Optimal. Leaf size=40 \[ \frac{2 b (c+d x)^{3/2}}{3 d^2}-\frac{2 \sqrt{c+d x} (b c-a d)}{d^2} \]
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Rubi [A] time = 0.0442325, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 b (c+d x)^{3/2}}{3 d^2}-\frac{2 \sqrt{c+d x} (b c-a d)}{d^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/Sqrt[c + d*x],x]
[Out]
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Rubi in Sympy [A] time = 7.89419, size = 36, normalized size = 0.9 \[ \frac{2 b \left (c + d x\right )^{\frac{3}{2}}}{3 d^{2}} + \frac{2 \sqrt{c + d x} \left (a d - b c\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(d*x+c)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0215717, size = 29, normalized size = 0.72 \[ \frac{2 \sqrt{c+d x} (3 a d-2 b c+b d x)}{3 d^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)/Sqrt[c + d*x],x]
[Out]
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Maple [A] time = 0.005, size = 26, normalized size = 0.7 \[{\frac{2\,bdx+6\,ad-4\,bc}{3\,{d}^{2}}\sqrt{dx+c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(d*x+c)^(1/2),x)
[Out]
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Maxima [A] time = 1.35306, size = 53, normalized size = 1.32 \[ \frac{2 \,{\left (3 \, \sqrt{d x + c} a + \frac{{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} b}{d}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/sqrt(d*x + c),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222045, size = 34, normalized size = 0.85 \[ \frac{2 \,{\left (b d x - 2 \, b c + 3 \, a d\right )} \sqrt{d x + c}}{3 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/sqrt(d*x + c),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.06103, size = 121, normalized size = 3.02 \[ \begin{cases} - \frac{\frac{2 a c}{\sqrt{c + d x}} + 2 a \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right ) + \frac{2 b c \left (- \frac{c}{\sqrt{c + d x}} - \sqrt{c + d x}\right )}{d} + \frac{2 b \left (\frac{c^{2}}{\sqrt{c + d x}} + 2 c \sqrt{c + d x} - \frac{\left (c + d x\right )^{\frac{3}{2}}}{3}\right )}{d}}{d} & \text{for}\: d \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\sqrt{c}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(d*x+c)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.216979, size = 53, normalized size = 1.32 \[ \frac{2 \,{\left (3 \, \sqrt{d x + c} a + \frac{{\left ({\left (d x + c\right )}^{\frac{3}{2}} - 3 \, \sqrt{d x + c} c\right )} b}{d}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/sqrt(d*x + c),x, algorithm="giac")
[Out]