Optimal. Leaf size=56 \[ \frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2} \]
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Rubi [A] time = 0.0237935, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {43} \[ \frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(a c-b c x)^5} \, dx &=\int \left (\frac{4 a^2}{c^5 (a-b x)^5}-\frac{4 a}{c^5 (a-b x)^4}+\frac{1}{c^5 (a-b x)^3}\right ) \, dx\\ &=\frac{a^2}{b c^5 (a-b x)^4}-\frac{4 a}{3 b c^5 (a-b x)^3}+\frac{1}{2 b c^5 (a-b x)^2}\\ \end{align*}
Mathematica [A] time = 0.0119657, size = 35, normalized size = 0.62 \[ \frac{a^2+2 a b x+3 b^2 x^2}{6 b c^5 (a-b x)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 51, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{5}} \left ({\frac{{a}^{2}}{b \left ( bx-a \right ) ^{4}}}+{\frac{4\,a}{3\,b \left ( bx-a \right ) ^{3}}}+{\frac{1}{2\,b \left ( bx-a \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06078, size = 105, normalized size = 1.88 \begin{align*} \frac{3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48695, size = 155, normalized size = 2.77 \begin{align*} \frac{3 \, b^{2} x^{2} + 2 \, a b x + a^{2}}{6 \,{\left (b^{5} c^{5} x^{4} - 4 \, a b^{4} c^{5} x^{3} + 6 \, a^{2} b^{3} c^{5} x^{2} - 4 \, a^{3} b^{2} c^{5} x + a^{4} b c^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.584813, size = 82, normalized size = 1.46 \begin{align*} \frac{a^{2} + 2 a b x + 3 b^{2} x^{2}}{6 a^{4} b c^{5} - 24 a^{3} b^{2} c^{5} x + 36 a^{2} b^{3} c^{5} x^{2} - 24 a b^{4} c^{5} x^{3} + 6 b^{5} c^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05837, size = 86, normalized size = 1.54 \begin{align*} \frac{\frac{6 \, a^{2}}{{\left (b c x - a c\right )}^{4} b} + \frac{8 \, a}{{\left (b c x - a c\right )}^{3} b c} + \frac{3}{{\left (b c x - a c\right )}^{2} b c^{2}}}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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