Optimal. Leaf size=43 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e} \]
[Out]
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Rubi [A] time = 0.0484752, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(-1 + 2*p)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 19.753, size = 39, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{2 p} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{- p} \log{\left (d + e x \right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(-1+2*p)/((c*e**2*x**2+2*c*d*e*x+c*d**2)**p),x)
[Out]
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Mathematica [A] time = 0.0133801, size = 32, normalized size = 0.74 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c (d+e x)^2\right )^{-p}}{e} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(-1 + 2*p)/(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]
[Out]
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Maple [A] time = 0.065, size = 74, normalized size = 1.7 \[{\frac{1}{{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}} \left ( x\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1+2\,p \right ) \ln \left ( ex+d \right ) }}+{\frac{d\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1+2\,p \right ) \ln \left ( ex+d \right ) }}}{e}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(-1+2*p)/((c*e^2*x^2+2*c*d*e*x+c*d^2)^p),x)
[Out]
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Maxima [A] time = 0.686852, size = 20, normalized size = 0.47 \[ \frac{c^{-p} \log \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(2*p - 1)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.231118, size = 20, normalized size = 0.47 \[ \frac{\log \left (e x + d\right )}{c^{p} e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(2*p - 1)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(-1+2*p)/((c*e**2*x**2+2*c*d*e*x+c*d**2)**p),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{2 \, p - 1}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^(2*p - 1)/(c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p,x, algorithm="giac")
[Out]