Optimal. Leaf size=70 \[ \frac{\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]
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Rubi [A] time = 0.107803, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
[In] Int[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 38.7638, size = 70, normalized size = 1. \[ - \frac{2 \log{\left (b + 2 c x \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{2}} + \frac{\log{\left (a + b x + c x^{2} \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{2}} + \frac{1}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)
[Out]
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Mathematica [A] time = 0.0774938, size = 65, normalized size = 0.93 \[ \frac{\frac{\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2}+\frac{1}{\left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{\left (b^2-4 a c\right )^2}}{d^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)),x]
[Out]
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Maple [A] time = 0.011, size = 78, normalized size = 1.1 \[ -2\,{\frac{\ln \left ( 2\,cx+b \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}}-{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) ^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a),x)
[Out]
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Maxima [A] time = 0.685791, size = 174, normalized size = 2.49 \[ \frac{1}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x +{\left (b^{4} - 4 \, a b^{2} c\right )} d^{3}} + \frac{\log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} - \frac{2 \, \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21508, size = 211, normalized size = 3.01 \[ \frac{b^{2} - 4 \, a c +{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (2 \, c x + b\right )}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{3} x^{2} + 4 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{3} x +{\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.3073, size = 119, normalized size = 1.7 \[ - \frac{1}{4 a b^{2} c d^{3} - b^{4} d^{3} + x^{2} \left (16 a c^{3} d^{3} - 4 b^{2} c^{2} d^{3}\right ) + x \left (16 a b c^{2} d^{3} - 4 b^{3} c d^{3}\right )} - \frac{2 \log{\left (\frac{b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} + \frac{\log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.217941, size = 150, normalized size = 2.14 \[ -\frac{2 \, c{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d^{3} - 8 \, a b^{2} c^{2} d^{3} + 16 \, a^{2} c^{3} d^{3}} + \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} + \frac{1}{{\left (b^{2} - 4 \, a c\right )}{\left (2 \, c x + b\right )}^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)),x, algorithm="giac")
[Out]