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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IntegrateAlgebraic [A] time = 0.02, size = 20, normalized size = 0.74 \begin {gather*} \frac {1}{2} \sqrt {c x^2} (2 a+b x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.92, 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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giac [A] time = 1.09, size = 17, normalized size = 0.63 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \sqrt {c} \mathrm {sgn}\relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 17, normalized size = 0.63 \begin {gather*} \frac {\left (b x +2 a \right ) \sqrt {c \,x^{2}}}{2} \end {gather*}
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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mupad [B] time = 0.19, size = 14, normalized size = 0.52 \begin {gather*} \frac {\sqrt {c}\,\relax |x|\,\left (2\,a+b\,x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 29, normalized size = 1.07 \begin {gather*} a \sqrt {c} \sqrt {x^{2}} + \frac {b \sqrt {c} x \sqrt {x^{2}}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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