Optimal. Leaf size=79 \[ \frac{1}{3} \sqrt{x} (b x+2)^{5/2}+\frac{5}{6} \sqrt{x} (b x+2)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{b x+2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
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Rubi [A] time = 0.0538055, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{3} \sqrt{x} (b x+2)^{5/2}+\frac{5}{6} \sqrt{x} (b x+2)^{3/2}+\frac{5}{2} \sqrt{x} \sqrt{b x+2}+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[(2 + b*x)^(5/2)/Sqrt[x],x]
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Rubi in Sympy [A] time = 8.12388, size = 73, normalized size = 0.92 \[ \frac{\sqrt{x} \left (b x + 2\right )^{\frac{5}{2}}}{3} + \frac{5 \sqrt{x} \left (b x + 2\right )^{\frac{3}{2}}}{6} + \frac{5 \sqrt{x} \sqrt{b x + 2}}{2} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+2)**(5/2)/x**(1/2),x)
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Mathematica [A] time = 0.0540694, size = 57, normalized size = 0.72 \[ \frac{1}{6} \sqrt{x} \sqrt{b x+2} \left (2 b^2 x^2+13 b x+33\right )+\frac{5 \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 + b*x)^(5/2)/Sqrt[x],x]
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Maple [A] time = 0.007, size = 84, normalized size = 1.1 \[{\frac{1}{3} \left ( bx+2 \right ) ^{{\frac{5}{2}}}\sqrt{x}}+{\frac{5}{6} \left ( bx+2 \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{5}{2}\sqrt{x}\sqrt{bx+2}}+{\frac{5}{2}\sqrt{x \left ( bx+2 \right ) }\ln \left ({(bx+1){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+2\,x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+2)^(5/2)/x^(1/2),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)^(5/2)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.251911, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, b^{2} x^{2} + 13 \, b x + 33\right )} \sqrt{b x + 2} \sqrt{b} \sqrt{x} + 15 \, \log \left (\sqrt{b x + 2} b \sqrt{x} +{\left (b x + 1\right )} \sqrt{b}\right )}{6 \, \sqrt{b}}, \frac{{\left (2 \, b^{2} x^{2} + 13 \, b x + 33\right )} \sqrt{b x + 2} \sqrt{-b} \sqrt{x} + 30 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b}}{b \sqrt{x}}\right )}{6 \, \sqrt{-b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + 2)^(5/2)/sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 59.0891, size = 97, normalized size = 1.23 \[ \frac{b^{3} x^{\frac{7}{2}}}{3 \sqrt{b x + 2}} + \frac{17 b^{2} x^{\frac{5}{2}}}{6 \sqrt{b x + 2}} + \frac{59 b x^{\frac{3}{2}}}{6 \sqrt{b x + 2}} + \frac{11 \sqrt{x}}{\sqrt{b x + 2}} + \frac{5 \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+2)**(5/2)/x**(1/2),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)^(5/2)/sqrt(x),x, algorithm="giac")
[Out]