )}}{2 \, {\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.14 \[ -\frac {c d}{\left (e x +d \right ) e^{2}}-\frac {a \,e^{2}-c \,d^{2}}{2 \left (e x +d \right )^{2} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 43, normalized size = 1.23 \[ -\frac {2 \, c d e x + c d^{2} + a e^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 30, normalized size = 0.86 \[ -\frac {\frac {a}{2}-\frac {c\,x^2}{2}}{d^2+2\,d\,e\,x+e^2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 44, normalized size = 1.26 \[ \frac {- a e^{2} - c d^{2} - 2 c d e x}{2 d^{2} e^{2} + 4 d e^{3} x + 2 e^{4} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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