Optimal. Leaf size=75 \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2} \]
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Rubi [A] time = 0.0664855, antiderivative size = 75, 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} \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}+\frac{b^3 x^2}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^5}{\left (a c+(b c+a d) x+b d x^2\right )^2} \, dx &=\int \frac{(a+b x)^3}{(c+d x)^2} \, dx\\ &=\int \left (-\frac{b^2 (2 b c-3 a d)}{d^3}+\frac{b^3 x}{d^2}+\frac{(-b c+a d)^3}{d^3 (c+d x)^2}+\frac{3 b (b c-a d)^2}{d^3 (c+d x)}\right ) \, dx\\ &=-\frac{b^2 (2 b c-3 a d) x}{d^3}+\frac{b^3 x^2}{2 d^2}+\frac{(b c-a d)^3}{d^4 (c+d x)}+\frac{3 b (b c-a d)^2 \log (c+d x)}{d^4}\\ \end{align*}
Mathematica [A] time = 0.0341325, size = 114, normalized size = 1.52 \[ \frac{3 a^2 b c d^2-a^3 d^3-3 a b^2 c^2 d+b^3 c^3}{d^4 (c+d x)}+\frac{3 \left (a^2 b d^2-2 a b^2 c d+b^3 c^2\right ) \log (c+d x)}{d^4}-\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{b^3 x^2}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.046, size = 149, normalized size = 2. \begin{align*}{\frac{{b}^{3}{x}^{2}}{2\,{d}^{2}}}+3\,{\frac{{b}^{2}ax}{{d}^{2}}}-2\,{\frac{{b}^{3}cx}{{d}^{3}}}+3\,{\frac{b\ln \left ( dx+c \right ){a}^{2}}{{d}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( dx+c \right ) ca}{{d}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( dx+c \right ){c}^{2}}{{d}^{4}}}-{\frac{{a}^{3}}{d \left ( dx+c \right ) }}+3\,{\frac{bc{a}^{2}}{{d}^{2} \left ( dx+c \right ) }}-3\,{\frac{a{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }}+{\frac{{b}^{3}{c}^{3}}{{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 = 1.05338, size = 158, normalized size = 2.11 \begin{align*} \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{d^{5} x + c d^{4}} + \frac{b^{3} d x^{2} - 2 \,{\left (2 \, b^{3} c - 3 \, a b^{2} d\right )} x}{2 \, d^{3}} + \frac{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\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 = 1.52327, size = 354, normalized size = 4.72 \begin{align*} \frac{b^{3} d^{3} x^{3} + 2 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} - 3 \,{\left (b^{3} c d^{2} - 2 \, a b^{2} d^{3}\right )} x^{2} - 2 \,{\left (2 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2}\right )} x + 6 \,{\left (b^{3} c^{3} - 2 \, a b^{2} c^{2} d + a^{2} b c d^{2} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (d x + c\right )}{2 \,{\left (d^{5} x + c d^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.13319, size = 100, normalized size = 1.33 \begin{align*} \frac{b^{3} x^{2}}{2 d^{2}} + \frac{3 b \left (a d - b c\right )^{2} \log{\left (c + d x \right )}}{d^{4}} - \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{c d^{4} + d^{5} x} + \frac{x \left (3 a b^{2} d - 2 b^{3} c\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17548, size = 159, normalized size = 2.12 \begin{align*} \frac{3 \,{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{d^{4}} + \frac{b^{3} d^{2} x^{2} - 4 \, b^{3} c d x + 6 \, a b^{2} d^{2} x}{2 \, d^{4}} + \frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{{\left (d x + c\right )} d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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