Optimal. Leaf size=78 \[ -\frac{3 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac{3 b (b c-a d)^2}{d^4 (c+d x)}+\frac{(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac{b^3 x}{d^3} \]
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Rubi [A] time = 0.0613338, antiderivative size = 78, 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, 43} \[ -\frac{3 b^2 (b c-a d) \log (c+d x)}{d^4}-\frac{3 b (b c-a d)^2}{d^4 (c+d x)}+\frac{(b c-a d)^3}{2 d^4 (c+d x)^2}+\frac{b^3 x}{d^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^6}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac{(a+b x)^3}{(c+d x)^3} \, dx\\ &=\int \left (\frac{b^3}{d^3}+\frac{(-b c+a d)^3}{d^3 (c+d x)^3}+\frac{3 b (b c-a d)^2}{d^3 (c+d x)^2}-\frac{3 b^2 (b c-a d)}{d^3 (c+d x)}\right ) \, dx\\ &=\frac{b^3 x}{d^3}+\frac{(b c-a d)^3}{2 d^4 (c+d x)^2}-\frac{3 b (b c-a d)^2}{d^4 (c+d x)}-\frac{3 b^2 (b c-a d) \log (c+d x)}{d^4}\\ \end{align*}
Mathematica [A] time = 0.038989, size = 114, normalized size = 1.46 \[ \frac{-3 a^2 b d^2 (c+2 d x)-a^3 d^3+3 a b^2 c d (3 c+4 d x)-6 b^2 (c+d x)^2 (b c-a d) \log (c+d x)+b^3 \left (-4 c^2 d x-5 c^3+4 c d^2 x^2+2 d^3 x^3\right )}{2 d^4 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 160, normalized size = 2.1 \begin{align*}{\frac{{b}^{3}x}{{d}^{3}}}+3\,{\frac{{b}^{2}\ln \left ( dx+c \right ) a}{{d}^{3}}}-3\,{\frac{{b}^{3}\ln \left ( dx+c \right ) c}{{d}^{4}}}-{\frac{{a}^{3}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{3\,bc{a}^{2}}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}}-{\frac{3\,a{b}^{2}{c}^{2}}{2\,{d}^{3} \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{3}{c}^{3}}{2\,{d}^{4} \left ( dx+c \right ) ^{2}}}-3\,{\frac{b{a}^{2}}{{d}^{2} \left ( dx+c \right ) }}+6\,{\frac{ac{b}^{2}}{{d}^{3} \left ( dx+c \right ) }}-3\,{\frac{{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04148, size = 169, normalized size = 2.17 \begin{align*} \frac{b^{3} x}{d^{3}} - \frac{5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} - \frac{3 \,{\left (b^{3} c - a b^{2} d\right )} \log \left (d x + c\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.62815, size = 375, normalized size = 4.81 \begin{align*} \frac{2 \, b^{3} d^{3} x^{3} + 4 \, b^{3} c d^{2} x^{2} - 5 \, b^{3} c^{3} + 9 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - a^{3} d^{3} - 2 \,{\left (2 \, b^{3} c^{2} d - 6 \, a b^{2} c d^{2} + 3 \, a^{2} b d^{3}\right )} x - 6 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d +{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (b^{3} c^{2} d - a b^{2} c d^{2}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.58782, size = 128, normalized size = 1.64 \begin{align*} \frac{b^{3} x}{d^{3}} + \frac{3 b^{2} \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{4}} - \frac{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3} + x \left (6 a^{2} b d^{3} - 12 a b^{2} c d^{2} + 6 b^{3} c^{2} d\right )}{2 c^{2} d^{4} + 4 c d^{5} x + 2 d^{6} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13169, size = 151, normalized size = 1.94 \begin{align*} \frac{b^{3} x}{d^{3}} - \frac{3 \,{\left (b^{3} c - a b^{2} d\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4}} - \frac{5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3} + 6 \,{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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