Optimal. Leaf size=33 \[ \frac {b x \log (x)}{c \sqrt {c x^2}}-\frac {a}{c \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {15, 43} \[ \frac {b x \log (x)}{c \sqrt {c x^2}}-\frac {a}{c \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {x (a+b x)}{\left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x^2} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a}{c \sqrt {c x^2}}+\frac {b x \log (x)}{c \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 22, normalized size = 0.67 \[ \frac {x^2 (b x \log (x)-a)}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 23, normalized size = 0.70 \[ \frac {\sqrt {c x^{2}} {\left (b x \log \relax (x) - a\right )}}{c^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 47, normalized size = 1.42 \[ -\frac {b \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right ) - \frac {2 \, a \sqrt {c}}{\sqrt {c} x - \sqrt {c x^{2}}}}{c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.64 \[ \frac {\left (b x \ln \relax (x )-a \right ) x^{2}}{\left (c \,x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 21, normalized size = 0.64 \[ \frac {b \log \relax (x)}{c^{\frac {3}{2}}} - \frac {a}{\sqrt {c x^{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 28, normalized size = 0.85 \[ -\frac {a+b\,x-b\,\ln \left (x+\relax |x|\right )\,\sqrt {x^2}}{c^{3/2}\,\sqrt {x^2}} \]
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 \[ \int \frac {x \left (a + b x\right )}{\left (c x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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