Optimal. Leaf size=25 \[ \frac {a+b x}{2 b \sqrt {\frac {c}{(a+b x)^2}}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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, 30} \[ \frac {a+b x}{2 b \sqrt {\frac {c}{(a+b x)^2}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\frac {c}{(a+b x)^2}}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\frac {c}{x^2}}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}(\int x \, dx,x,a+b x)}{b \sqrt {\frac {c}{(a+b x)^2}} (a+b x)}\\ &=\frac {a+b x}{2 b \sqrt {\frac {c}{(a+b x)^2}}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.28 \[ \frac {x (2 a+b x)}{2 (a+b x) \sqrt {\frac {c}{(a+b x)^2}}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 48, normalized size = 1.92 \[ \frac {{\left (b^{2} x^{3} + 3 \, a b x^{2} + 2 \, a^{2} x\right )} \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 23, normalized size = 0.92 \[ \frac {b x^{2} + 2 \, a x}{2 \, \sqrt {c} \mathrm {sgn}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 1.16 \[ \frac {\left (b x +2 a \right ) x}{2 \left (b x +a \right ) \sqrt {\frac {c}{\left (b x +a \right )^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 15, normalized size = 0.60 \[ \frac {b x^{2} + 2 \, a x}{2 \, \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.22, size = 42, normalized size = 1.68 \[ \sqrt {\frac {c}{{\left (a+b\,x\right )}^2}}\,\left (\frac {a^2\,x}{c}+\frac {b^2\,x^3}{2\,c}+\frac {3\,a\,b\,x^2}{2\,c}\right ) \]
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 \[ \int \frac {1}{\sqrt {\frac {c}{\left (a + b x\right )^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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