Optimal. Leaf size=49 \[ \frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}-\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{b x}{d} \]
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Rubi [A] time = 0.122266, 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 \[ \frac{2 c (b c-a d) \log \left (c+d \sqrt{x}\right )}{d^3}-\frac{2 \sqrt{x} (b c-a d)}{d^2}+\frac{b x}{d} \]
Antiderivative was successfully verified.
[In] Int[(a + b*Sqrt[x])/(c + d*Sqrt[x]),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{2 b \int ^{\sqrt{x}} x\, dx}{d} - \frac{2 c \left (a d - b c\right ) \log{\left (c + d \sqrt{x} \right )}}{d^{3}} + \left (2 a d - 2 b c\right ) \int ^{\sqrt{x}} \frac{1}{d^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**(1/2))/(c+d*x**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0436009, size = 51, normalized size = 1.04 \[ \frac{2 \left (b c^2-a c d\right ) \log \left (c+d \sqrt{x}\right )}{d^3}+\frac{2 \sqrt{x} (a d-b c)}{d^2}+\frac{b x}{d} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*Sqrt[x])/(c + d*Sqrt[x]),x]
[Out]
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Maple [A] time = 0.007, size = 59, normalized size = 1.2 \[{\frac{bx}{d}}+2\,{\frac{a\sqrt{x}}{d}}-2\,{\frac{b\sqrt{x}c}{{d}^{2}}}-2\,{\frac{c\ln \left ( c+d\sqrt{x} \right ) a}{{d}^{2}}}+2\,{\frac{{c}^{2}\ln \left ( c+d\sqrt{x} \right ) b}{{d}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^(1/2))/(c+d*x^(1/2)),x)
[Out]
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Maxima [A] time = 1.70584, size = 63, normalized size = 1.29 \[ \frac{b d x - 2 \,{\left (b c - a d\right )} \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)/(d*sqrt(x) + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239749, size = 65, normalized size = 1.33 \[ \frac{b d^{2} x + 2 \,{\left (b c^{2} - a c d\right )} \log \left (d \sqrt{x} + c\right ) - 2 \,{\left (b c d - a d^{2}\right )} \sqrt{x}}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)/(d*sqrt(x) + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.817719, size = 82, normalized size = 1.67 \[ \begin{cases} - \frac{2 a c \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{2}} + \frac{2 a \sqrt{x}}{d} + \frac{2 b c^{2} \log{\left (\frac{c}{d} + \sqrt{x} \right )}}{d^{3}} - \frac{2 b c \sqrt{x}}{d^{2}} + \frac{b x}{d} & \text{for}\: d \neq 0 \\\frac{a x + \frac{2 b x^{\frac{3}{2}}}{3}}{c} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**(1/2))/(c+d*x**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.215998, size = 66, normalized size = 1.35 \[ \frac{b d x - 2 \, b c \sqrt{x} + 2 \, a d \sqrt{x}}{d^{2}} + \frac{2 \,{\left (b c^{2} - a c d\right )}{\rm ln}\left ({\left | d \sqrt{x} + c \right |}\right )}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*sqrt(x) + a)/(d*sqrt(x) + c),x, algorithm="giac")
[Out]