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 = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.02, 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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fricas [B] time = 0.40, size = 60, normalized size = 2.14 \[ -\frac {3 \, b^{2} x^{2} + a^{2}}{3 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 29, normalized size = 1.04 \[ -\frac {3 \, b^{2} x^{2} + a^{2}}{3 \, {\left (b x - a\right )}^{3} b c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 1.86 \[ \frac {-\frac {4 a^{2}}{3 \left (b x -a \right )^{3} b}-\frac {2 a}{\left (b x -a \right )^{2} b}-\frac {1}{\left (b x -a \right ) b}}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.33, size = 60, normalized size = 2.14 \[ -\frac {3 \, b^{2} x^{2} + a^{2}}{3 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 58, normalized size = 2.07 \[ \frac {b\,x^2+\frac {a^2}{3\,b}}{a^3\,c^4-3\,a^2\,b\,c^4\,x+3\,a\,b^2\,c^4\,x^2-b^3\,c^4\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.35, size = 61, normalized size = 2.18 \[ \frac {- a^{2} - 3 b^{2} x^{2}}{- 3 a^{3} b c^{4} + 9 a^{2} b^{2} c^{4} x - 9 a b^{3} c^{4} x^{2} + 3 b^{4} c^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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