Optimal. Leaf size=56 \[ \frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2} \]
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Rubi [A] time = 0.0954756, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{1}{(c+d x) (b c-a d)}+\frac{b \log (a+b x)}{(b c-a d)^2}-\frac{b \log (c+d x)}{(b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 19.3187, size = 46, normalized size = 0.82 \[ \frac{b \log{\left (a + b x \right )}}{\left (a d - b c\right )^{2}} - \frac{b \log{\left (c + d x \right )}}{\left (a d - b c\right )^{2}} - \frac{1}{\left (c + d x\right ) \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.0450136, size = 53, normalized size = 0.95 \[ \frac{b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c}{(c+d x) (b c-a d)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/(a*c + (b*c + a*d)*x + b*d*x^2)^2,x]
[Out]
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Maple [A] time = 0.016, size = 58, normalized size = 1. \[{\frac{b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{2}}}-{\frac{1}{ \left ( ad-bc \right ) \left ( dx+c \right ) }}-{\frac{b\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(a*c+(a*d+b*c)*x+x^2*b*d)^2,x)
[Out]
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Maxima [A] time = 0.742067, size = 122, normalized size = 2.18 \[ \frac{b \log \left (b x + a\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} - \frac{b \log \left (d x + c\right )}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}} + \frac{1}{b c^{2} - a c d +{\left (b c d - a d^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.21468, size = 124, normalized size = 2.21 \[ \frac{b c - a d +{\left (b d x + b c\right )} \log \left (b x + a\right ) -{\left (b d x + b c\right )} \log \left (d x + c\right )}{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.12425, size = 233, normalized size = 4.16 \[ - \frac{b \log{\left (x + \frac{- \frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{\left (a d - b c\right )^{2}} + \frac{b \log{\left (x + \frac{\frac{a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac{3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac{3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac{b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{\left (a d - b c\right )^{2}} - \frac{1}{a c d - b c^{2} + x \left (a d^{2} - b c d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(a*c+(a*d+b*c)*x+b*d*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.212752, size = 126, normalized size = 2.25 \[ \frac{b^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}} - \frac{b d{\rm ln}\left ({\left | d x + c \right |}\right )}{b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}} + \frac{1}{{\left (b c - a d\right )}{\left (d x + c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/(b*d*x^2 + a*c + (b*c + a*d)*x)^2,x, algorithm="giac")
[Out]