Optimal. Leaf size=24 \[ -\frac {(d+e x)^{m-1}}{c e (1-m)} \]
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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} \begin {gather*} -\frac {(d+e x)^{m-1}}{c e (1-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 32
Rubi steps
\begin {align*} \int \frac {(d+e x)^m}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac {(d+e x)^{-2+m}}{c} \, dx\\ &=\frac {\int (d+e x)^{-2+m} \, dx}{c}\\ &=-\frac {(d+e x)^{-1+m}}{c e (1-m)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{m-1}}{c e (m-1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^m}{c d^2+2 c d e x+c e^2 x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.41, size = 36, normalized size = 1.50 \begin {gather*} \frac {{\left (e x + d\right )}^{m}}{c d e m - c d e + {\left (c e^{2} m - c e^{2}\right )} x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (e x + d\right )}^{m}}{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 0.92 \begin {gather*} \frac {\left (e x +d \right )^{m -1}}{\left (m -1\right ) c e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 27, normalized size = 1.12 \begin {gather*} \frac {{\left (e x + d\right )}^{m}}{c e^{2} {\left (m - 1\right )} x + c d e {\left (m - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (d+e\,x\right )}^m}{c\,e^2\,\left (x+\frac {d}{e}\right )\,\left (m-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 63, normalized size = 2.62 \begin {gather*} \begin {cases} \text {NaN} & \text {for}\: d = 0 \wedge e = 0 \wedge m = 1 \\0^{m} \tilde {\infty } x & \text {for}\: d = - e x \\\frac {d^{m} x}{c d^{2}} & \text {for}\: e = 0 \\\frac {\log {\left (\frac {d}{e} + x \right )}}{c e} & \text {for}\: m = 1 \\\frac {\left (d + e x\right )^{m}}{c d e m - c d e + c e^{2} m x - c e^{2} x} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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