Optimal. Leaf size=30 \[ -\frac {a^2}{4 x^4}-\frac {2 a b}{3 x^3}-\frac {b^2}{2 x^2} \]
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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 = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} -\frac {a^2}{4 x^4}-\frac {2 a b}{3 x^3}-\frac {b^2}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^5} \, dx &=\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx\\ &=-\frac {a^2}{4 x^4}-\frac {2 a b}{3 x^3}-\frac {b^2}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{4 x^4}-\frac {2 a b}{3 x^3}-\frac {b^2}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 25, normalized size = 0.83
method | result | size |
norman | \(\frac {-\frac {1}{2} x^{2} b^{2}-\frac {2}{3} a b x -\frac {1}{4} a^{2}}{x^{4}}\) | \(24\) |
risch | \(\frac {-\frac {1}{2} x^{2} b^{2}-\frac {2}{3} a b x -\frac {1}{4} a^{2}}{x^{4}}\) | \(24\) |
gosper | \(-\frac {6 x^{2} b^{2}+8 a b x +3 a^{2}}{12 x^{4}}\) | \(25\) |
default | \(-\frac {a^{2}}{4 x^{4}}-\frac {2 a b}{3 x^{3}}-\frac {b^{2}}{2 x^{2}}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 24, normalized size = 0.80 \begin {gather*} -\frac {6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.44, size = 24, normalized size = 0.80 \begin {gather*} -\frac {6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 26, normalized size = 0.87 \begin {gather*} \frac {- 3 a^{2} - 8 a b x - 6 b^{2} x^{2}}{12 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 24, normalized size = 0.80 \begin {gather*} -\frac {6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 24, normalized size = 0.80 \begin {gather*} -\frac {\frac {a^2}{4}+\frac {2\,a\,b\,x}{3}+\frac {b^2\,x^2}{2}}{x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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