Optimal. Leaf size=33 \[ \frac {1}{2} a x \sqrt {c x^2}+\frac {1}{3} b x^2 \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 43} \begin {gather*} \frac {1}{2} a x \sqrt {c x^2}+\frac {1}{3} b x^2 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \sqrt {c x^2} (a+b x) \, dx &=\frac {\sqrt {c x^2} \int x (a+b x) \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (a x+b x^2\right ) \, dx}{x}\\ &=\frac {1}{2} a x \sqrt {c x^2}+\frac {1}{3} b x^2 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{6} x \sqrt {c x^2} (3 a+2 b x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{6} x \sqrt {c x^2} (3 a+2 b x) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 20, normalized size = 0.61 \begin {gather*} \frac {1}{6} \, {\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 22, normalized size = 0.67 \begin {gather*} \frac {1}{6} \, {\left (2 \, b x^{3} \mathrm {sgn}\relax (x) + 3 \, a x^{2} \mathrm {sgn}\relax (x)\right )} \sqrt {c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.58 \begin {gather*} \frac {\left (2 b x +3 a \right ) \sqrt {c \,x^{2}}\, x}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 25, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2}} a x + \frac {\left (c x^{2}\right )^{\frac {3}{2}} b}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 20, normalized size = 0.61 \begin {gather*} \frac {\sqrt {c}\,\left (2\,b\,\sqrt {x^6}+3\,a\,x\,\relax |x|\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 34, normalized size = 1.03 \begin {gather*} \frac {a \sqrt {c} x \sqrt {x^{2}}}{2} + \frac {b \sqrt {c} x^{2} \sqrt {x^{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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