Optimal. Leaf size=26 \[ -\frac {(a+b x)^2}{2 a x \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 37}
\begin {gather*} -\frac {(a+b x)^2}{2 a x \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx &=\frac {x \int \frac {a+b x}{x^3} \, dx}{\sqrt {c x^2}}\\ &=-\frac {(a+b x)^2}{2 a x \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 0.88 \begin {gather*} \frac {c x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 19, normalized size = 0.73
method | result | size |
gosper | \(-\frac {2 b x +a}{2 x \sqrt {c \,x^{2}}}\) | \(19\) |
default | \(-\frac {2 b x +a}{2 x \sqrt {c \,x^{2}}}\) | \(19\) |
risch | \(\frac {-b x -\frac {a}{2}}{x \sqrt {c \,x^{2}}}\) | \(20\) |
trager | \(\frac {\left (-1+x \right ) \left (a x +2 b x +a \right ) \sqrt {c \,x^{2}}}{2 c \,x^{3}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 19, normalized size = 0.73 \begin {gather*} -\frac {b}{\sqrt {c} x} - \frac {a}{2 \, \sqrt {c} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 21, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, c x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 24, normalized size = 0.92 \begin {gather*} - \frac {a}{2 x \sqrt {c x^{2}}} - \frac {b}{\sqrt {c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.02, size = 18, normalized size = 0.69 \begin {gather*} -\frac {2 \, b x + a}{2 \, \sqrt {c} x^{2} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 25, normalized size = 0.96 \begin {gather*} -\frac {2\,b\,x^3+a\,x^2}{2\,\sqrt {c}\,x\,{\left (x^2\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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