Optimal. Leaf size=50 \[ \frac{(b d-a e)^2 \log (d+e x)}{e^3}-\frac{b x (b d-a e)}{e^2}+\frac{(a+b x)^2}{2 e} \]
[Out]
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Rubi [A] time = 0.0674319, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(b d-a e)^2 \log (d+e x)}{e^3}-\frac{b x (b d-a e)}{e^2}+\frac{(a+b x)^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x),x]
[Out]
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Rubi in Sympy [A] time = 24.6695, size = 41, normalized size = 0.82 \[ \frac{b x \left (a e - b d\right )}{e^{2}} + \frac{\left (a + b x\right )^{2}}{2 e} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d),x)
[Out]
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Mathematica [A] time = 0.0324213, size = 43, normalized size = 0.86 \[ \frac{b e x (4 a e-2 b d+b e x)+2 (b d-a e)^2 \log (d+e x)}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)/(d + e*x),x]
[Out]
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Maple [A] time = 0.005, size = 74, normalized size = 1.5 \[{\frac{{b}^{2}{x}^{2}}{2\,e}}+2\,{\frac{abx}{e}}-{\frac{{b}^{2}xd}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){a}^{2}}{e}}-2\,{\frac{\ln \left ( ex+d \right ) dab}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{2}{d}^{2}}{{e}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)/(e*x+d),x)
[Out]
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Maxima [A] time = 0.685643, size = 81, normalized size = 1.62 \[ \frac{b^{2} e x^{2} - 2 \,{\left (b^{2} d - 2 \, a b e\right )} x}{2 \, e^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.199191, size = 84, normalized size = 1.68 \[ \frac{b^{2} e^{2} x^{2} - 2 \,{\left (b^{2} d e - 2 \, a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.53329, size = 44, normalized size = 0.88 \[ \frac{b^{2} x^{2}}{2 e} + \frac{x \left (2 a b e - b^{2} d\right )}{e^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)/(e*x+d),x)
[Out]
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GIAC/XCAS [A] time = 0.211165, size = 82, normalized size = 1.64 \[{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (b^{2} x^{2} e - 2 \, b^{2} d x + 4 \, a b x e\right )} e^{\left (-2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)/(e*x + d),x, algorithm="giac")
[Out]