Optimal. Leaf size=15 \[ -\frac {c}{2 e (d+e x)^2} \]
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Rubi [A] time = 0.01, antiderivative size = 15, 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} \begin {gather*} -\frac {c}{2 e (d+e x)^2} \end {gather*}
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)^5} \, dx &=\frac {\int \frac {c d e^2+c e^3 x}{(d+e x)^4} \, dx}{e^2}\\ &=c \int \frac {1}{(d+e x)^3} \, dx\\ &=-\frac {c}{2 e (d+e x)^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} -\frac {c}{2 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c d^2+2 c d e x+c e^2 x^2}{(d+e x)^5} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.37, size = 25, normalized size = 1.67 \begin {gather*} -\frac {c}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 13, normalized size = 0.87 \begin {gather*} -\frac {c e^{\left (-1\right )}}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.93 \begin {gather*} -\frac {c}{2 \left (e x +d \right )^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 25, normalized size = 1.67 \begin {gather*} -\frac {c}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.40, size = 24, normalized size = 1.60 \begin {gather*} -\frac {c}{2\,e\,\left (d^2+2\,d\,e\,x+e^2\,x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.22, size = 26, normalized size = 1.73 \begin {gather*} - \frac {c}{2 d^{2} e + 4 d e^{2} x + 2 e^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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