Optimal. Leaf size=41 \[ \frac{\left (a e^2+c d^2\right ) \log (d+e x)}{e^3}-\frac{c d x}{e^2}+\frac{c x^2}{2 e} \]
[Out]
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Rubi [A] time = 0.0769249, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a e^2+c d^2\right ) \log (d+e x)}{e^3}-\frac{c d x}{e^2}+\frac{c x^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(d + e*x),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{c \int x\, dx}{e} - \frac{d \int c\, dx}{e^{2}} + \frac{\left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.0202616, size = 38, normalized size = 0.93 \[ \frac{2 \left (a e^2+c d^2\right ) \log (d+e x)+c e x (e x-2 d)}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(d + e*x),x]
[Out]
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Maple [A] time = 0.004, size = 44, normalized size = 1.1 \[{\frac{c{x}^{2}}{2\,e}}-{\frac{cdx}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ) a}{e}}+{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{{e}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d),x)
[Out]
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Maxima [A] time = 0.707066, size = 53, normalized size = 1.29 \[ \frac{c e x^{2} - 2 \, c d x}{2 \, e^{2}} + \frac{{\left (c d^{2} + a e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.20853, size = 53, normalized size = 1.29 \[ \frac{c e^{2} x^{2} - 2 \, c d e x + 2 \,{\left (c d^{2} + a e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.24985, size = 36, normalized size = 0.88 \[ - \frac{c d x}{e^{2}} + \frac{c x^{2}}{2 e} + \frac{\left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.207355, size = 53, normalized size = 1.29 \[{\left (c d^{2} + a e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c x^{2} e - 2 \, c d x\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(e*x + d),x, algorithm="giac")
[Out]