Optimal. Leaf size=24 \[ -\frac {(d+e x)^{m-3}}{c^2 e (3-m)} \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {27, 12, 32} \[ -\frac {(d+e x)^{m-3}}{c^2 e (3-m)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^{-4+m}}{c^2} \, dx\\ &=\frac {\int (d+e x)^{-4+m} \, dx}{c^2}\\ &=-\frac {(d+e x)^{-3+m}}{c^2 e (3-m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.88 \[ \frac {(d+e x)^{m-3}}{c^2 e (m-3)} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.04, size = 100, normalized size = 4.17 \[ \frac {{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 3 \, c^{2} d^{3} e + {\left (c^{2} e^{4} m - 3 \, c^{2} e^{4}\right )} x^{3} + 3 \, {\left (c^{2} d e^{3} m - 3 \, c^{2} d e^{3}\right )} x^{2} + 3 \, {\left (c^{2} d^{2} e^{2} m - 3 \, c^{2} d^{2} e^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{m}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 40, normalized size = 1.67 \[ \frac {\left (e x +d \right )^{m -1}}{\left (e^{2} x^{2}+2 d x e +d^{2}\right ) \left (m -3\right ) c^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 65, normalized size = 2.71 \[ \frac {{\left (e x + d\right )}^{m}}{c^{2} e^{4} {\left (m - 3\right )} x^{3} + 3 \, c^{2} d e^{3} {\left (m - 3\right )} x^{2} + 3 \, c^{2} d^{2} e^{2} {\left (m - 3\right )} x + c^{2} d^{3} e {\left (m - 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 50, normalized size = 2.08 \[ \frac {{\left (d+e\,x\right )}^m}{c^2\,e^4\,\left (m-3\right )\,\left (x^3+\frac {d^3}{e^3}+\frac {3\,d\,x^2}{e}+\frac {3\,d^2\,x}{e^2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.13, size = 136, normalized size = 5.67 \[ \begin {cases} \frac {x}{c^{2} d} & \text {for}\: e = 0 \wedge m = 3 \\\frac {d^{m} x}{c^{2} d^{4}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c^{2} e} & \text {for}\: m = 3 \\\frac {\left (d + e x\right )^{m}}{c^{2} d^{3} e m - 3 c^{2} d^{3} e + 3 c^{2} d^{2} e^{2} m x - 9 c^{2} d^{2} e^{2} x + 3 c^{2} d e^{3} m x^{2} - 9 c^{2} d e^{3} x^{2} + c^{2} e^{4} m x^{3} - 3 c^{2} e^{4} x^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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