Optimal. Leaf size=47 \[ \frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 c (b-c)} \]
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Rubi [A] time = 0.0550301, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2103, 32} \[ \frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 c (b-c)} \]
Antiderivative was successfully verified.
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Rule 2103
Rule 32
Rubi steps
\begin{align*} \int \frac{x}{\sqrt{a+b x}+\sqrt{a+c x}} \, dx &=\frac{\int \sqrt{a+b x} \, dx}{b-c}-\frac{\int \sqrt{a+c x} \, dx}{b-c}\\ &=\frac{2 (a+b x)^{3/2}}{3 b (b-c)}-\frac{2 (a+c x)^{3/2}}{3 (b-c) c}\\ \end{align*}
Mathematica [A] time = 0.0760489, size = 39, normalized size = 0.83 \[ \frac{2 \left (\frac{(a+b x)^{3/2}}{b}-\frac{(a+c x)^{3/2}}{c}\right )}{3 (b-c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 40, normalized size = 0.9 \begin{align*}{\frac{2}{3\,b \left ( b-c \right ) } \left ( bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{2}{ \left ( 3\,b-3\,c \right ) c} \left ( cx+a \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{x}{\sqrt{b x + a} + \sqrt{c x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20339, size = 109, normalized size = 2.32 \begin{align*} \frac{2 \,{\left ({\left (b c x + a c\right )} \sqrt{b x + a} -{\left (b c x + a b\right )} \sqrt{c x + a}\right )}}{3 \,{\left (b^{2} c - b c^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{a + b x} + \sqrt{a + c x}}\, dx \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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