Optimal. Leaf size=13 \[ -\frac {c}{e (d+e x)} \]
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Rubi [A] time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {24, 21, 32} \[ -\frac {c}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 21
Rule 24
Rule 32
Rubi steps
\begin {align*} \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^4} \, dx &=\frac {\int \frac {c d e^2+c e^3 x}{(d+e x)^3} \, dx}{e^2}\\ &=c \int \frac {1}{(d+e x)^2} \, dx\\ &=-\frac {c}{e (d+e x)}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {c}{e (d+e x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 14, normalized size = 1.08 \[ -\frac {c}{e^{2} x + d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 34, normalized size = 2.62 \[ -\frac {{\left (c x^{2} e^{4} + 2 \, c d x e^{3} + c d^{2} e^{2}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 14, normalized size = 1.08 \[ -\frac {c}{\left (e x +d \right ) e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 14, normalized size = 1.08 \[ -\frac {c}{e^{2} x + d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 13, normalized size = 1.00 \[ -\frac {c}{e\,\left (d+e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 10, normalized size = 0.77 \[ - \frac {c}{d e + e^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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