Optimal. Leaf size=59 \[ -\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3} \]
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Rubi [A] time = 0.0431223, antiderivative size = 59, 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} \[ -\frac{2 d (b c-a d)}{b^3 (a+b x)}-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}+\frac{d^2 \log (a+b x)}{b^3} \]
Antiderivative was successfully verified.
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Rule 626
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a c+(b c+a d) x+b d x^2\right )^2}{(a+b x)^5} \, dx &=\int \frac{(c+d x)^2}{(a+b x)^3} \, dx\\ &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^3}+\frac{2 d (b c-a d)}{b^2 (a+b x)^2}+\frac{d^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{2 b^3 (a+b x)^2}-\frac{2 d (b c-a d)}{b^3 (a+b x)}+\frac{d^2 \log (a+b x)}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0241648, size = 49, normalized size = 0.83 \[ \frac{2 d^2 \log (a+b x)-\frac{(b c-a d) (3 a d+b (c+4 d x))}{(a+b x)^2}}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 92, normalized size = 1.6 \begin{align*} -{\frac{{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{acd}{{b}^{2} \left ( bx+a \right ) ^{2}}}-{\frac{{c}^{2}}{2\,b \left ( bx+a \right ) ^{2}}}+{\frac{{d}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}}+2\,{\frac{a{d}^{2}}{{b}^{3} \left ( bx+a \right ) }}-2\,{\frac{cd}{{b}^{2} \left ( bx+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994966, size = 107, normalized size = 1.81 \begin{align*} -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} + \frac{d^{2} \log \left (b x + a\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60843, size = 207, normalized size = 3.51 \begin{align*} -\frac{b^{2} c^{2} + 2 \, a b c d - 3 \, a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x - 2 \,{\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )}{2 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.853165, size = 80, normalized size = 1.36 \begin{align*} \frac{3 a^{2} d^{2} - 2 a b c d - b^{2} c^{2} + x \left (4 a b d^{2} - 4 b^{2} c d\right )}{2 a^{2} b^{3} + 4 a b^{4} x + 2 b^{5} x^{2}} + \frac{d^{2} \log{\left (a + b x \right )}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19446, size = 149, normalized size = 2.53 \begin{align*} -\frac{d^{2} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{\frac{b^{5} c^{2}}{{\left (b x + a\right )}^{2}} + \frac{4 \, b^{4} c d}{b x + a} - \frac{2 \, a b^{4} c d}{{\left (b x + a\right )}^{2}} - \frac{4 \, a b^{3} d^{2}}{b x + a} + \frac{a^{2} b^{3} d^{2}}{{\left (b x + a\right )}^{2}}}{2 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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