Optimal. Leaf size=45 \[ \frac{(b B-A c) (c d-b e) \log (b+c x)}{b c^2}+\frac{A d \log (x)}{b}+\frac{B e x}{c} \]
[Out]
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Rubi [A] time = 0.109571, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(b B-A c) (c d-b e) \log (b+c x)}{b c^2}+\frac{A d \log (x)}{b}+\frac{B e x}{c} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(d + e*x))/(b*x + c*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{A d \log{\left (x \right )}}{b} + \frac{e \int B\, dx}{c} + \frac{\left (A c - B b\right ) \left (b e - c d\right ) \log{\left (b + c x \right )}}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(e*x+d)/(c*x**2+b*x),x)
[Out]
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Mathematica [A] time = 0.0379705, size = 46, normalized size = 1.02 \[ \frac{-(b B-A c) (b e-c d) \log (b+c x)+A c^2 d \log (x)+b B c e x}{b c^2} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(d + e*x))/(b*x + c*x^2),x]
[Out]
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Maple [A] time = 0.008, size = 68, normalized size = 1.5 \[{\frac{Bex}{c}}+{\frac{Ad\ln \left ( x \right ) }{b}}+{\frac{\ln \left ( cx+b \right ) Ae}{c}}-{\frac{\ln \left ( cx+b \right ) Ad}{b}}-{\frac{b\ln \left ( cx+b \right ) Be}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) Bd}{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)/(c*x^2+b*x),x)
[Out]
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Maxima [A] time = 0.716573, size = 77, normalized size = 1.71 \[ \frac{B e x}{c} + \frac{A d \log \left (x\right )}{b} + \frac{{\left ({\left (B b c - A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} \log \left (c x + b\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.308324, size = 77, normalized size = 1.71 \[ \frac{B b c e x + A c^{2} d \log \left (x\right ) +{\left ({\left (B b c - A c^{2}\right )} d -{\left (B b^{2} - A b c\right )} e\right )} \log \left (c x + b\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.3948, size = 88, normalized size = 1.96 \[ \frac{A d \log{\left (x \right )}}{b} + \frac{B e x}{c} - \frac{\left (- A c + B b\right ) \left (b e - c d\right ) \log{\left (x + \frac{A b c d + \frac{b \left (- A c + B b\right ) \left (b e - c d\right )}{c}}{- A b c e + 2 A c^{2} d + B b^{2} e - B b c d} \right )}}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)/(c*x**2+b*x),x)
[Out]
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GIAC/XCAS [A] time = 0.277633, size = 80, normalized size = 1.78 \[ \frac{B x e}{c} + \frac{A d{\rm ln}\left ({\left | x \right |}\right )}{b} + \frac{{\left (B b c d - A c^{2} d - B b^{2} e + A b c e\right )}{\rm ln}\left ({\left | c x + b \right |}\right )}{b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(e*x + d)/(c*x^2 + b*x),x, algorithm="giac")
[Out]