Optimal. Leaf size=65 \[ -\frac {\sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {\sqrt {c x^2} \log (x)}{a^2 x}+\frac {\sqrt {c x^2}}{a x (a+b x)} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 44} \begin {gather*} -\frac {\sqrt {c x^2} \log (a+b x)}{a^2 x}+\frac {\sqrt {c x^2} \log (x)}{a^2 x}+\frac {\sqrt {c x^2}}{a x (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 44
Rubi steps
\begin {align*} \int \frac {\sqrt {c x^2}}{x^2 (a+b x)^2} \, dx &=\frac {\sqrt {c x^2} \int \frac {1}{x (a+b x)^2} \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (\frac {1}{a^2 x}-\frac {b}{a (a+b x)^2}-\frac {b}{a^2 (a+b x)}\right ) \, dx}{x}\\ &=\frac {\sqrt {c x^2}}{a x (a+b x)}+\frac {\sqrt {c x^2} \log (x)}{a^2 x}-\frac {\sqrt {c x^2} \log (a+b x)}{a^2 x}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.69 \begin {gather*} \frac {c x (\log (x) (a+b x)-(a+b x) \log (a+b x)+a)}{a^2 \sqrt {c x^2} (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 48, normalized size = 0.74 \begin {gather*} \sqrt {c x^2} \left (-\frac {\log (a+b x)}{a^2 x}+\frac {\log (x)}{a^2 x}+\frac {1}{a x (a+b x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 42, normalized size = 0.65 \begin {gather*} \frac {\sqrt {c x^{2}} {\left ({\left (b x + a\right )} \log \left (\frac {x}{b x + a}\right ) + a\right )}}{a^{2} b x^{2} + a^{3} x} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 52, normalized size = 0.80 \begin {gather*} \frac {\sqrt {c \,x^{2}}\, \left (b x \ln \relax (x )-b x \ln \left (b x +a \right )+a \ln \relax (x )-a \ln \left (b x +a \right )+a \right )}{\left (b x +a \right ) a^{2} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 38, normalized size = 0.58 \begin {gather*} \frac {\sqrt {c}}{a b x + a^{2}} - \frac {\sqrt {c} \log \left (b x + a\right )}{a^{2}} + \frac {\sqrt {c} \log \relax (x)}{a^{2}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c\,x^2}}{x^2\,{\left (a+b\,x\right )}^2} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c x^{2}}}{x^{2} \left (a + b x\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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