Optimal. Leaf size=86 \[ \frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{15 \sqrt{x} \sqrt{b x+2}}{2 b^3}+\frac{5 x^{3/2} \sqrt{b x+2}}{2 b^2}-\frac{2 x^{5/2}}{b \sqrt{b x+2}} \]
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Rubi [A] time = 0.0673449, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}-\frac{15 \sqrt{x} \sqrt{b x+2}}{2 b^3}+\frac{5 x^{3/2} \sqrt{b x+2}}{2 b^2}-\frac{2 x^{5/2}}{b \sqrt{b x+2}} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/(2 + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 10.4864, size = 82, normalized size = 0.95 \[ - \frac{2 x^{\frac{5}{2}}}{b \sqrt{b x + 2}} + \frac{5 x^{\frac{3}{2}} \sqrt{b x + 2}}{2 b^{2}} - \frac{15 \sqrt{x} \sqrt{b x + 2}}{2 b^{3}} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]
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 = 0.0801026, size = 59, normalized size = 0.69 \[ \frac{15 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{7/2}}+\frac{\sqrt{x} \left (b^2 x^2-5 b x-30\right )}{2 b^3 \sqrt{b x+2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/(2 + b*x)^(3/2),x]
[Out]
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Maple [A] time = 0.036, size = 106, normalized size = 1.2 \[{\frac{bx-7}{2\,{b}^{3}}\sqrt{x}\sqrt{bx+2}}+{1 \left ({\frac{15}{2}\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){b}^{-{\frac{7}{2}}}}-8\,{\frac{1}{{b}^{4}}\sqrt{b \left ( x+2\,{b}^{-1} \right ) ^{2}-2\,x-4\,{b}^{-1}} \left ( x+2\,{b}^{-1} \right ) ^{-1}} \right ) \sqrt{x \left ( bx+2 \right ) }{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}} \]
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(x^(5/2)/(b*x + 2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229675, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b x + 2} \sqrt{x} \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right ) +{\left (b^{2} x^{3} - 5 \, b x^{2} - 30 \, x\right )} \sqrt{b}}{2 \, \sqrt{b x + 2} b^{\frac{7}{2}} \sqrt{x}}, \frac{30 \, \sqrt{b x + 2} \sqrt{x} \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (b^{2} x^{3} - 5 \, b x^{2} - 30 \, x\right )} \sqrt{-b}}{2 \, \sqrt{b x + 2} \sqrt{-b} b^{3} \sqrt{x}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(b*x + 2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 80.6611, size = 80, normalized size = 0.93 \[ \frac{x^{\frac{5}{2}}}{2 b \sqrt{b x + 2}} - \frac{5 x^{\frac{3}{2}}}{2 b^{2} \sqrt{b x + 2}} - \frac{15 \sqrt{x}}{b^{3} \sqrt{b x + 2}} + \frac{15 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{7}{2}}} \]
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 [A] time = 0.227263, size = 161, normalized size = 1.87 \[ \frac{{\left (\sqrt{{\left (b x + 2\right )} b - 2 \, b} \sqrt{b x + 2}{\left (\frac{b x + 2}{b^{3}} - \frac{9}{b^{3}}\right )} - \frac{15 \,{\rm ln}\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2}\right )}{b^{\frac{5}{2}}} - \frac{64}{{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )} b^{\frac{3}{2}}}\right )}{\left | b \right |}}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/(b*x + 2)^(3/2),x, algorithm="giac")
[Out]