Optimal. Leaf size=65 \[ -\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2} \]
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Rubi [A] time = 0.0331657, antiderivative size = 65, 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} \[ -\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{d^2}{2 b^3 (a+b x)^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(c+d x)^2}{(a+b x)^5} \, dx &=\int \left (\frac{(b c-a d)^2}{b^2 (a+b x)^5}+\frac{2 d (b c-a d)}{b^2 (a+b x)^4}+\frac{d^2}{b^2 (a+b x)^3}\right ) \, dx\\ &=-\frac{(b c-a d)^2}{4 b^3 (a+b x)^4}-\frac{2 d (b c-a d)}{3 b^3 (a+b x)^3}-\frac{d^2}{2 b^3 (a+b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0187041, size = 56, normalized size = 0.86 \[ -\frac{a^2 d^2+2 a b d (c+2 d x)+b^2 \left (3 c^2+8 c d x+6 d^2 x^2\right )}{12 b^3 (a+b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 71, normalized size = 1.1 \begin{align*} -{\frac{{d}^{2}}{2\,{b}^{3} \left ( bx+a \right ) ^{2}}}+{\frac{2\,d \left ( ad-bc \right ) }{3\,{b}^{3} \left ( bx+a \right ) ^{3}}}-{\frac{{a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2}}{4\,{b}^{3} \left ( bx+a \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962005, size = 132, normalized size = 2.03 \begin{align*} -\frac{6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00984, size = 201, normalized size = 3.09 \begin{align*} -\frac{6 \, b^{2} d^{2} x^{2} + 3 \, b^{2} c^{2} + 2 \, a b c d + a^{2} d^{2} + 4 \,{\left (2 \, b^{2} c d + a b d^{2}\right )} x}{12 \,{\left (b^{7} x^{4} + 4 \, a b^{6} x^{3} + 6 \, a^{2} b^{5} x^{2} + 4 \, a^{3} b^{4} x + a^{4} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.938739, size = 104, normalized size = 1.6 \begin{align*} - \frac{a^{2} d^{2} + 2 a b c d + 3 b^{2} c^{2} + 6 b^{2} d^{2} x^{2} + x \left (4 a b d^{2} + 8 b^{2} c d\right )}{12 a^{4} b^{3} + 48 a^{3} b^{4} x + 72 a^{2} b^{5} x^{2} + 48 a b^{6} x^{3} + 12 b^{7} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06059, size = 130, normalized size = 2. \begin{align*} -\frac{\frac{3 \, c^{2}}{{\left (b x + a\right )}^{4}} + \frac{8 \, c d}{{\left (b x + a\right )}^{3} b} - \frac{6 \, a c d}{{\left (b x + a\right )}^{4} b} + \frac{6 \, d^{2}}{{\left (b x + a\right )}^{2} b^{2}} - \frac{8 \, a d^{2}}{{\left (b x + a\right )}^{3} b^{2}} + \frac{3 \, a^{2} d^{2}}{{\left (b x + a\right )}^{4} b^{2}}}{12 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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