Optimal. Leaf size=27 \[ a \sqrt {c x^2}+\frac {1}{2} b x \sqrt {c x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {15}
\begin {gather*} a \sqrt {c x^2}+\frac {1}{2} b x \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx &=\frac {\sqrt {c x^2} \int (a+b x) \, dx}{x}\\ &=a \sqrt {c x^2}+\frac {1}{2} b x \sqrt {c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 0.89 \begin {gather*} \frac {c x^2 (2 a+b x)}{2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 17, normalized size = 0.63
method | result | size |
gosper | \(\frac {\left (b x +2 a \right ) \sqrt {c \,x^{2}}}{2}\) | \(17\) |
default | \(\frac {\left (b x +2 a \right ) \sqrt {c \,x^{2}}}{2}\) | \(17\) |
risch | \(a \sqrt {c \,x^{2}}+\frac {b x \sqrt {c \,x^{2}}}{2}\) | \(22\) |
trager | \(\frac {\left (b x +2 a +b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{2 x}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 16, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2}} {\left (b x + 2 \, a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 22, normalized size = 0.81 \begin {gather*} a \sqrt {c x^{2}} + \frac {b x \sqrt {c x^{2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.90, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \sqrt {c} \mathrm {sgn}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 14, normalized size = 0.52 \begin {gather*} \frac {\sqrt {c}\,\left |x\right |\,\left (2\,a+b\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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