Optimal. Leaf size=28 \[ \frac{(a+b x)^3}{6 a b c^4 (a-b x)^3} \]
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Rubi [A] time = 0.0051261, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ \frac{(a+b x)^3}{6 a b c^4 (a-b x)^3} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{(a c-b c x)^4} \, dx &=\frac{(a+b x)^3}{6 a b c^4 (a-b x)^3}\\ \end{align*}
Mathematica [A] time = 0.0182353, size = 31, normalized size = 1.11 \[ -\frac{a^2+3 b^2 x^2}{3 b c^4 (b x-a)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 52, normalized size = 1.9 \begin{align*}{\frac{1}{{c}^{4}} \left ( -{\frac{4\,{a}^{2}}{3\,b \left ( bx-a \right ) ^{3}}}-{\frac{1}{b \left ( bx-a \right ) }}-2\,{\frac{a}{b \left ( bx-a \right ) ^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.0375, size = 81, normalized size = 2.89 \begin{align*} -\frac{3 \, b^{2} x^{2} + a^{2}}{3 \,{\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55696, size = 116, normalized size = 4.14 \begin{align*} -\frac{3 \, b^{2} x^{2} + a^{2}}{3 \,{\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.507735, size = 61, normalized size = 2.18 \begin{align*} - \frac{a^{2} + 3 b^{2} x^{2}}{- 3 a^{3} b c^{4} + 9 a^{2} b^{2} c^{4} x - 9 a b^{3} c^{4} x^{2} + 3 b^{4} c^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06528, size = 39, normalized size = 1.39 \begin{align*} -\frac{3 \, b^{2} x^{2} + a^{2}}{3 \,{\left (b x - a\right )}^{3} b c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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