Optimal. Leaf size=57 \[ \frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \]
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Rubi [A] time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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} \[ \frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3} \]
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)^6} \, dx &=\int \left (\frac {4 a^2}{c^6 (a-b x)^6}-\frac {4 a}{c^6 (a-b x)^5}+\frac {1}{c^6 (a-b x)^4}\right ) \, dx\\ &=\frac {4 a^2}{5 b c^6 (a-b x)^5}-\frac {a}{b c^6 (a-b x)^4}+\frac {1}{3 b c^6 (a-b x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 0.67 \[ -\frac {2 a^2+5 a b x+5 b^2 x^2}{15 b c^6 (b x-a)^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 95, normalized size = 1.67 \[ -\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 36, normalized size = 0.63 \[ -\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b x - a\right )}^{5} b c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.91 \[ \frac {-\frac {4 a^{2}}{5 \left (b x -a \right )^{5} b}-\frac {a}{\left (b x -a \right )^{4} b}-\frac {1}{3 \left (b x -a \right )^{3} b}}{c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 95, normalized size = 1.67 \[ -\frac {5 \, b^{2} x^{2} + 5 \, a b x + 2 \, a^{2}}{15 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 91, normalized size = 1.60 \[ \frac {\frac {a\,x}{3}+\frac {b\,x^2}{3}+\frac {2\,a^2}{15\,b}}{a^5\,c^6-5\,a^4\,b\,c^6\,x+10\,a^3\,b^2\,c^6\,x^2-10\,a^2\,b^3\,c^6\,x^3+5\,a\,b^4\,c^6\,x^4-b^5\,c^6\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.51, size = 100, normalized size = 1.75 \[ \frac {- 2 a^{2} - 5 a b x - 5 b^{2} x^{2}}{- 15 a^{5} b c^{6} + 75 a^{4} b^{2} c^{6} x - 150 a^{3} b^{3} c^{6} x^{2} + 150 a^{2} b^{4} c^{6} x^{3} - 75 a b^{5} c^{6} x^{4} + 15 b^{6} c^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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