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 = 0.02, antiderivative size = 56, 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 = {45}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 45
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*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.62 \begin {gather*} \frac {a^2+2 a b x+3 b^2 x^2}{6 b c^5 (a-b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 48, normalized size = 0.86
method | result | size |
risch | \(\frac {\frac {x^{2} b}{2}+\frac {a x}{3}+\frac {a^{2}}{6 b}}{c^{5} \left (-b x +a \right )^{4}}\) | \(32\) |
gosper | \(\frac {3 x^{2} b^{2}+2 a b x +a^{2}}{6 \left (-b x +a \right )^{4} c^{5} b}\) | \(34\) |
norman | \(\frac {\frac {a^{2}}{6 b c}+\frac {b \,x^{2}}{2 c}+\frac {a x}{3 c}}{c^{4} \left (-b x +a \right )^{4}}\) | \(41\) |
default | \(\frac {\frac {1}{2 b \left (-b x +a \right )^{2}}-\frac {4 a}{3 b \left (-b x +a \right )^{3}}+\frac {a^{2}}{b \left (-b x +a \right )^{4}}}{c^{5}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 78, normalized size = 1.39 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 78, normalized size = 1.39 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 85, normalized size = 1.52 \begin {gather*} - \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 64, normalized size = 1.14 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 76, normalized size = 1.36 \begin {gather*} \frac {\frac {a\,x}{3}+\frac {b\,x^2}{2}+\frac {a^2}{6\,b}}{a^4\,c^5-4\,a^3\,b\,c^5\,x+6\,a^2\,b^2\,c^5\,x^2-4\,a\,b^3\,c^5\,x^3+b^4\,c^5\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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