Optimal. Leaf size=126 \[ -\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{x^{3/2} \sqrt{b x+2}}{8 b^2}+\frac{1}{5} x^{7/2} (b x+2)^{3/2}+\frac{3}{20} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{20 b} \]
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Rubi [A] time = 0.10299, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}}+\frac{3 \sqrt{x} \sqrt{b x+2}}{8 b^3}-\frac{x^{3/2} \sqrt{b x+2}}{8 b^2}+\frac{1}{5} x^{7/2} (b x+2)^{3/2}+\frac{3}{20} x^{7/2} \sqrt{b x+2}+\frac{x^{5/2} \sqrt{b x+2}}{20 b} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)*(2 + b*x)^(3/2),x]
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Rubi in Sympy [A] time = 16.2817, size = 119, normalized size = 0.94 \[ \frac{x^{\frac{5}{2}} \left (b x + 2\right )^{\frac{5}{2}}}{5 b} - \frac{x^{\frac{3}{2}} \left (b x + 2\right )^{\frac{5}{2}}}{4 b^{2}} + \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{4 b^{3}} - \frac{\sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{8 b^{3}} - \frac{3 \sqrt{x} \sqrt{b x + 2}}{8 b^{3}} - \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{4 b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)*(b*x+2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0857385, size = 78, normalized size = 0.62 \[ \frac{\sqrt{x} \sqrt{b x+2} \left (8 b^4 x^4+22 b^3 x^3+2 b^2 x^2-5 b x+15\right )}{40 b^3}-\frac{3 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{4 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)*(2 + b*x)^(3/2),x]
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Maple [A] time = 0.009, size = 123, normalized size = 1. \[{\frac{1}{5\,b}{x}^{{\frac{5}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}-{\frac{1}{4\,{b}^{2}}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}}+{\frac{1}{4\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}-{\frac{1}{8\,{b}^{3}} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}-{\frac{3}{8\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}-{\frac{3}{8}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)*(b*x+2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)*x^(5/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220873, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (8 \, b^{4} x^{4} + 22 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - 5 \, b x + 15\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 15 \, \log \left (-\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{40 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, b^{4} x^{4} + 22 \, b^{3} x^{3} + 2 \, b^{2} x^{2} - 5 \, b x + 15\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} - 30 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{40 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)*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(x**(5/2)*(b*x+2)**(3/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(3/2)*x^(5/2),x, algorithm="giac")
[Out]