Optimal. Leaf size=107 \[ -\frac{d^2}{(a+b x) (b c-a d)^3}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)} \]
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Rubi [A] time = 0.0712196, antiderivative size = 107, 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} \[ -\frac{d^2}{(a+b x) (b c-a d)^3}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}-\frac{1}{3 (a+b x)^3 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )} \, dx &=\int \frac{1}{(a+b x)^4 (c+d x)} \, dx\\ &=\int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx\\ &=-\frac{1}{3 (b c-a d) (a+b x)^3}+\frac{d}{2 (b c-a d)^2 (a+b x)^2}-\frac{d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}\\ \end{align*}
Mathematica [A] time = 0.042251, size = 107, normalized size = 1. \[ -\frac{d^2}{(a+b x) (b c-a d)^3}-\frac{d^3 \log (a+b x)}{(b c-a d)^4}+\frac{d^3 \log (c+d x)}{(b c-a d)^4}+\frac{d}{2 (a+b x)^2 (b c-a d)^2}+\frac{1}{3 (a+b x)^3 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 103, normalized size = 1. \begin{align*}{\frac{{d}^{3}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{4}}}+{\frac{1}{ \left ( 3\,ad-3\,bc \right ) \left ( bx+a \right ) ^{3}}}+{\frac{d}{2\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{d}^{3}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{4}}}+{\frac{{d}^{2}}{ \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 = 1.24775, size = 487, normalized size = 4.55 \begin{align*} -\frac{d^{3} \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{d^{3} \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} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \,{\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{6 \,{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} +{\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \,{\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \,{\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60064, size = 856, normalized size = 8. \begin{align*} -\frac{2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, 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 - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x + 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \,{\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )}{6 \,{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4} +{\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} x^{3} + 3 \,{\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.87301, size = 570, normalized size = 5.33 \begin{align*} \frac{d^{3} \log{\left (x + \frac{- \frac{a^{5} d^{8}}{\left (a d - b c\right )^{4}} + \frac{5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} - \frac{10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} + \frac{10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} - \frac{5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} + \frac{b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} - \frac{d^{3} \log{\left (x + \frac{\frac{a^{5} d^{8}}{\left (a d - b c\right )^{4}} - \frac{5 a^{4} b c d^{7}}{\left (a d - b c\right )^{4}} + \frac{10 a^{3} b^{2} c^{2} d^{6}}{\left (a d - b c\right )^{4}} - \frac{10 a^{2} b^{3} c^{3} d^{5}}{\left (a d - b c\right )^{4}} + \frac{5 a b^{4} c^{4} d^{4}}{\left (a d - b c\right )^{4}} + a d^{4} - \frac{b^{5} c^{5} d^{3}}{\left (a d - b c\right )^{4}} + b c d^{3}}{2 b d^{4}} \right )}}{\left (a d - b c\right )^{4}} + \frac{11 a^{2} d^{2} - 7 a b c d + 2 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (15 a b d^{2} - 3 b^{2} c d\right )}{6 a^{6} d^{3} - 18 a^{5} b c d^{2} + 18 a^{4} b^{2} c^{2} d - 6 a^{3} b^{3} c^{3} + x^{3} \left (6 a^{3} b^{3} d^{3} - 18 a^{2} b^{4} c d^{2} + 18 a b^{5} c^{2} d - 6 b^{6} c^{3}\right ) + x^{2} \left (18 a^{4} b^{2} d^{3} - 54 a^{3} b^{3} c d^{2} + 54 a^{2} b^{4} c^{2} d - 18 a b^{5} c^{3}\right ) + x \left (18 a^{5} b d^{3} - 54 a^{4} b^{2} c d^{2} + 54 a^{3} b^{3} c^{2} d - 18 a^{2} b^{4} c^{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2146, size = 328, normalized size = 3.07 \begin{align*} -\frac{b d^{3} \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{d^{4} \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{2 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 18 \, a^{2} b c d^{2} - 11 \, 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 - 6 \, a b^{2} c d^{2} + 5 \, a^{2} b d^{3}\right )} x}{6 \,{\left (b c - a d\right )}^{4}{\left (b x + a\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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