Optimal. Leaf size=109 \[ \frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2} \]
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Rubi [A] time = 0.0778673, antiderivative size = 109, 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, Rules used = {626, 44} \[ \frac{d^2}{(c+d x) (b c-a d)^3}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}+\frac{2 b d}{(a+b x) (b c-a d)^3}-\frac{b}{2 (a+b x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) \left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=\int \frac{1}{(a+b x)^3 (c+d x)^2} \, dx\\ &=\int \left (\frac{b^2}{(b c-a d)^2 (a+b x)^3}-\frac{2 b^2 d}{(b c-a d)^3 (a+b x)^2}+\frac{3 b^2 d^2}{(b c-a d)^4 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)^2}-\frac{3 b d^3}{(b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{b}{2 (b c-a d)^2 (a+b x)^2}+\frac{2 b d}{(b c-a d)^3 (a+b x)}+\frac{d^2}{(b c-a d)^3 (c+d x)}+\frac{3 b d^2 \log (a+b x)}{(b c-a d)^4}-\frac{3 b d^2 \log (c+d x)}{(b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.0723044, size = 98, normalized size = 0.9 \[ \frac{\frac{2 d^2 (b c-a d)}{c+d x}+\frac{4 b d (b c-a d)}{a+b x}-\frac{b (b c-a d)^2}{(a+b x)^2}+6 b d^2 \log (a+b x)-6 b d^2 \log (c+d x)}{2 (b c-a d)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 109, normalized size = 1. \begin{align*} -{\frac{{d}^{2}}{ \left ( ad-bc \right ) ^{3} \left ( dx+c \right ) }}-3\,{\frac{{d}^{2}b\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}-{\frac{b}{2\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}+3\,{\frac{{d}^{2}b\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}}-2\,{\frac{bd}{ \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.1036, size = 521, normalized size = 4.78 \begin{align*} \frac{3 \, b d^{2} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac{3 \, b d^{2} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac{6 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 5 \, a b c d + 2 \, a^{2} d^{2} + 3 \,{\left (b^{2} c d + 3 \, a b d^{2}\right )} x}{2 \,{\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} +{\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} +{\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} +{\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.61282, size = 991, normalized size = 9.09 \begin{align*} -\frac{b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} +{\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right ) + 6 \,{\left (b^{3} d^{3} x^{3} + a^{2} b c d^{2} +{\left (b^{3} c d^{2} + 2 \, a b^{2} d^{3}\right )} x^{2} +{\left (2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (a^{2} b^{4} c^{5} - 4 \, a^{3} b^{3} c^{4} d + 6 \, a^{4} b^{2} c^{3} d^{2} - 4 \, a^{5} b c^{2} d^{3} + a^{6} c d^{4} +{\left (b^{6} c^{4} d - 4 \, a b^{5} c^{3} d^{2} + 6 \, a^{2} b^{4} c^{2} d^{3} - 4 \, a^{3} b^{3} c d^{4} + a^{4} b^{2} d^{5}\right )} x^{3} +{\left (b^{6} c^{5} - 2 \, a b^{5} c^{4} d - 2 \, a^{2} b^{4} c^{3} d^{2} + 8 \, a^{3} b^{3} c^{2} d^{3} - 7 \, a^{4} b^{2} c d^{4} + 2 \, a^{5} b d^{5}\right )} x^{2} +{\left (2 \, a b^{5} c^{5} - 7 \, a^{2} b^{4} c^{4} d + 8 \, a^{3} b^{3} c^{3} d^{2} - 2 \, a^{4} b^{2} c^{2} d^{3} - 2 \, a^{5} b c d^{4} + a^{6} d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.87319, size = 632, normalized size = 5.8 \begin{align*} - \frac{3 b d^{2} \log{\left (x + \frac{- \frac{3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} + \frac{15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} - \frac{30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} + \frac{30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} - \frac{15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} + \frac{3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} + \frac{3 b d^{2} \log{\left (x + \frac{\frac{3 a^{5} b d^{7}}{\left (a d - b c\right )^{4}} - \frac{15 a^{4} b^{2} c d^{6}}{\left (a d - b c\right )^{4}} + \frac{30 a^{3} b^{3} c^{2} d^{5}}{\left (a d - b c\right )^{4}} - \frac{30 a^{2} b^{4} c^{3} d^{4}}{\left (a d - b c\right )^{4}} + \frac{15 a b^{5} c^{4} d^{3}}{\left (a d - b c\right )^{4}} + 3 a b d^{3} - \frac{3 b^{6} c^{5} d^{2}}{\left (a d - b c\right )^{4}} + 3 b^{2} c d^{2}}{6 b^{2} d^{3}} \right )}}{\left (a d - b c\right )^{4}} - \frac{2 a^{2} d^{2} + 5 a b c d - b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (9 a b d^{2} + 3 b^{2} c d\right )}{2 a^{5} c d^{3} - 6 a^{4} b c^{2} d^{2} + 6 a^{3} b^{2} c^{3} d - 2 a^{2} b^{3} c^{4} + x^{3} \left (2 a^{3} b^{2} d^{4} - 6 a^{2} b^{3} c d^{3} + 6 a b^{4} c^{2} d^{2} - 2 b^{5} c^{3} d\right ) + x^{2} \left (4 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 6 a^{2} b^{3} c^{2} d^{2} + 2 a b^{4} c^{3} d - 2 b^{5} c^{4}\right ) + x \left (2 a^{5} d^{4} - 2 a^{4} b c d^{3} - 6 a^{3} b^{2} c^{2} d^{2} + 10 a^{2} b^{3} c^{3} d - 4 a b^{4} c^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21765, size = 342, normalized size = 3.14 \begin{align*} \frac{3 \, b^{2} d^{2} \log \left ({\left | b x + a \right |}\right )}{b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}} - \frac{3 \, b d^{3} \log \left ({\left | d x + c \right |}\right )}{b^{4} c^{4} d - 4 \, a b^{3} c^{3} d^{2} + 6 \, a^{2} b^{2} c^{2} d^{3} - 4 \, a^{3} b c d^{4} + a^{4} d^{5}} - \frac{b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (b^{3} c^{2} d + 2 \, a b^{2} c d^{2} - 3 \, a^{2} b d^{3}\right )} x}{2 \,{\left (b c - a d\right )}^{4}{\left (b x + a\right )}^{2}{\left (d x + c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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