Optimal. Leaf size=95 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a+b x}+\frac{1}{3} x^{3/2} (a+b x)^{3/2} \]
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Rubi [A] time = 0.070938, antiderivative size = 95, 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 \[ -\frac{a^3 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{a^2 \sqrt{x} \sqrt{a+b x}}{8 b}+\frac{1}{4} a x^{3/2} \sqrt{a+b x}+\frac{1}{3} x^{3/2} (a+b x)^{3/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(a + b*x)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 11.472, size = 85, normalized size = 0.89 \[ - \frac{a^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} - \frac{a^{2} \sqrt{x} \sqrt{a + b x}}{8 b} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{12 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*x**(1/2),x)
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Mathematica [A] time = 0.0477117, size = 78, normalized size = 0.82 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (3 a^2+14 a b x+8 b^2 x^2\right )-3 a^3 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(a + b*x)^(3/2),x]
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Maple [A] time = 0.009, size = 96, normalized size = 1. \[{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{a}{4}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{{a}^{2}}{8\,b}\sqrt{x}\sqrt{bx+a}}-{\frac{{a}^{3}}{16}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*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 + a)^(3/2)*sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.221572, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, b^{2} x^{2} + 14 \, a b x + 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{3}{2}}}, -\frac{3 \, a^{3} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (8 \, b^{2} x^{2} + 14 \, a b x + 3 \, a^{2}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 24.9127, size = 124, normalized size = 1.31 \[ \frac{a^{\frac{5}{2}} \sqrt{x}}{8 b \sqrt{1 + \frac{b x}{a}}} + \frac{17 a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} + \frac{11 \sqrt{a} b x^{\frac{5}{2}}}{12 \sqrt{1 + \frac{b x}{a}}} - \frac{a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{3}{2}}} + \frac{b^{2} x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)*x**(1/2),x)
[Out]
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(3/2)*sqrt(x),x, algorithm="giac")
[Out]