Optimal. Leaf size=122 \[ \frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}-\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{(a+b x)^5}{5 d} \]
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Rubi [A] time = 0.0533321, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{b x (b c-a d)^4}{d^5}-\frac{(a+b x)^2 (b c-a d)^3}{2 d^4}+\frac{(a+b x)^3 (b c-a d)^2}{3 d^3}-\frac{(a+b x)^4 (b c-a d)}{4 d^2}-\frac{(b c-a d)^5 \log (c+d x)}{d^6}+\frac{(a+b x)^5}{5 d} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{c+d x} \, dx &=\int \left (\frac{b (b c-a d)^4}{d^5}-\frac{b (b c-a d)^3 (a+b x)}{d^4}+\frac{b (b c-a d)^2 (a+b x)^2}{d^3}-\frac{b (b c-a d) (a+b x)^3}{d^2}+\frac{b (a+b x)^4}{d}+\frac{(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx\\ &=\frac{b (b c-a d)^4 x}{d^5}-\frac{(b c-a d)^3 (a+b x)^2}{2 d^4}+\frac{(b c-a d)^2 (a+b x)^3}{3 d^3}-\frac{(b c-a d) (a+b x)^4}{4 d^2}+\frac{(a+b x)^5}{5 d}-\frac{(b c-a d)^5 \log (c+d x)}{d^6}\\ \end{align*}
Mathematica [A] time = 0.0663722, size = 167, normalized size = 1.37 \[ \frac{b d x \left (100 a^2 b^2 d^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )+300 a^3 b d^3 (d x-2 c)+300 a^4 d^4+25 a b^3 d \left (6 c^2 d x-12 c^3-4 c d^2 x^2+3 d^3 x^3\right )+b^4 \left (20 c^2 d^2 x^2-30 c^3 d x+60 c^4-15 c d^3 x^3+12 d^4 x^4\right )\right )-60 (b c-a d)^5 \log (c+d x)}{60 d^6} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 302, normalized size = 2.5 \begin{align*}{\frac{{b}^{5}{x}^{5}}{5\,d}}+{\frac{5\,a{b}^{4}{x}^{4}}{4\,d}}-{\frac{{b}^{5}{x}^{4}c}{4\,{d}^{2}}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{3}}{3\,d}}-{\frac{5\,a{b}^{4}{x}^{3}c}{3\,{d}^{2}}}+{\frac{{b}^{5}{x}^{3}{c}^{2}}{3\,{d}^{3}}}+5\,{\frac{{a}^{3}{b}^{2}{x}^{2}}{d}}-5\,{\frac{{a}^{2}{b}^{3}{x}^{2}c}{{d}^{2}}}+{\frac{5\,a{b}^{4}{x}^{2}{c}^{2}}{2\,{d}^{3}}}-{\frac{{b}^{5}{x}^{2}{c}^{3}}{2\,{d}^{4}}}+5\,{\frac{{a}^{4}bx}{d}}-10\,{\frac{{a}^{3}{b}^{2}cx}{{d}^{2}}}+10\,{\frac{{a}^{2}{b}^{3}{c}^{2}x}{{d}^{3}}}-5\,{\frac{a{b}^{4}{c}^{3}x}{{d}^{4}}}+{\frac{{b}^{5}{c}^{4}x}{{d}^{5}}}+{\frac{\ln \left ( dx+c \right ){a}^{5}}{d}}-5\,{\frac{\ln \left ( dx+c \right ){a}^{4}bc}{{d}^{2}}}+10\,{\frac{\ln \left ( dx+c \right ){a}^{3}{b}^{2}{c}^{2}}{{d}^{3}}}-10\,{\frac{\ln \left ( dx+c \right ){a}^{2}{b}^{3}{c}^{3}}{{d}^{4}}}+5\,{\frac{\ln \left ( dx+c \right ) a{b}^{4}{c}^{4}}{{d}^{5}}}-{\frac{\ln \left ( dx+c \right ){b}^{5}{c}^{5}}{{d}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.974107, size = 348, normalized size = 2.85 \begin{align*} \frac{12 \, b^{5} d^{4} x^{5} - 15 \,{\left (b^{5} c d^{3} - 5 \, a b^{4} d^{4}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{2} - 5 \, a b^{4} c d^{3} + 10 \, a^{2} b^{3} d^{4}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d - 5 \, a b^{4} c^{2} d^{2} + 10 \, a^{2} b^{3} c d^{3} - 10 \, a^{3} b^{2} d^{4}\right )} x^{2} + 60 \,{\left (b^{5} c^{4} - 5 \, a b^{4} c^{3} d + 10 \, a^{2} b^{3} c^{2} d^{2} - 10 \, a^{3} b^{2} c d^{3} + 5 \, a^{4} b d^{4}\right )} x}{60 \, d^{5}} - \frac{{\left (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}\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 = 1.84718, size = 537, normalized size = 4.4 \begin{align*} \frac{12 \, b^{5} d^{5} x^{5} - 15 \,{\left (b^{5} c d^{4} - 5 \, a b^{4} d^{5}\right )} x^{4} + 20 \,{\left (b^{5} c^{2} d^{3} - 5 \, a b^{4} c d^{4} + 10 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d^{2} - 5 \, a b^{4} c^{2} d^{3} + 10 \, a^{2} b^{3} c d^{4} - 10 \, a^{3} b^{2} d^{5}\right )} x^{2} + 60 \,{\left (b^{5} c^{4} d - 5 \, a b^{4} c^{3} d^{2} + 10 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4} + 5 \, a^{4} b d^{5}\right )} x - 60 \,{\left (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}\right )} \log \left (d x + c\right )}{60 \, d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.662197, size = 202, normalized size = 1.66 \begin{align*} \frac{b^{5} x^{5}}{5 d} + \frac{x^{4} \left (5 a b^{4} d - b^{5} c\right )}{4 d^{2}} + \frac{x^{3} \left (10 a^{2} b^{3} d^{2} - 5 a b^{4} c d + b^{5} c^{2}\right )}{3 d^{3}} + \frac{x^{2} \left (10 a^{3} b^{2} d^{3} - 10 a^{2} b^{3} c d^{2} + 5 a b^{4} c^{2} d - b^{5} c^{3}\right )}{2 d^{4}} + \frac{x \left (5 a^{4} b d^{4} - 10 a^{3} b^{2} c d^{3} + 10 a^{2} b^{3} c^{2} d^{2} - 5 a b^{4} c^{3} d + b^{5} c^{4}\right )}{d^{5}} + \frac{\left (a d - b c\right )^{5} \log{\left (c + d x \right )}}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.05788, size = 369, normalized size = 3.02 \begin{align*} \frac{12 \, b^{5} d^{4} x^{5} - 15 \, b^{5} c d^{3} x^{4} + 75 \, a b^{4} d^{4} x^{4} + 20 \, b^{5} c^{2} d^{2} x^{3} - 100 \, a b^{4} c d^{3} x^{3} + 200 \, a^{2} b^{3} d^{4} x^{3} - 30 \, b^{5} c^{3} d x^{2} + 150 \, a b^{4} c^{2} d^{2} x^{2} - 300 \, a^{2} b^{3} c d^{3} x^{2} + 300 \, a^{3} b^{2} d^{4} x^{2} + 60 \, b^{5} c^{4} x - 300 \, a b^{4} c^{3} d x + 600 \, a^{2} b^{3} c^{2} d^{2} x - 600 \, a^{3} b^{2} c d^{3} x + 300 \, a^{4} b d^{4} x}{60 \, d^{5}} - \frac{{\left (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}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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