Optimal. Leaf size=57 \[ -\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2} \]
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Rubi [A] time = 0.101375, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{1}{(a+b x) (b c-a d)}-\frac{d \log (a+b x)}{(b c-a d)^2}+\frac{d \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
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Rubi in Sympy [A] time = 25.9366, size = 46, normalized size = 0.81 \[ - \frac{d \log{\left (a + b x \right )}}{\left (a d - b c\right )^{2}} + \frac{d \log{\left (c + d x \right )}}{\left (a d - b c\right )^{2}} + \frac{1}{\left (a + b x\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x+a)/(a*c+(a*d+b*c)*x+b*d*x**2),x)
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Mathematica [A] time = 0.0436742, size = 53, normalized size = 0.93 \[ \frac{d (a+b x) \log (c+d x)-d (a+b x) \log (a+b x)+a d-b c}{(a+b x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b*x)*(a*c + (b*c + a*d)*x + b*d*x^2)),x]
[Out]
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Maple [A] time = 0.015, size = 57, normalized size = 1. \[{\frac{1}{ \left ( ad-bc \right ) \left ( bx+a \right ) }}-{\frac{d\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}}+{\frac{d\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x+a)/(a*c+(a*d+b*c)*x+x^2*b*d),x)
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Maxima [A] time = 0.751067, size = 124, normalized size = 2.18 \[ -\frac{d \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{d \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{1}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)),x, algorithm="maxima")
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Fricas [A] time = 0.208568, size = 126, normalized size = 2.21 \[ -\frac{b c - a d +{\left (b d x + a d\right )} \log \left (b x + a\right ) -{\left (b d x + a d\right )} \log \left (d x + c\right )}{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} +{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)),x, algorithm="fricas")
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Sympy [A] time = 2.98165, size = 233, normalized size = 4.09 \[ \frac{d \log{\left (x + \frac{- \frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} - \frac{d \log{\left (x + \frac{\frac{a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac{b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{\left (a d - b c\right )^{2}} + \frac{1}{a^{2} d - a b c + x \left (a b d - b^{2} c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x+a)/(a*c+(a*d+b*c)*x+b*d*x**2),x)
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GIAC/XCAS [A] time = 0.211698, size = 127, normalized size = 2.23 \[ -\frac{b d{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} + \frac{d^{2}{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} - \frac{1}{{\left (b c - a d\right )}{\left (b x + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)),x, algorithm="giac")
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