Optimal. Leaf size=130 \[ \frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}-\frac{1}{4 (a+b x)^4 (b c-a d)} \]
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Rubi [A] time = 0.0922863, antiderivative size = 130, 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} \[ \frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}-\frac{1}{4 (a+b x)^4 (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 626
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^4 \left (a c+(b c+a d) x+b d x^2\right )} \, dx &=\int \frac{1}{(a+b x)^5 (c+d x)} \, dx\\ &=\int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx\\ &=-\frac{1}{4 (b c-a d) (a+b x)^4}+\frac{d}{3 (b c-a d)^2 (a+b x)^3}-\frac{d^2}{2 (b c-a d)^3 (a+b x)^2}+\frac{d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}\\ \end{align*}
Mathematica [A] time = 0.0507717, size = 130, normalized size = 1. \[ \frac{d^3}{(a+b x) (b c-a d)^4}-\frac{d^2}{2 (a+b x)^2 (b c-a d)^3}+\frac{d^4 \log (a+b x)}{(b c-a d)^5}-\frac{d^4 \log (c+d x)}{(b c-a d)^5}+\frac{d}{3 (a+b x)^3 (b c-a d)^2}+\frac{1}{4 (a+b x)^4 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 125, normalized size = 1. \begin{align*}{\frac{{d}^{4}\ln \left ( dx+c \right ) }{ \left ( ad-bc \right ) ^{5}}}+{\frac{1}{ \left ( 4\,ad-4\,bc \right ) \left ( bx+a \right ) ^{4}}}+{\frac{d}{3\, \left ( ad-bc \right ) ^{2} \left ( bx+a \right ) ^{3}}}+{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{4} \left ( bx+a \right ) }}-{\frac{{d}^{4}\ln \left ( bx+a \right ) }{ \left ( ad-bc \right ) ^{5}}}+{\frac{{d}^{2}}{2\, \left ( ad-bc \right ) ^{3} \left ( bx+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27895, size = 753, normalized size = 5.79 \begin{align*} \frac{d^{4} \log \left (b x + a\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} - \frac{d^{4} \log \left (d x + c\right )}{b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}} + \frac{12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \,{\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \,{\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{12 \,{\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} +{\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \,{\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \,{\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \,{\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.73601, size = 1320, normalized size = 10.15 \begin{align*} -\frac{3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x - 12 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (b x + a\right ) + 12 \,{\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \log \left (d x + c\right )}{12 \,{\left (a^{4} b^{5} c^{5} - 5 \, a^{5} b^{4} c^{4} d + 10 \, a^{6} b^{3} c^{3} d^{2} - 10 \, a^{7} b^{2} c^{2} d^{3} + 5 \, a^{8} b c d^{4} - a^{9} d^{5} +{\left (b^{9} c^{5} - 5 \, a b^{8} c^{4} d + 10 \, a^{2} b^{7} c^{3} d^{2} - 10 \, a^{3} b^{6} c^{2} d^{3} + 5 \, a^{4} b^{5} c d^{4} - a^{5} b^{4} d^{5}\right )} x^{4} + 4 \,{\left (a b^{8} c^{5} - 5 \, a^{2} b^{7} c^{4} d + 10 \, a^{3} b^{6} c^{3} d^{2} - 10 \, a^{4} b^{5} c^{2} d^{3} + 5 \, a^{5} b^{4} c d^{4} - a^{6} b^{3} d^{5}\right )} x^{3} + 6 \,{\left (a^{2} b^{7} c^{5} - 5 \, a^{3} b^{6} c^{4} d + 10 \, a^{4} b^{5} c^{3} d^{2} - 10 \, a^{5} b^{4} c^{2} d^{3} + 