Optimal. Leaf size=49 \[ \frac{(b d-a e)^2 \log (a+b x)}{b^3}+\frac{e x (b d-a e)}{b^2}+\frac{(d+e x)^2}{2 b} \]
[Out]
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Rubi [A] time = 0.0519659, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{(b d-a e)^2 \log (a+b x)}{b^3}+\frac{e x (b d-a e)}{b^2}+\frac{(d+e x)^2}{2 b} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)*(d + e*x)^2)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (d + e x\right )^{2}}{2 b} - \frac{\left (a e - b d\right ) \int e\, dx}{b^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)*(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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Mathematica [A] time = 0.0293552, size = 43, normalized size = 0.88 \[ \frac{b e x (-2 a e+4 b d+b e x)+2 (b d-a e)^2 \log (a+b x)}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)*(d + e*x)^2)/(a^2 + 2*a*b*x + b^2*x^2),x]
[Out]
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Maple [A] time = 0.004, size = 74, normalized size = 1.5 \[{\frac{{e}^{2}{x}^{2}}{2\,b}}-{\frac{{e}^{2}xa}{{b}^{2}}}+2\,{\frac{dex}{b}}+{\frac{\ln \left ( bx+a \right ){a}^{2}{e}^{2}}{{b}^{3}}}-2\,{\frac{\ln \left ( bx+a \right ) ade}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ){d}^{2}}{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)*(e*x+d)^2/(b^2*x^2+2*a*b*x+a^2),x)
[Out]
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Maxima [A] time = 0.70926, size = 82, normalized size = 1.67 \[ \frac{b e^{2} x^{2} + 2 \,{\left (2 \, b d e - a e^{2}\right )} x}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.269893, size = 85, normalized size = 1.73 \[ \frac{b^{2} e^{2} x^{2} + 2 \,{\left (2 \, b^{2} d e - a b e^{2}\right )} x + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (b x + a\right )}{2 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.55446, size = 44, normalized size = 0.9 \[ \frac{e^{2} x^{2}}{2 b} - \frac{x \left (a e^{2} - 2 b d e\right )}{b^{2}} + \frac{\left (a e - b d\right )^{2} \log{\left (a + b x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)*(e*x+d)**2/(b**2*x**2+2*a*b*x+a**2),x)
[Out]
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GIAC/XCAS [A] time = 0.27723, size = 80, normalized size = 1.63 \[ \frac{b x^{2} e^{2} + 4 \, b d x e - 2 \, a x e^{2}}{2 \, b^{2}} + \frac{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(e*x + d)^2/(b^2*x^2 + 2*a*b*x + a^2),x, algorithm="giac")
[Out]