Optimal. Leaf size=29 \[ a c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, 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 c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^{3/2} (a+b x)}{x^3} \, dx &=\frac {\left (c \sqrt {c x^2}\right ) \int (a+b x) \, dx}{x}\\ &=a c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.72 \begin {gather*} \frac {1}{2} c \sqrt {c x^2} (2 a+b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.69
method | result | size |
gosper | \(\frac {\left (b x +2 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{2 x^{2}}\) | \(20\) |
default | \(\frac {\left (b x +2 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{2 x^{2}}\) | \(20\) |
risch | \(a c \sqrt {c \,x^{2}}+\frac {b c x \sqrt {c \,x^{2}}}{2}\) | \(24\) |
trager | \(\frac {c \left (b x +2 a +b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{2 x}\) | \(25\) |
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 = 1.29, size = 18, normalized size = 0.62 \begin {gather*} \frac {1}{2} \, {\left (b c x + 2 \, a c\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 26, normalized size = 0.90 \begin {gather*} \frac {a \left (c x^{2}\right )^{\frac {3}{2}}}{x^{2}} + \frac {b \left (c x^{2}\right )^{\frac {3}{2}}}{2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.65, size = 17, normalized size = 0.59 \begin {gather*} \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \mathrm {sgn}\left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 14, normalized size = 0.48 \begin {gather*} \frac {c^{3/2}\,\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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