Optimal. Leaf size=133 \[ -\frac{5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac{10 b^3 x (b c-a d)^2}{d^5}-\frac{10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac{5 b (b c-a d)^4}{d^6 (c+d x)}+\frac{(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac{b^5 (c+d x)^3}{3 d^6} \]
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Rubi [A] time = 0.135829, antiderivative size = 133, 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} \[ -\frac{5 b^4 (c+d x)^2 (b c-a d)}{2 d^6}+\frac{10 b^3 x (b c-a d)^2}{d^5}-\frac{10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}-\frac{5 b (b c-a d)^4}{d^6 (c+d x)}+\frac{(b c-a d)^5}{2 d^6 (c+d x)^2}+\frac{b^5 (c+d x)^3}{3 d^6} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^8}{\left (a c+(b c+a d) x+b d x^2\right )^3} \, dx &=\int \frac{(a+b x)^5}{(c+d x)^3} \, dx\\ &=\int \left (\frac{10 b^3 (b c-a d)^2}{d^5}+\frac{(-b c+a d)^5}{d^5 (c+d x)^3}+\frac{5 b (b c-a d)^4}{d^5 (c+d x)^2}-\frac{10 b^2 (b c-a d)^3}{d^5 (c+d x)}-\frac{5 b^4 (b c-a d) (c+d x)}{d^5}+\frac{b^5 (c+d x)^2}{d^5}\right ) \, dx\\ &=\frac{10 b^3 (b c-a d)^2 x}{d^5}+\frac{(b c-a d)^5}{2 d^6 (c+d x)^2}-\frac{5 b (b c-a d)^4}{d^6 (c+d x)}-\frac{5 b^4 (b c-a d) (c+d x)^2}{2 d^6}+\frac{b^5 (c+d x)^3}{3 d^6}-\frac{10 b^2 (b c-a d)^3 \log (c+d x)}{d^6}\\ \end{align*}
Mathematica [A] time = 0.0803342, size = 230, normalized size = 1.73 \[ \frac{30 a^2 b^3 d^2 \left (-4 c^2 d x-5 c^3+4 c d^2 x^2+2 d^3 x^3\right )+30 a^3 b^2 c d^3 (3 c+4 d x)-15 a^4 b d^4 (c+2 d x)-3 a^5 d^5+15 a b^4 d \left (-11 c^2 d^2 x^2+2 c^3 d x+7 c^4-4 c d^3 x^3+d^4 x^4\right )-60 b^2 (c+d x)^2 (b c-a d)^3 \log (c+d x)+b^5 \left (63 c^3 d^2 x^2+20 c^2 d^3 x^3+6 c^4 d x-27 c^5-5 c d^4 x^4+2 d^5 x^5\right )}{6 d^6 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 346, normalized size = 2.6 \begin{align*}{\frac{{b}^{5}{x}^{3}}{3\,{d}^{3}}}+{\frac{5\,{b}^{4}{x}^{2}a}{2\,{d}^{3}}}-{\frac{3\,{b}^{5}{x}^{2}c}{2\,{d}^{4}}}+10\,{\frac{{a}^{2}{b}^{3}x}{{d}^{3}}}-15\,{\frac{a{b}^{4}cx}{{d}^{4}}}+6\,{\frac{{b}^{5}{c}^{2}x}{{d}^{5}}}+10\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{3}}{{d}^{3}}}-30\,{\frac{{b}^{3}\ln \left ( dx+c \right ) c{a}^{2}}{{d}^{4}}}+30\,{\frac{{b}^{4}\ln \left ( dx+c \right ) a{c}^{2}}{{d}^{5}}}-10\,{\frac{{b}^{5}\ln \left ( dx+c \right ){c}^{3}}{{d}^{6}}}-{\frac{{a}^{5}}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{5\,c{a}^{4}b}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{c}^{2}{a}^{3}{b}^{2}}{{d}^{3} \left ( dx+c \right ) ^{2}}}+5\,{\frac{{a}^{2}{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) ^{2}}}-{\frac{5\,a{b}^{4}{c}^{4}}{2\,{d}^{5} \left ( dx+c \right ) ^{2}}}+{\frac{{b}^{5}{c}^{5}}{2\,{d}^{6} \left ( dx+c \right ) ^{2}}}-5\,{\frac{{a}^{4}b}{{d}^{2} \left ( dx+c \right ) }}+20\,{\frac{{a}^{3}{b}^{2}c}{{d}^{3} \left ( dx+c \right ) }}-30\,{\frac{{a}^{2}{b}^{3}{c}^{2}}{{d}^{4} \left ( dx+c \right ) }}+20\,{\frac{a{b}^{4}{c}^{3}}{{d}^{5} \left ( dx+c \right ) }}-5\,{\frac{{b}^{5}{c}^{4}}{{d}^{6} \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.11074, size = 366, normalized size = 2.75 \begin{align*} -\frac{9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \,{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \,{\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} + \frac{2 \, b^{5} d^{2} x^{3} - 3 \,{\left (3 \, b^{5} c d - 5 \, a b^{4} d^{2}\right )} x^{2} + 6 \,{\left (6 \, b^{5} c^{2} - 15 \, a b^{4} c d + 10 \, a^{2} b^{3} d^{2}\right )} x}{6 \, d^{5}} - \frac{10 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left (d x + c\right )}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.59612, size = 840, normalized size = 6.32 \begin{align*} \frac{2 \, b^{5} d^{5} x^{5} - 27 \, b^{5} c^{5} + 105 \, a b^{4} c^{4} d - 150 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 15 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5} - 5 \,{\left (b^{5} c d^{4} - 3 \, a b^{4} d^{5}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{3} - 3 \, a b^{4} c d^{4} + 3 \, a^{2} b^{3} d^{5}\right )} x^{3} + 3 \,{\left (21 \, b^{5} c^{3} d^{2} - 55 \, a b^{4} c^{2} d^{3} + 40 \, a^{2} b^{3} c d^{4}\right )} x^{2} + 6 \,{\left (b^{5} c^{4} d + 5 \, a b^{4} c^{3} d^{2} - 20 \, a^{2} b^{3} c^{2} d^{3} + 20 \, a^{3} b^{2} c d^{4} - 5 \, a^{4} b d^{5}\right )} x - 60 \,{\left (b^{5} c^{5} - 3 \, a b^{4} c^{4} d + 3 \, a^{2} b^{3} c^{3} d^{2} - a^{3} b^{2} c^{2} d^{3} +{\left (b^{5} c^{3} d^{2} - 3 \, a b^{4} c^{2} d^{3} + 3 \, a^{2} b^{3} c d^{4} - a^{3} b^{2} d^{5}\right )} x^{2} + 2 \,{\left (b^{5} c^{4} d - 3 \, a b^{4} c^{3} d^{2} + 3 \, a^{2} b^{3} c^{2} d^{3} - a^{3} b^{2} c d^{4}\right )} x\right )} \log \left (d x + c\right )}{6 \,{\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.39264, size = 253, normalized size = 1.9 \begin{align*} \frac{b^{5} x^{3}}{3 d^{3}} + \frac{10 b^{2} \left (a d - b c\right )^{3} \log{\left (c + d x \right )}}{d^{6}} - \frac{a^{5} d^{5} + 5 a^{4} b c d^{4} - 30 a^{3} b^{2} c^{2} d^{3} + 50 a^{2} b^{3} c^{3} d^{2} - 35 a b^{4} c^{4} d + 9 b^{5} c^{5} + x \left (10 a^{4} b d^{5} - 40 a^{3} b^{2} c d^{4} + 60 a^{2} b^{3} c^{2} d^{3} - 40 a b^{4} c^{3} d^{2} + 10 b^{5} c^{4} d\right )}{2 c^{2} d^{6} + 4 c d^{7} x + 2 d^{8} x^{2}} + \frac{x^{2} \left (5 a b^{4} d - 3 b^{5} c\right )}{2 d^{4}} + \frac{x \left (10 a^{2} b^{3} d^{2} - 15 a b^{4} c d + 6 b^{5} c^{2}\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25914, size = 356, normalized size = 2.68 \begin{align*} -\frac{10 \,{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} - \frac{9 \, b^{5} c^{5} - 35 \, a b^{4} c^{4} d + 50 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} + a^{5} d^{5} + 10 \,{\left (b^{5} c^{4} d - 4 \, a b^{4} c^{3} d^{2} + 6 \, a^{2} b^{3} c^{2} d^{3} - 4 \, a^{3} b^{2} c d^{4} + a^{4} b d^{5}\right )} x}{2 \,{\left (d x + c\right )}^{2} d^{6}} + \frac{2 \, b^{5} d^{6} x^{3} - 9 \, b^{5} c d^{5} x^{2} + 15 \, a b^{4} d^{6} x^{2} + 36 \, b^{5} c^{2} d^{4} x - 90 \, a b^{4} c d^{5} x + 60 \, a^{2} b^{3} d^{6} x}{6 \, d^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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