Optimal. Leaf size=32 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]
[Out]
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Rubi [A] time = 0.0550035, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{2 e p} \]
Antiderivative was successfully verified.
[In] Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p/(d + e*x),x]
[Out]
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Rubi in Sympy [A] time = 18.8986, size = 27, normalized size = 0.84 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*e**2*x**2+2*c*d*e*x+c*d**2)**p/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.00854067, size = 21, normalized size = 0.66 \[ \frac{\left (c (d+e x)^2\right )^p}{2 e p} \]
Antiderivative was successfully verified.
[In] Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p/(d + e*x),x]
[Out]
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Maple [A] time = 0.002, size = 31, normalized size = 1. \[{\frac{ \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{2\,ep}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*e^2*x^2+2*c*d*e*x+c*d^2)^p/(e*x+d),x)
[Out]
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Maxima [A] time = 0.685717, size = 27, normalized size = 0.84 \[ \frac{{\left (e x + d\right )}^{2 \, p} c^{p}}{2 \, e p} \]
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)^p/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.236119, size = 41, normalized size = 1.28 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, e p} \]
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)^p/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.987994, size = 48, normalized size = 1.5 \[ \begin{cases} \frac{x}{d} & \text{for}\: e = 0 \wedge p = 0 \\\frac{x \left (c d^{2}\right )^{p}}{d} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{e} & \text{for}\: p = 0 \\\frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{2 e p} & \text{otherwise} \end{cases} \]
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)**p/(e*x+d),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{e x + d}\,{d x} \]
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)^p/(e*x + d),x, algorithm="giac")
[Out]