Optimal. Leaf size=68 \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]
[Out]
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Rubi [A] time = 0.170699, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{(b B-A c) \log (b+c x)}{b (c d-b e)}-\frac{(B d-A e) \log (d+e x)}{d (c d-b e)}+\frac{A \log (x)}{b d} \]
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 [A] time = 24.2591, size = 51, normalized size = 0.75 \[ \frac{A \log{\left (x \right )}}{b d} - \frac{\left (A e - B d\right ) \log{\left (d + e x \right )}}{d \left (b e - c d\right )} + \frac{\left (A c - B b\right ) \log{\left (b + c x \right )}}{b \left (b e - c d\right )} \]
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 = 0.0694552, size = 63, normalized size = 0.93 \[ \frac{\log (b+c x) (A c d-b B d)+b (B d-A e) \log (d+e x)+A \log (x) (b e-c d)}{b d (b e-c d)} \]
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 = 0.011, size = 94, normalized size = 1.4 \[{\frac{A\ln \left ( x \right ) }{bd}}+{\frac{\ln \left ( cx+b \right ) Ac}{ \left ( be-cd \right ) b}}-{\frac{\ln \left ( cx+b \right ) B}{be-cd}}-{\frac{\ln \left ( ex+d \right ) Ae}{d \left ( be-cd \right ) }}+{\frac{\ln \left ( ex+d \right ) B}{be-cd}} \]
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 = 0.722578, size = 92, normalized size = 1.35 \[ \frac{{\left (B b - A c\right )} \log \left (c x + b\right )}{b c d - b^{2} e} - \frac{{\left (B d - A e\right )} \log \left (e x + d\right )}{c d^{2} - b d e} + \frac{A \log \left (x\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*(e*x + d)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 1.49535, size = 88, normalized size = 1.29 \[ \frac{{\left (B b - A c\right )} d \log \left (c x + b\right ) -{\left (B b d - A b e\right )} \log \left (e x + d\right ) +{\left (A c d - A b e\right )} \log \left (x\right )}{b c d^{2} - b^{2} d e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*(e*x + d)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
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 = 0.295542, size = 184, normalized size = 2.71 \[ -\frac{A{\rm ln}\left ({\left | c x^{2} e + c d x + b x e + b d \right |}\right )}{2 \, b d} + \frac{A{\rm ln}\left ({\left | x \right |}\right )}{b d} + \frac{{\left (2 \, B b d - A c d - A b e\right )}{\rm ln}\left (\frac{{\left | 2 \, c x e + c d + b e -{\left | c d - b e \right |} \right |}}{{\left | 2 \, c x e + c d + b e +{\left | c d - b e \right |} \right |}}\right )}{2 \, b d{\left | c d - b e \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)*(e*x + d)),x, algorithm="giac")
[Out]