Optimal. Leaf size=35 \[ \frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \]
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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 {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^2} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int x (a+b x) \, dx}{x}\\ &=\frac {\left (c \sqrt {c x^2}\right ) \int \left (a x+b x^2\right ) \, dx}{x}\\ &=\frac {1}{2} a c x \sqrt {c x^2}+\frac {1}{3} b c x^2 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 23, normalized size = 0.66 \begin {gather*} \frac {1}{6} c x \sqrt {c x^2} (3 a+2 b x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.76, size = 19, normalized size = 0.54 \begin {gather*} \frac {\left (\frac {a}{2}+\frac {b x}{3}\right ) {\left (c x^2\right )}^{\frac {3}{2}}}{x} \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 | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{6 x}\) | \(21\) |
default | \(\frac {\left (2 b x +3 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{6 x}\) | \(21\) |
risch | \(\frac {a c x \sqrt {c \,x^{2}}}{2}+\frac {b c \,x^{2} \sqrt {c \,x^{2}}}{3}\) | \(28\) |
trager | \(\frac {c \left (2 x^{2} b +3 a x +2 b x +3 a +2 b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{6 x}\) | \(38\) |
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 = 0.30, size = 22, normalized size = 0.63 \begin {gather*} \frac {1}{6} \, {\left (2 \, b c x^{2} + 3 \, a c x\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 24, normalized size = 0.69 \begin {gather*} \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{2 x} + \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 26, normalized size = 0.74 \begin {gather*} \sqrt {c} c \left (\frac {1}{2} a x^{2} \mathrm {sign}\left (x\right )+\frac {1}{3} b x^{3} \mathrm {sign}\left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 20, normalized size = 0.57 \begin {gather*} \frac {c^{3/2}\,\left (2\,b\,\sqrt {x^6}+3\,a\,x\,\left |x\right |\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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