Optimal. Leaf size=35 \[ \frac {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45}
\begin {gather*} \frac {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)}{\sqrt {c x^2}} \, dx &=\frac {x \int x (a+b x) \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (a x+b x^2\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a x^3}{2 \sqrt {c x^2}}+\frac {b x^4}{3 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 0.69 \begin {gather*} \frac {x^3 (3 a+2 b x)}{6 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.74, size = 19, normalized size = 0.54 \begin {gather*} \frac {x^3 \left (\frac {a}{2}+\frac {b x}{3}\right )}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 0.60
method | result | size |
gosper | \(\frac {x^{3} \left (2 b x +3 a \right )}{6 \sqrt {c \,x^{2}}}\) | \(21\) |
default | \(\frac {x^{3} \left (2 b x +3 a \right )}{6 \sqrt {c \,x^{2}}}\) | \(21\) |
risch | \(\frac {a \,x^{3}}{2 \sqrt {c \,x^{2}}}+\frac {b \,x^{4}}{3 \sqrt {c \,x^{2}}}\) | \(28\) |
trager | \(\frac {\left (2 x^{2} b +3 a x +2 b x +3 a +2 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{6 c x}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 26, normalized size = 0.74 \begin {gather*} \frac {\sqrt {c x^{2}} b x^{2}}{3 \, c} + \frac {a x^{2}}{2 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 23, normalized size = 0.66 \begin {gather*} \frac {{\left (2 \, b x^{2} + 3 \, a x\right )} \sqrt {c x^{2}}}{6 \, c} \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 = 0.83 \begin {gather*} \frac {a x^{3}}{2 \sqrt {c x^{2}}} + \frac {b x^{4}}{3 \sqrt {c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 25, normalized size = 0.71 \begin {gather*} \frac {\frac {1}{3} b x^{3}+\frac {1}{2} a x^{2}}{\sqrt {c} \mathrm {sign}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 23, normalized size = 0.66 \begin {gather*} \frac {2\,b\,\sqrt {x^6}+3\,a\,x\,\sqrt {x^2}}{6\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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