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 = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 32
Rule 2103
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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.44, size = 50, normalized size = 1.06 \[ \frac {2 \, {\left ({\left (b c x + a c\right )} \sqrt {b x + a} - {\left (b c x + a b\right )} \sqrt {c x + a}\right )}}{3 \, {\left (b^{2} c - b c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 107, normalized size = 2.28 \[ -\frac {2 \, {\left ({\left (\frac {{\left (b x + a\right )} b^{2} c {\left | b \right |}}{b^{5} c - b^{4} c^{2}} + \frac {a b^{3} {\left | b \right |} - a b^{2} c {\left | b \right |}}{b^{5} c - b^{4} c^{2}}\right )} \sqrt {a b^{2} + {\left (b x + a\right )} b c - a b c} - \frac {{\left (b x + a\right )}^{\frac {3}{2}}}{b - c}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 0.85 \[ \frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right ) b}-\frac {2 \left (c x +a \right )^{\frac {3}{2}}}{3 \left (b -c \right ) c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {b x + a} + \sqrt {c x + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.91, size = 79, normalized size = 1.68 \[ \frac {2\,x\,\sqrt {a+b\,x}}{3\,\left (b-c\right )}-\frac {2\,x\,\sqrt {a+c\,x}}{3\,\left (b-c\right )}+\frac {2\,a\,\sqrt {a+b\,x}}{3\,b\,\left (b-c\right )}-\frac {2\,a\,\sqrt {a+c\,x}}{3\,c\,\left (b-c\right )} \]
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 \[ \int \frac {x}{\sqrt {a + b x} + \sqrt {a + c x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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