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} \[ a c \sqrt {c x^2}+\frac {1}{2} b c x \sqrt {c x^2} \]
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 \[ \frac {1}{2} c \sqrt {c x^2} (2 a+b x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 18, normalized size = 0.62 \[ \frac {1}{2} \, {\left (b c x + 2 \, a c\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 17, normalized size = 0.59 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} c^{\frac {3}{2}} \mathrm {sgn}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.69 \[ \frac {\left (b x +2 a \right ) \left (c \,x^{2}\right )^{\frac {3}{2}}}{2 x^{2}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \frac {c^{3/2}\,\relax |x|\,\left (2\,a+b\,x\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 32, normalized size = 1.10 \[ \frac {a c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{x^{2}} + \frac {b c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}}{2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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