Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-1}}{c e (1-m)} \]
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Rubi [A] time = 0.0324684, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{(d+e x)^{m-1}}{c e (1-m)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]
[Out]
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Rubi in Sympy [A] time = 20.2682, size = 15, normalized size = 0.62 \[ - \frac{\left (d + e x\right )^{m - 1}}{c e \left (- m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**m/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)
[Out]
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Mathematica [A] time = 0.0207131, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-1}}{c e (m-1)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^m/(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 22, normalized size = 0.9 \[{\frac{ \left ( ex+d \right ) ^{-1+m}}{ce \left ( -1+m \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^m/(c*e^2*x^2+2*c*d*e*x+c*d^2),x)
[Out]
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Maxima [A] time = 0.694647, size = 36, normalized size = 1.5 \[ \frac{{\left (e x + d\right )}^{m}}{c e^{2}{\left (m - 1\right )} x + c d e{\left (m - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236705, size = 49, normalized size = 2.04 \[ \frac{{\left (e x + d\right )}^{m}}{c d e m - c d e +{\left (c e^{2} m - c e^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.08711, size = 63, normalized size = 2.62 \[ \begin{cases} \mathrm{NaN} & \text{for}\: d = 0 \wedge e = 0 \wedge m = 1 \\0^{m} \tilde{\infty } x & \text{for}\: d = - e x \\\frac{d^{m} x}{c d^{2}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c e} & \text{for}\: m = 1 \\\frac{\left (d + e x\right )^{m}}{c d e m - c d e + c e^{2} m x - c e^{2} x} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**m/(c*e**2*x**2+2*c*d*e*x+c*d**2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^m/(c*e^2*x^2 + 2*c*d*e*x + c*d^2),x, algorithm="giac")
[Out]