Optimal. Leaf size=75 \[ \frac{d^2 x (3 b c-2 a d)}{b^3}-\frac{(b c-a d)^3}{b^4 (a+b x)}+\frac{3 d (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^3 x^2}{2 b^2} \]
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Rubi [A] time = 0.0614668, antiderivative size = 75, 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} \[ \frac{d^2 x (3 b c-2 a d)}{b^3}-\frac{(b c-a d)^3}{b^4 (a+b x)}+\frac{3 d (b c-a d)^2 \log (a+b x)}{b^4}+\frac{d^3 x^2}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^3}{(a+b x)^2} \, dx &=\int \left (\frac{d^2 (3 b c-2 a d)}{b^3}+\frac{d^3 x}{b^2}+\frac{(b c-a d)^3}{b^3 (a+b x)^2}+\frac{3 d (b c-a d)^2}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{d^2 (3 b c-2 a d) x}{b^3}+\frac{d^3 x^2}{2 b^2}-\frac{(b c-a d)^3}{b^4 (a+b x)}+\frac{3 d (b c-a d)^2 \log (a+b x)}{b^4}\\ \end{align*}
Mathematica [A] time = 0.0501473, size = 72, normalized size = 0.96 \[ \frac{2 b d^2 x (3 b c-2 a d)-\frac{2 (b c-a d)^3}{a+b x}+6 d (b c-a d)^2 \log (a+b x)+b^2 d^3 x^2}{2 b^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 149, normalized size = 2. \begin{align*}{\frac{{d}^{3}{x}^{2}}{2\,{b}^{2}}}-2\,{\frac{a{d}^{3}x}{{b}^{3}}}+3\,{\frac{{d}^{2}xc}{{b}^{2}}}+{\frac{{a}^{3}{d}^{3}}{{b}^{4} \left ( bx+a \right ) }}-3\,{\frac{{a}^{2}c{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}+3\,{\frac{a{c}^{2}d}{{b}^{2} \left ( bx+a \right ) }}-{\frac{{c}^{3}}{b \left ( bx+a \right ) }}+3\,{\frac{{d}^{3}\ln \left ( bx+a \right ){a}^{2}}{{b}^{4}}}-6\,{\frac{{d}^{2}\ln \left ( bx+a \right ) ac}{{b}^{3}}}+3\,{\frac{d\ln \left ( bx+a \right ){c}^{2}}{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05798, size = 159, normalized size = 2.12 \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}}{b^{5} x + a b^{4}} + \frac{b d^{3} x^{2} + 2 \,{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x}{2 \, b^{3}} + \frac{3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (b x + a\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19866, 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 (2 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + 2 \,{\left (3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x + 6 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.851266, size = 100, normalized size = 1.33 \begin{align*} \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{a b^{4} + b^{5} x} + \frac{d^{3} x^{2}}{2 b^{2}} - \frac{x \left (2 a d^{3} - 3 b c d^{2}\right )}{b^{3}} + \frac{3 d \left (a d - b c\right )^{2} \log{\left (a + b x \right )}}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15804, size = 225, normalized size = 3. \begin{align*} \frac{{\left (d^{3} + \frac{6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )}}{{\left (b x + a\right )} b}\right )}{\left (b x + a\right )}^{2}}{2 \, b^{4}} - \frac{3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{4}} - \frac{\frac{b^{5} c^{3}}{b x + a} - \frac{3 \, a b^{4} c^{2} d}{b x + a} + \frac{3 \, a^{2} b^{3} c d^{2}}{b x + a} - \frac{a^{3} b^{2} d^{3}}{b x + a}}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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