5 \, a^{6} b^{3} c d^{4} - a^{7} b^{2} d^{5}\right )} x^{2} + 4 \,{\left (a^{3} b^{6} c^{5} - 5 \, a^{4} b^{5} c^{4} d + 10 \, a^{5} b^{4} c^{3} d^{2} - 10 \, a^{6} b^{3} c^{2} d^{3} + 5 \, a^{7} b^{2} c d^{4} - a^{8} b d^{5}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.33716, size = 802, normalized size = 6.17 \begin{align*} \frac{d^{4} \log{\left (x + \frac{- \frac{a^{6} d^{10}}{\left (a d - b c\right )^{5}} + \frac{6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} - \frac{15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} + \frac{20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} - \frac{15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} + \frac{6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} - \frac{b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} - \frac{d^{4} \log{\left (x + \frac{\frac{a^{6} d^{10}}{\left (a d - b c\right )^{5}} - \frac{6 a^{5} b c d^{9}}{\left (a d - b c\right )^{5}} + \frac{15 a^{4} b^{2} c^{2} d^{8}}{\left (a d - b c\right )^{5}} - \frac{20 a^{3} b^{3} c^{3} d^{7}}{\left (a d - b c\right )^{5}} + \frac{15 a^{2} b^{4} c^{4} d^{6}}{\left (a d - b c\right )^{5}} - \frac{6 a b^{5} c^{5} d^{5}}{\left (a d - b c\right )^{5}} + a d^{5} + \frac{b^{6} c^{6} d^{4}}{\left (a d - b c\right )^{5}} + b c d^{4}}{2 b d^{5}} \right )}}{\left (a d - b c\right )^{5}} + \frac{25 a^{3} d^{3} - 23 a^{2} b c d^{2} + 13 a b^{2} c^{2} d - 3 b^{3} c^{3} + 12 b^{3} d^{3} x^{3} + x^{2} \left (42 a b^{2} d^{3} - 6 b^{3} c d^{2}\right ) + x \left (52 a^{2} b d^{3} - 20 a b^{2} c d^{2} + 4 b^{3} c^{2} d\right )}{12 a^{8} d^{4} - 48 a^{7} b c d^{3} + 72 a^{6} b^{2} c^{2} d^{2} - 48 a^{5} b^{3} c^{3} d + 12 a^{4} b^{4} c^{4} + x^{4} \left (12 a^{4} b^{4} d^{4} - 48 a^{3} b^{5} c d^{3} + 72 a^{2} b^{6} c^{2} d^{2} - 48 a b^{7} c^{3} d + 12 b^{8} c^{4}\right ) + x^{3} \left (48 a^{5} b^{3} d^{4} - 192 a^{4} b^{4} c d^{3} + 288 a^{3} b^{5} c^{2} d^{2} - 192 a^{2} b^{6} c^{3} d + 48 a b^{7} c^{4}\right ) + x^{2} \left (72 a^{6} b^{2} d^{4} - 288 a^{5} b^{3} c d^{3} + 432 a^{4} b^{4} c^{2} d^{2} - 288 a^{3} b^{5} c^{3} d + 72 a^{2} b^{6} c^{4}\right ) + x \left (48 a^{7} b d^{4} - 192 a^{6} b^{2} c d^{3} + 288 a^{5} b^{3} c^{2} d^{2} - 192 a^{4} b^{4} c^{3} d + 48 a^{3} b^{5} c^{4}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19591, size = 456, normalized size = 3.51 \begin{align*} \frac{b d^{4} \log \left ({\left | b x + a \right |}\right )}{b^{6} c^{5} - 5 \, a b^{5} c^{4} d + 10 \, a^{2} b^{4} c^{3} d^{2} - 10 \, a^{3} b^{3} c^{2} d^{3} + 5 \, a^{4} b^{2} c d^{4} - a^{5} b d^{5}} - \frac{d^{5} \log \left ({\left | d x + c \right |}\right )}{b^{5} c^{5} d - 5 \, a b^{4} c^{4} d^{2} + 10 \, a^{2} b^{3} c^{3} d^{3} - 10 \, a^{3} b^{2} c^{2} d^{4} + 5 \, a^{4} b c d^{5} - a^{5} d^{6}} - \frac{3 \, b^{4} c^{4} - 16 \, a b^{3} c^{3} d + 36 \, a^{2} b^{2} c^{2} d^{2} - 48 \, a^{3} b c d^{3} + 25 \, a^{4} d^{4} - 12 \,{\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} x^{3} + 6 \,{\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} x^{2} - 4 \,{\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} x}{12 \,{\left (b c - a d\right )}^{5}{\left (b x + a\right )}^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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