Optimal. Leaf size=86 \[ -\frac{a d+b c+2 b d x}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3} \]
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Rubi [A] time = 0.0261368, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {614, 616, 31} \[ -\frac{a d+b c+2 b d x}{(b c-a d)^2 \left (x (a d+b c)+a c+b d x^2\right )}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Rule 614
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=-\frac{b c+a d+2 b d x}{(b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )}-\frac{(2 b d) \int \frac{1}{a c+(b c+a d) x+b d x^2} \, dx}{(b c-a d)^2}\\ &=-\frac{b c+a d+2 b d x}{(b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )}+\frac{\left (2 b^2 d^2\right ) \int \frac{1}{b c+b d x} \, dx}{(b c-a d)^3}-\frac{\left (2 b^2 d^2\right ) \int \frac{1}{a d+b d x} \, dx}{(b c-a d)^3}\\ &=-\frac{b c+a d+2 b d x}{(b c-a d)^2 \left (a c+(b c+a d) x+b d x^2\right )}-\frac{2 b d \log (a+b x)}{(b c-a d)^3}+\frac{2 b d \log (c+d x)}{(b c-a d)^3}\\ \end{align*}
Mathematica [A] time = 0.0663112, size = 66, normalized size = 0.77 \[ \frac{\frac{b (a d-b c)}{a+b x}+\frac{d (a d-b c)}{c+d x}-2 b d \log (a+b x)+2 b d \log (c+d x)}{(b c-a d)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 82, normalized size = 1. \begin{align*} -{\frac{d}{ \left ( ad-bc \right ) ^{2} \left ( dx+c \right ) }}-2\,{\frac{bd\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{3}}}-{\frac{b}{ \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) }}+2\,{\frac{bd\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1048, size = 281, normalized size = 3.27 \begin{align*} -\frac{2 \, b d \log \left (b x + a\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} + \frac{2 \, b d \log \left (d x + c\right )}{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}} - \frac{2 \, b d x + b c + a d}{a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} +{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61919, size = 486, normalized size = 5.65 \begin{align*} -\frac{b^{2} c^{2} - a^{2} d^{2} + 2 \,{\left (b^{2} c d - a b d^{2}\right )} x + 2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) - 2 \,{\left (b^{2} d^{2} x^{2} + a b c d +{\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3} +{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} x^{2} +{\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.93193, size = 405, normalized size = 4.71 \begin{align*} - \frac{2 b d \log{\left (x + \frac{- \frac{2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} + \frac{8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} - \frac{12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} + \frac{8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} - \frac{2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} + \frac{2 b d \log{\left (x + \frac{\frac{2 a^{4} b d^{5}}{\left (a d - b c\right )^{3}} - \frac{8 a^{3} b^{2} c d^{4}}{\left (a d - b c\right )^{3}} + \frac{12 a^{2} b^{3} c^{2} d^{3}}{\left (a d - b c\right )^{3}} - \frac{8 a b^{4} c^{3} d^{2}}{\left (a d - b c\right )^{3}} + 2 a b d^{2} + \frac{2 b^{5} c^{4} d}{\left (a d - b c\right )^{3}} + 2 b^{2} c d}{4 b^{2} d^{2}} \right )}}{\left (a d - b c\right )^{3}} - \frac{a d + b c + 2 b d x}{a^{3} c d^{2} - 2 a^{2} b c^{2} d + a b^{2} c^{3} + x^{2} \left (a^{2} b d^{3} - 2 a b^{2} c d^{2} + b^{3} c^{2} d\right ) + x \left (a^{3} d^{3} - a^{2} b c d^{2} - a b^{2} c^{2} d + b^{3} c^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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