Optimal. Leaf size=36 \[ d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \]
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Rubi [A] time = 0.057868, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ d^3 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )+d^3 (b+2 c x)^2 \]
Antiderivative was successfully verified.
[In] Int[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2),x]
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Rubi in Sympy [A] time = 17.7539, size = 34, normalized size = 0.94 \[ d^{3} \left (b + 2 c x\right )^{2} + d^{3} \left (- 4 a c + b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0274497, size = 33, normalized size = 0.92 \[ d^3 \left (\left (b^2-4 a c\right ) \log (a+x (b+c x))+4 c x (b+c x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(b*d + 2*c*d*x)^3/(a + b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 57, normalized size = 1.6 \[ 4\,{x}^{2}{c}^{2}{d}^{3}-4\,\ln \left ( c{x}^{2}+bx+a \right ) ac{d}^{3}+\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{d}^{3}+4\,xbc{d}^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*d*x+b*d)^3/(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.694869, size = 58, normalized size = 1.61 \[ 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/(c*x^2 + b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203616, size = 58, normalized size = 1.61 \[ 4 \, c^{2} d^{3} x^{2} + 4 \, b c d^{3} x +{\left (b^{2} - 4 \, a c\right )} d^{3} \log \left (c x^{2} + b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/(c*x^2 + b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.16274, size = 44, normalized size = 1.22 \[ 4 b c d^{3} x + 4 c^{2} d^{3} x^{2} - d^{3} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.215881, size = 72, normalized size = 2. \[{\left (b^{2} d^{3} - 4 \, a c d^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right ) + \frac{4 \,{\left (c^{4} d^{3} x^{2} + b c^{3} d^{3} x\right )}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*d*x + b*d)^3/(c*x^2 + b*x + a),x, algorithm="giac")
[Out]