Optimal. Leaf size=48 \[ \frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.059282, antiderivative size = 48, 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 \[ \frac{4 b \sqrt{x}}{c^2 \sqrt{b x+c x^2}}+\frac{2 x^{3/2}}{c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 6.06826, size = 42, normalized size = 0.88 \[ \frac{4 b \sqrt{x}}{c^{2} \sqrt{b x + c x^{2}}} + \frac{2 x^{\frac{3}{2}}}{c \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0213057, size = 28, normalized size = 0.58 \[ \frac{2 \sqrt{x} (2 b+c x)}{c^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 32, normalized size = 0.7 \[ 2\,{\frac{ \left ( cx+b \right ) \left ( cx+2\,b \right ){x}^{3/2}}{{c}^{2} \left ( c{x}^{2}+bx \right ) ^{3/2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(c*x^2+b*x)^(3/2),x)
[Out]
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Maxima [A] time = 0.735935, size = 47, normalized size = 0.98 \[ \frac{2 \, \sqrt{c x + b} x}{c^{2} x + b c} + \frac{4 \, b}{\sqrt{c x + b} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.225112, size = 39, normalized size = 0.81 \[ \frac{2 \,{\left (c x^{2} + 2 \, b x\right )}}{\sqrt{c x^{2} + b x} c^{2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.20994, size = 42, normalized size = 0.88 \[ \frac{2 \,{\left (\sqrt{c x + b} + \frac{b}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{4 \, \sqrt{b}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]