Optimal. Leaf size=30 \[ \frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45}
\begin {gather*} \frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x}{(a+b x)^4} \, dx &=\int \left (-\frac {a}{b (a+b x)^4}+\frac {1}{b (a+b x)^3}\right ) \, dx\\ &=\frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 20, normalized size = 0.67 \begin {gather*} -\frac {a+3 b x}{6 b^2 (a+b x)^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 27, normalized size = 0.90
method | result | size |
gosper | \(-\frac {3 b x +a}{6 \left (b x +a \right )^{3} b^{2}}\) | \(19\) |
norman | \(\frac {-\frac {x}{2 b}-\frac {a}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(22\) |
risch | \(\frac {-\frac {x}{2 b}-\frac {a}{6 b^{2}}}{\left (b x +a \right )^{3}}\) | \(22\) |
default | \(\frac {a}{3 b^{2} \left (b x +a \right )^{3}}-\frac {1}{2 b^{2} \left (b x +a \right )^{2}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 43, normalized size = 1.43 \begin {gather*} -\frac {3 \, b x + a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.86, size = 43, normalized size = 1.43 \begin {gather*} -\frac {3 \, b x + a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 44, normalized size = 1.47 \begin {gather*} \frac {- a - 3 b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 18, normalized size = 0.60 \begin {gather*} -\frac {3 \, b x + a}{6 \, {\left (b x + a\right )}^{3} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 44, normalized size = 1.47 \begin {gather*} -\frac {\frac {a}{6\,b^2}+\frac {x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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