Optimal. Leaf size=22 \[ \frac {\sqrt {c x^2} \log (a+b x)}{b x} \]
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Rubi [A] time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 31} \[ \frac {\sqrt {c x^2} \log (a+b x)}{b x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 31
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x (a+b x)} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{a+b x} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \log (a+b x)}{b x}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 0.95 \[ \frac {c x \log (a+b x)}{b \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 20, normalized size = 0.91 \[ \frac {\sqrt {c x^{2}} \log \left (b x + a\right )}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 28, normalized size = 1.27 \[ \sqrt {c} {\left (\frac {\log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b} - \frac {\log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.95 \[ \frac {\sqrt {c \,x^{2}}\, \ln \left (b x +a \right )}{b x} \]
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 = -0.05 \[ \int \frac {\sqrt {c\,x^2}}{x\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}}}{x \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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