Optimal. Leaf size=28 \[ \frac {(a+b x) \sqrt {\frac {c}{(a+b x)^2}} \log (a+b x)}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac {(a+b x) \sqrt {\frac {c}{(a+b x)^2}} \log (a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 29
Rule 247
Rubi steps
\begin {align*} \int \sqrt {\frac {c}{(a+b x)^2}} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {\frac {c}{x^2}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\left (\sqrt {\frac {c}{(a+b x)^2}} (a+b x)\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\sqrt {\frac {c}{(a+b x)^2}} (a+b x) \log (a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 1.00 \[ \frac {(a+b x) \sqrt {\frac {c}{(a+b x)^2}} \log (a+b x)}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 37, normalized size = 1.32 \[ \frac {{\left (b x + a\right )} \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}} \log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 20, normalized size = 0.71 \[ \frac {\sqrt {c} \log \left ({\left | b x + a \right |}\right ) \mathrm {sgn}\left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.96 \[ \frac {\left (b x +a \right ) \sqrt {\frac {c}{\left (b x +a \right )^{2}}}\, \ln \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 13, normalized size = 0.46 \[ \frac {\sqrt {c} \log \left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {\frac {c}{{\left (a+b\,x\right )}^2}} \,d x \]
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 \[ \int \sqrt {\frac {c}{\left (a + b x\right )^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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