Optimal. Leaf size=44 \[ \frac{b^2-4 a c}{16 c^2 d^3 (b+2 c x)^2}+\frac{\log (b+2 c x)}{8 c^2 d^3} \]
[Out]
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Rubi [A] time = 0.0840983, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{b^2-4 a c}{16 c^2 d^3 (b+2 c x)^2}+\frac{\log (b+2 c x)}{8 c^2 d^3} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^3,x]
[Out]
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Rubi in Sympy [A] time = 16.0867, size = 41, normalized size = 0.93 \[ \frac{\log{\left (b + 2 c x \right )}}{8 c^{2} d^{3}} + \frac{- a c + \frac{b^{2}}{4}}{4 c^{2} d^{3} \left (b + 2 c x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**3,x)
[Out]
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Mathematica [A] time = 0.0293466, size = 37, normalized size = 0.84 \[ \frac{\frac{b^2-4 a c}{(b+2 c x)^2}+2 \log (b+2 c x)}{16 c^2 d^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^3,x]
[Out]
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Maple [A] time = 0.008, size = 53, normalized size = 1.2 \[ -{\frac{a}{4\,c{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{{b}^{2}}{16\,{c}^{2}{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{\ln \left ( 2\,cx+b \right ) }{8\,{c}^{2}{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)/(2*c*d*x+b*d)^3,x)
[Out]
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Maxima [A] time = 0.678234, size = 81, normalized size = 1.84 \[ \frac{b^{2} - 4 \, a c}{16 \,{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}} + \frac{\log \left (2 \, c x + b\right )}{8 \, c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.214134, size = 95, normalized size = 2.16 \[ \frac{b^{2} - 4 \, a c + 2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (2 \, c x + b\right )}{16 \,{\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.05699, size = 60, normalized size = 1.36 \[ - \frac{4 a c - b^{2}}{16 b^{2} c^{2} d^{3} + 64 b c^{3} d^{3} x + 64 c^{4} d^{3} x^{2}} + \frac{\log{\left (b + 2 c x \right )}}{8 c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.214637, size = 55, normalized size = 1.25 \[ \frac{{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{8 \, c^{2} d^{3}} + \frac{b^{2} - 4 \, a c}{16 \,{\left (2 \, c x + b\right )}^{2} c^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^3,x, algorithm="giac")
[Out]