Optimal. Leaf size=73 \[ -\frac{2 d \log (x) (c d-b e)}{b^3}+\frac{2 d (c d-b e) \log (b+c x)}{b^3}-\frac{(c d-b e)^2}{b^2 c (b+c x)}-\frac{d^2}{b^2 x} \]
[Out]
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Rubi [A] time = 0.151421, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2 d \log (x) (c d-b e)}{b^3}+\frac{2 d (c d-b e) \log (b+c x)}{b^3}-\frac{(c d-b e)^2}{b^2 c (b+c x)}-\frac{d^2}{b^2 x} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2/(b*x + c*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 19.1059, size = 63, normalized size = 0.86 \[ - \frac{d^{2}}{b^{2} x} - \frac{\left (b e - c d\right )^{2}}{b^{2} c \left (b + c x\right )} + \frac{2 d \left (b e - c d\right ) \log{\left (x \right )}}{b^{3}} - \frac{2 d \left (b e - c d\right ) \log{\left (b + c x \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2/(c*x**2+b*x)**2,x)
[Out]
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Mathematica [A] time = 0.126047, size = 67, normalized size = 0.92 \[ \frac{-\frac{b (c d-b e)^2}{c (b+c x)}+2 d \log (x) (b e-c d)+2 d (c d-b e) \log (b+c x)-\frac{b d^2}{x}}{b^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2/(b*x + c*x^2)^2,x]
[Out]
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Maple [A] time = 0.016, size = 106, normalized size = 1.5 \[ -{\frac{{d}^{2}}{{b}^{2}x}}+2\,{\frac{d\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{2}\ln \left ( x \right ) c}{{b}^{3}}}-{\frac{{e}^{2}}{c \left ( cx+b \right ) }}+2\,{\frac{de}{b \left ( cx+b \right ) }}-{\frac{c{d}^{2}}{{b}^{2} \left ( cx+b \right ) }}-2\,{\frac{d\ln \left ( cx+b \right ) e}{{b}^{2}}}+2\,{\frac{{d}^{2}\ln \left ( cx+b \right ) c}{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2/(c*x^2+b*x)^2,x)
[Out]
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Maxima [A] time = 0.695563, size = 126, normalized size = 1.73 \[ -\frac{b c d^{2} +{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2}\right )} x}{b^{2} c^{2} x^{2} + b^{3} c x} + \frac{2 \,{\left (c d^{2} - b d e\right )} \log \left (c x + b\right )}{b^{3}} - \frac{2 \,{\left (c d^{2} - b d e\right )} \log \left (x\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229284, size = 201, normalized size = 2.75 \[ -\frac{b^{2} c d^{2} +{\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e + b^{3} e^{2}\right )} x - 2 \,{\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} +{\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (c x + b\right ) + 2 \,{\left ({\left (c^{3} d^{2} - b c^{2} d e\right )} x^{2} +{\left (b c^{2} d^{2} - b^{2} c d e\right )} x\right )} \log \left (x\right )}{b^{3} c^{2} x^{2} + b^{4} c x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.56398, size = 173, normalized size = 2.37 \[ - \frac{b c d^{2} + x \left (b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right )}{b^{3} c x + b^{2} c^{2} x^{2}} + \frac{2 d \left (b e - c d\right ) \log{\left (x + \frac{2 b^{2} d e - 2 b c d^{2} - 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} - \frac{2 d \left (b e - c d\right ) \log{\left (x + \frac{2 b^{2} d e - 2 b c d^{2} + 2 b d \left (b e - c d\right )}{4 b c d e - 4 c^{2} d^{2}} \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2/(c*x**2+b*x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.207511, size = 136, normalized size = 1.86 \[ -\frac{2 \,{\left (c d^{2} - b d e\right )}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \,{\left (c^{2} d^{2} - b c d e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b^{3} c} - \frac{2 \, c^{2} d^{2} x - 2 \, b c d x e + b c d^{2} + b^{2} x e^{2}}{{\left (c x^{2} + b x\right )} b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2/(c*x^2 + b*x)^2,x, algorithm="giac")
[Out]