Optimal. Leaf size=32 \[ \frac {b \sqrt {c x^2} \log (x)}{x}-\frac {a \sqrt {c x^2}}{x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 32, 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} \[ \frac {b \sqrt {c x^2} \log (x)}{x}-\frac {a \sqrt {c x^2}}{x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx &=\frac {\sqrt {c x^2} \int \frac {a+b x}{x^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {a}{x^2}+\frac {b}{x}\right ) \, dx}{x}\\ &=-\frac {a \sqrt {c x^2}}{x^2}+\frac {b \sqrt {c x^2} \log (x)}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.62 \[ \frac {c (b x \log (x)-a)}{\sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 20, normalized size = 0.62 \[ \frac {\sqrt {c x^{2}} {\left (b x \log \relax (x) - a\right )}}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 20, normalized size = 0.62 \[ {\left (b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\relax (x) - \frac {a \mathrm {sgn}\relax (x)}{x}\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.66 \[ \frac {\sqrt {c \,x^{2}}\, \left (b x \ln \relax (x )-a \right )}{x^{2}} \]
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 [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {c\,x^2}\,\left (a+b\,x\right )}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}} \left (a + b x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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