Optimal. Leaf size=71 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{1}{3 x^{3/2} (b x+2)^{3/2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.0449746, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 \sqrt{b x+2}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{b x+2}}+\frac{1}{3 x^{3/2} (b x+2)^{3/2}}+\frac{2 b \sqrt{b x+2}}{3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^(5/2)*(2 + b*x)^(5/2)),x]
[Out]
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Rubi in Sympy [A] time = 5.42585, size = 66, normalized size = 0.93 \[ \frac{2 b \sqrt{b x + 2}}{3 \sqrt{x}} - \frac{2 \sqrt{b x + 2}}{3 x^{\frac{3}{2}}} + \frac{1}{x^{\frac{3}{2}} \sqrt{b x + 2}} + \frac{1}{3 x^{\frac{3}{2}} \left (b x + 2\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**(5/2)/(b*x+2)**(5/2),x)
[Out]
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Mathematica [A] time = 0.0260629, size = 40, normalized size = 0.56 \[ \frac{2 b^3 x^3+6 b^2 x^2+3 b x-1}{3 x^{3/2} (b x+2)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^(5/2)*(2 + b*x)^(5/2)),x]
[Out]
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Maple [A] time = 0.006, size = 35, normalized size = 0.5 \[{\frac{2\,{b}^{3}{x}^{3}+6\,{b}^{2}{x}^{2}+3\,bx-1}{3}{x}^{-{\frac{3}{2}}} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^(5/2)/(b*x+2)^(5/2),x)
[Out]
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Maxima [A] time = 1.34373, size = 74, normalized size = 1.04 \[ \frac{3 \, \sqrt{b x + 2} b}{8 \, \sqrt{x}} - \frac{{\left (b^{3} - \frac{9 \,{\left (b x + 2\right )} b^{2}}{x}\right )} x^{\frac{3}{2}}}{24 \,{\left (b x + 2\right )}^{\frac{3}{2}}} - \frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{24 \, x^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + 2)^(5/2)*x^(5/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.212732, size = 61, normalized size = 0.86 \[ \frac{2 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 3 \, b x - 1}{3 \,{\left (b x^{2} + 2 \, x\right )} \sqrt{b x + 2} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + 2)^(5/2)*x^(5/2)),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**(5/2)/(b*x+2)**(5/2),x)
[Out]
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GIAC/XCAS [A] time = 0.231582, size = 213, normalized size = 3. \[ \frac{{\left (4 \,{\left (b x + 2\right )} b^{2}{\left | b \right |} - 9 \, b^{2}{\left | b \right |}\right )} \sqrt{b x + 2}}{12 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}} + \frac{3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} b^{\frac{7}{2}} + 18 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} b^{\frac{9}{2}} + 16 \, b^{\frac{11}{2}}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3}{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x + 2)^(5/2)*x^(5/2)),x, algorithm="giac")
[Out]