Optimal. Leaf size=49 \[ -\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}+\frac{b x}{d} \]
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Rubi [A] time = 0.0487507, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {376, 77} \[ -\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}+\frac{b x}{d} \]
Antiderivative was successfully verified.
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Rule 376
Rule 77
Rubi steps
\begin{align*} \int \frac{a+b \sqrt{x}}{c+d \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x (a+b x)}{c+d x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{-b c+a d}{d^2}+\frac{b x}{d}+\frac{c (b c-a d)}{d^2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 (b c-a d) \sqrt{x}}{d^2}+\frac{b x}{d}+\frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}\\ \end{align*}
Mathematica [A] time = 0.0406529, size = 41, normalized size = 0.84 \[ \frac{2 (a d-b c) \left (d \sqrt{x}-c \log \left (c+d \sqrt{x}\right )\right )}{d^3}+\frac{b x}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 59, normalized size = 1.2 \begin{align*}{\frac{bx}{d}}+2\,{\frac{a\sqrt{x}}{d}}-2\,{\frac{b\sqrt{x}c}{{d}^{2}}}-2\,{\frac{c\ln \left ( c+d\sqrt{x} \right ) a}{{d}^{2}}}+2\,{\frac{{c}^{2}\ln \left ( c+d\sqrt{x} \right ) b}{{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966631, size = 63, normalized size = 1.29 \begin{align*} \frac{b d x - 2 \,{\left (b c - a d\right )} \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30843, size = 111, normalized size = 2.27 \begin{align*} \frac{b d^{2} x + 2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right ) - 2 \,{\left (b c d - a d^{2}\right )} \sqrt{x}}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.271116, size = 82, normalized size = 1.67 \begin{align*} \begin{cases} - \frac{2 a c \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{2}} + \frac{2 a \sqrt{x}}{d} + \frac{2 b c^{2} \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{3}} - \frac{2 b c \sqrt{x}}{d^{2}} + \frac{b x}{d} & \text{for}\: d \neq 0 \\\frac{a x + \frac{2 b x^{\frac{3}{2}}}{3}}{c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0981, size = 66, normalized size = 1.35 \begin{align*} \frac{b d x - 2 \, b c \sqrt{x} + 2 \, a d \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left ({\left | d \sqrt{x} + c \right |}\right )}{d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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