Optimal. Leaf size=35 \[ \frac {a x \log (x)}{c \sqrt {c x^2}}+\frac {b x^2}{c \sqrt {c x^2}} \]
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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, 43} \begin {gather*} \frac {a x \log (x)}{c \sqrt {c x^2}}+\frac {b x^2}{c \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (b+\frac {a}{x}\right ) \, dx}{c \sqrt {c x^2}}\\ &=\frac {b x^2}{c \sqrt {c x^2}}+\frac {a x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 0.60 \begin {gather*} \frac {x^3 (a \log (x)+b x)}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 23, normalized size = 0.66 \begin {gather*} \frac {a x^3 \log (x)+b x^4}{\left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 22, normalized size = 0.63 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a \log \relax (x)\right )}}{c^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 40, normalized size = 1.14 \begin {gather*} -\frac {\frac {a \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{\sqrt {c}} - \frac {\sqrt {c x^{2}} b}{c}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.57 \begin {gather*} \frac {\left (a \ln \relax (x )+b x \right ) x^{3}}{\left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 23, normalized size = 0.66 \begin {gather*} \frac {b x^{2}}{\sqrt {c x^{2}} c} + \frac {a \log \relax (x)}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 30, normalized size = 0.86 \begin {gather*} \frac {b\,\relax |x|}{c^{3/2}}+\frac {a\,\ln \left (x+\relax |x|\right )}{c^{3/2}}-\frac {a\,x}{c^{3/2}\,\sqrt {x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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