Optimal. Leaf size=19 \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]
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Rubi [A] time = 0.0234116, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{c (d+e x)^{m+3}}{e (m+3)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^m*(c*d^2 + 2*c*d*e*x + c*e^2*x^2),x]
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Rubi in Sympy [A] time = 16.8165, size = 14, normalized size = 0.74 \[ \frac{c \left (d + e x\right )^{m + 3}}{e \left (m + 3\right )} \]
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 = 0.022809, size = 20, normalized size = 1.05 \[ \frac{c (d+e x)^{m+3}}{e m+3 e} \]
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 = 0.006, size = 36, normalized size = 1.9 \[{\frac{ \left ( ex+d \right ) ^{1+m}c \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ) }{e \left ( 3+m \right ) }} \]
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 [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23119, size = 66, normalized size = 3.47 \[ \frac{{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )}{\left (e x + d\right )}^{m}}{e m + 3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.83499, size = 116, normalized size = 6.11 \[ \begin{cases} \frac{c x}{d} & \text{for}\: e = 0 \wedge m = -3 \\c d^{2} d^{m} x & \text{for}\: e = 0 \\\frac{c \log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: m = -3 \\\frac{c d^{3} \left (d + e x\right )^{m}}{e m + 3 e} + \frac{3 c d^{2} e x \left (d + e x\right )^{m}}{e m + 3 e} + \frac{3 c d e^{2} x^{2} \left (d + e x\right )^{m}}{e m + 3 e} + \frac{c e^{3} x^{3} \left (d + e x\right )^{m}}{e m + 3 e} & \text{otherwise} \end{cases} \]
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)
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GIAC/XCAS [A] time = 0.218172, size = 112, normalized size = 5.89 \[ \frac{c x^{3} e^{\left (m{\rm ln}\left (x e + d\right ) + 3\right )} + 3 \, c d x^{2} e^{\left (m{\rm ln}\left (x e + d\right ) + 2\right )} + 3 \, c d^{2} x e^{\left (m{\rm ln}\left (x e + d\right ) + 1\right )} + c d^{3} e^{\left (m{\rm ln}\left (x e + d\right )\right )}}{m e + 3 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)*(e*x + d)^m,x, algorithm="giac")
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