Optimal. Leaf size=42 \[ \frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \]
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Rubi [A] time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {15, 36, 29, 31} \[ \frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 31
Rule 36
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^2 (a+b x)} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x (a+b x)} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \frac {1}{x} \, dx}{a x}-\frac {\left (b \sqrt {c x^2}\right ) \int \frac {1}{a+b x} \, dx}{a x}\\ &=\frac {\sqrt {c x^2} \log (x)}{a x}-\frac {\sqrt {c x^2} \log (a+b x)}{a x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.62 \[ \frac {c x (\log (x)-\log (a+b x))}{a \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 64, normalized size = 1.52 \[ \left [\frac {\sqrt {c x^{2}} \log \left (\frac {x}{b x + a}\right )}{a x}, \frac {2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )} \sqrt {-c}}{a c x}\right )}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 26, normalized size = 0.62 \[ \frac {\sqrt {c \,x^{2}}\, \left (\ln \relax (x )-\ln \left (b x +a \right )\right )}{a x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 24, normalized size = 0.57 \[ -\frac {\sqrt {c} \log \left (b x + a\right )}{a} + \frac {\sqrt {c} \log \relax (x)}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {c\,x^2}}{x^2\,\left (a+b\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {c x^{2}}}{x^{2} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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