Optimal. Leaf size=47 \[ \frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (b x+c)^{3/2}}{3 b (a-c)} \]
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Rubi [A] time = 0.0466709, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {6689} \[ \frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (b x+c)^{3/2}}{3 b (a-c)} \]
Antiderivative was successfully verified.
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Rule 6689
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+b x}+\sqrt{c+b x}} \, dx &=\frac{\int \left (\sqrt{a+b x}-\sqrt{c+b x}\right ) \, dx}{a-c}\\ &=\frac{2 (a+b x)^{3/2}}{3 b (a-c)}-\frac{2 (c+b x)^{3/2}}{3 b (a-c)}\\ \end{align*}
Mathematica [A] time = 0.0480739, size = 35, normalized size = 0.74 \[ \frac{2 \left ((a+b x)^{3/2}-(b x+c)^{3/2}\right )}{3 b (a-c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 40, normalized size = 0.9 \begin{align*}{\frac{2}{3\,b \left ( a-c \right ) } \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{2}{3\,b \left ( a-c \right ) } \left ( bx+c \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x + a} + \sqrt{b x + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.966095, size = 72, normalized size = 1.53 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} -{\left (b x + c\right )}^{\frac{3}{2}}\right )}}{3 \,{\left (a b - b c\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.647716, size = 136, normalized size = 2.89 \begin{align*} \begin{cases} \frac{2 a}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{4 b x}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{2 c}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} + \frac{2 \sqrt{a + b x} \sqrt{b x + c}}{3 b \sqrt{a + b x} + 3 b \sqrt{b x + c}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{a} + \sqrt{c}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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