Optimal. Leaf size=38 \[ \frac {\sqrt {c x^2}}{b}-\frac {a \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac {\sqrt {c x^2}}{b}-\frac {a \sqrt {c x^2} \log (a+b x)}{b^2 x} \]
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} \, dx &=\frac {\sqrt {c x^2} \int \frac {x}{a+b x} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx}{x}\\ &=\frac {\sqrt {c x^2}}{b}-\frac {a \sqrt {c x^2} \log (a+b x)}{b^2 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {c x (b x-a \log (a+b x))}{b^2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 27, normalized size = 0.71 \[ \frac {\sqrt {c x^{2}} {\left (b x - a \log \left (b x + a\right )\right )}}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 37, normalized size = 0.97 \[ \sqrt {c} {\left (\frac {x \mathrm {sgn}\relax (x)}{b} - \frac {a \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}} + \frac {a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\relax (x)}{b^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.76 \[ -\frac {\sqrt {c \,x^{2}}\, \left (a \ln \left (b x +a \right )-b x \right )}{b^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.45, size = 74, normalized size = 1.95 \[ -\frac {\left (-1\right )^{\frac {2 \, c x}{b}} a \sqrt {c} \log \left (\frac {2 \, c x}{b}\right )}{b^{2}} - \frac {\left (-1\right )^{\frac {2 \, a c x}{b}} a \sqrt {c} \log \left (-\frac {2 \, a c x}{b {\left | b x + a \right |}}\right )}{b^{2}} + \frac {\sqrt {c x^{2}}}{b} \]
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 \[ \int \frac {\sqrt {c\,x^2}}{a+b\,x} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}}}{a + b x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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