Optimal. Leaf size=130 \[ -\frac{5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac{5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac{10 b^2 x (b c-a d)^3}{d^5}+\frac{(b c-a d)^5}{d^6 (c+d x)}+\frac{5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac{b^5 (c+d x)^4}{4 d^6} \]
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Rubi [A] time = 0.139165, antiderivative size = 130, 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} \[ -\frac{5 b^4 (c+d x)^3 (b c-a d)}{3 d^6}+\frac{5 b^3 (c+d x)^2 (b c-a d)^2}{d^6}-\frac{10 b^2 x (b c-a d)^3}{d^5}+\frac{(b c-a d)^5}{d^6 (c+d x)}+\frac{5 b (b c-a d)^4 \log (c+d x)}{d^6}+\frac{b^5 (c+d x)^4}{4 d^6} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{(c+d x)^2} \, dx &=\int \left (-\frac{10 b^2 (b c-a d)^3}{d^5}+\frac{(-b c+a d)^5}{d^5 (c+d x)^2}+\frac{5 b (b c-a d)^4}{d^5 (c+d x)}+\frac{10 b^3 (b c-a d)^2 (c+d x)}{d^5}-\frac{5 b^4 (b c-a d) (c+d x)^2}{d^5}+\frac{b^5 (c+d x)^3}{d^5}\right ) \, dx\\ &=-\frac{10 b^2 (b c-a d)^3 x}{d^5}+\frac{(b c-a d)^5}{d^6 (c+d x)}+\frac{5 b^3 (b c-a d)^2 (c+d x)^2}{d^6}-\frac{5 b^4 (b c-a d) (c+d x)^3}{3 d^6}+\frac{b^5 (c+d x)^4}{4 d^6}+\frac{5 b (b c-a d)^4 \log (c+d x)}{d^6}\\ \end{align*}
Mathematica [A] time = 0.0717922, size = 228, normalized size = 1.75 \[ \frac{60 a^2 b^3 d^2 \left (-4 c^2 d x+2 c^3-3 c d^2 x^2+d^3 x^3\right )+120 a^3 b^2 d^3 \left (-c^2+c d x+d^2 x^2\right )+60 a^4 b c d^4-12 a^5 d^5+20 a b^4 d \left (6 c^2 d^2 x^2+9 c^3 d x-3 c^4-2 c d^3 x^3+d^4 x^4\right )+60 b (c+d x) (b c-a d)^4 \log (c+d x)+b^5 \left (-30 c^3 d^2 x^2+10 c^2 d^3 x^3-48 c^4 d x+12 c^5-5 c d^4 x^4+3 d^5 x^5\right )}{12 d^6 (c+d x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 326, normalized size = 2.5 \begin{align*}{\frac{{b}^{5}{x}^{4}}{4\,{d}^{2}}}+{\frac{5\,a{b}^{4}{x}^{3}}{3\,{d}^{2}}}-{\frac{2\,{b}^{5}{x}^{3}c}{3\,{d}^{3}}}+5\,{\frac{{a}^{2}{b}^{3}{x}^{2}}{{d}^{2}}}-5\,{\frac{a{b}^{4}{x}^{2}c}{{d}^{3}}}+{\frac{3\,{b}^{5}{x}^{2}{c}^{2}}{2\,{d}^{4}}}+10\,{\frac{{a}^{3}{b}^{2}x}{{d}^{2}}}-20\,{\frac{{a}^{2}{b}^{3}cx}{{d}^{3}}}+15\,{\frac{a{b}^{4}{c}^{2}x}{{d}^{4}}}-4\,{\frac{{b}^{5}{c}^{3}x}{{d}^{5}}}-{\frac{{a}^{5}}{d \left ( dx+c \right ) }}+5\,{\frac{{a}^{4}bc}{{d}^{2} \left ( dx+c \right ) }}-10\,{\frac{{a}^{3}{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+10\,{\frac{{a}^{2}{b}^{3}{c}^{3}}{{d}^{4} \left ( dx+c \right ) }}-5\,{\frac{a{b}^{4}{c}^{4}}{{d}^{5} \left ( dx+c \right ) }}+{\frac{{b}^{5}{c}^{5}}{{d}^{6} \left ( dx+c \right ) }}+5\,{\frac{b\ln \left ( dx+c \right ){a}^{4}}{{d}^{2}}}-20\,{\frac{{b}^{2}\ln \left ( dx+c \right ){a}^{3}c}{{d}^{3}}}+30\,{\frac{{b}^{3}\ln \left ( dx+c \right ){a}^{2}{c}^{2}}{{d}^{4}}}-20\,{\frac{{b}^{4}\ln \left ( dx+c \right ) a{c}^{3}}{{d}^{5}}}+5\,{\frac{{b}^{5}\ln \left ( dx+c \right ){c}^{4}}{{d}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.97263, size = 356, normalized size = 2.74 \begin{align*} \frac{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}}{d^{7} x + c d^{6}} + \frac{3 \, b^{5} d^{3} x^{4} - 4 \,{\left (2 \, b^{5} c d^{2} - 5 \, a b^{4} d^{3}\right )} x^{3} + 6 \,{\left (3 \, b^{5} c^{2} d - 10 \, a b^{4} c d^{2} + 10 \, a^{2} b^{3} d^{3}\right )} x^{2} - 12 \,{\left (4 \, b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 20 \, a^{2} b^{3} c d^{2} - 10 \, a^{3} b^{2} d^{3}\right )} x}{12 \, d^{5}} + \frac{5 \,{\left (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}\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 = 1.75834, size = 767, normalized size = 5.9 \begin{align*} \frac{3 \, b^{5} d^{5} x^{5} + 12 \, b^{5} c^{5} - 60 \, a b^{4} c^{4} d + 120 \, a^{2} b^{3} c^{3} d^{2} - 120 \, a^{3} b^{2} c^{2} d^{3} + 60 \, a^{4} b c d^{4} - 12 \, a^{5} d^{5} - 5 \,{\left (b^{5} c d^{4} - 4 \, a b^{4} d^{5}\right )} x^{4} + 10 \,{\left (b^{5} c^{2} d^{3} - 4 \, a b^{4} c d^{4} + 6 \, a^{2} b^{3} d^{5}\right )} x^{3} - 30 \,{\left (b^{5} c^{3} d^{2} - 4 \, a b^{4} c^{2} d^{3} + 6 \, a^{2} b^{3} c d^{4} - 4 \, a^{3} b^{2} d^{5}\right )} x^{2} - 12 \,{\left (4 \, b^{5} c^{4} d - 15 \, a b^{4} c^{3} d^{2} + 20 \, a^{2} b^{3} c^{2} d^{3} - 10 \, a^{3} b^{2} c d^{4}\right )} x + 60 \,{\left (b^{5} c^{5} - 4 \, a b^{4} c^{4} d + 6 \, a^{2} b^{3} c^{3} d^{2} - 4 \, a^{3} b^{2} c^{2} d^{3} + a^{4} b c d^{4} +{\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\right )} \log \left (d x + c\right )}{12 \,{\left (d^{7} x + c d^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14194, size = 224, normalized size = 1.72 \begin{align*} \frac{b^{5} x^{4}}{4 d^{2}} + \frac{5 b \left (a d - b c\right )^{4} \log{\left (c + d x \right )}}{d^{6}} - \frac{a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}}{c d^{6} + d^{7} x} + \frac{x^{3} \left (5 a b^{4} d - 2 b^{5} c\right )}{3 d^{3}} + \frac{x^{2} \left (10 a^{2} b^{3} d^{2} - 10 a b^{4} c d + 3 b^{5} c^{2}\right )}{2 d^{4}} + \frac{x \left (10 a^{3} b^{2} d^{3} - 20 a^{2} b^{3} c d^{2} + 15 a b^{4} c^{2} d - 4 b^{5} c^{3}\right )}{d^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07567, size = 458, normalized size = 3.52 \begin{align*} \frac{{\left (3 \, b^{5} - \frac{20 \,{\left (b^{5} c d - a b^{4} d^{2}\right )}}{{\left (d x + c\right )} d} + \frac{60 \,{\left (b^{5} c^{2} d^{2} - 2 \, a b^{4} c d^{3} + a^{2} b^{3} d^{4}\right )}}{{\left (d x + c\right )}^{2} d^{2}} - \frac{120 \,{\left (b^{5} c^{3} d^{3} - 3 \, a b^{4} c^{2} d^{4} + 3 \, a^{2} b^{3} c d^{5} - a^{3} b^{2} d^{6}\right )}}{{\left (d x + c\right )}^{3} d^{3}}\right )}{\left (d x + c\right )}^{4}}{12 \, d^{6}} - \frac{5 \,{\left (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}\right )} \log \left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d^{6}} + \frac{\frac{b^{5} c^{5} d^{4}}{d x + c} - \frac{5 \, a b^{4} c^{4} d^{5}}{d x + c} + \frac{10 \, a^{2} b^{3} c^{3} d^{6}}{d x + c} - \frac{10 \, a^{3} b^{2} c^{2} d^{7}}{d x + c} + \frac{5 \, a^{4} b c d^{8}}{d x + c} - \frac{a^{5} d^{9}}{d x + c}}{d^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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