Optimal. Leaf size=52 \[ \frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.0590356, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)/Sqrt[b*x + c*x^2],x]
[Out]
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Rubi in Sympy [A] time = 6.18155, size = 46, normalized size = 0.88 \[ - \frac{4 b \sqrt{b x + c x^{2}}}{3 c^{2} \sqrt{x}} + \frac{2 \sqrt{x} \sqrt{b x + c x^{2}}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0247225, size = 30, normalized size = 0.58 \[ \frac{2 (c x-2 b) \sqrt{x (b+c x)}}{3 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)/Sqrt[b*x + c*x^2],x]
[Out]
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Maple [A] time = 0.004, size = 33, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -cx+2\,b \right ) }{3\,{c}^{2}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)/(c*x^2+b*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.711942, size = 41, normalized size = 0.79 \[ \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )}}{3 \, \sqrt{c x + b} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218899, size = 54, normalized size = 1.04 \[ \frac{2 \,{\left (c^{2} x^{3} - b c x^{2} - 2 \, b^{2} x\right )}}{3 \, \sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{3}{2}}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.211046, size = 43, normalized size = 0.83 \[ \frac{4 \, b^{\frac{3}{2}}}{3 \, c^{2}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b\right )}}{3 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(3/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]