Optimal. Leaf size=102 \[ \frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{5/2}}-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b^2}-\frac{a x^{3/2} \sqrt{a-b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a-b x} \]
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Rubi [A] time = 0.0754344, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )}{8 b^{5/2}}-\frac{a^2 \sqrt{x} \sqrt{a-b x}}{8 b^2}-\frac{a x^{3/2} \sqrt{a-b x}}{12 b}+\frac{1}{3} x^{5/2} \sqrt{a-b x} \]
Antiderivative was successfully verified.
[In] Int[x^(3/2)*Sqrt[a - b*x],x]
[Out]
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Rubi in Sympy [A] time = 11.0076, size = 88, normalized size = 0.86 \[ - \frac{a^{3} \operatorname{atan}{\left (\frac{\sqrt{a - b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{5}{2}}} + \frac{a^{2} \sqrt{x} \sqrt{a - b x}}{8 b^{2}} - \frac{a \sqrt{x} \left (a - b x\right )^{\frac{3}{2}}}{4 b^{2}} - \frac{x^{\frac{3}{2}} \left (a - b x\right )^{\frac{3}{2}}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(-b*x+a)**(1/2),x)
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Mathematica [A] time = 0.0548508, size = 77, normalized size = 0.75 \[ \frac{3 a^3 \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a-b x}}\right )+\sqrt{b} \sqrt{x} \sqrt{a-b x} \left (-3 a^2-2 a b x+8 b^2 x^2\right )}{24 b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(3/2)*Sqrt[a - b*x],x]
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Maple [A] time = 0.007, size = 108, normalized size = 1.1 \[ -{\frac{1}{3\,b}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{a}{4\,{b}^{2}} \left ( -bx+a \right ) ^{{\frac{3}{2}}}\sqrt{x}}+{\frac{{a}^{2}}{8\,{b}^{2}}\sqrt{x}\sqrt{-bx+a}}+{\frac{{a}^{3}}{16}\sqrt{x \left ( -bx+a \right ) }\arctan \left ({1\sqrt{b} \left ( x-{\frac{a}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+ax}}}} \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{-bx+a}}}{\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(-b*x+a)^(1/2),x)
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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(sqrt(-b*x + a)*x^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.22113, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{3} \log \left (-2 \, \sqrt{-b x + a} b \sqrt{x} -{\left (2 \, b x - a\right )} \sqrt{-b}\right ) + 2 \,{\left (8 \, b^{2} x^{2} - 2 \, a b x - 3 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{-b} \sqrt{x}}{48 \, \sqrt{-b} b^{2}}, -\frac{3 \, a^{3} \arctan \left (\frac{\sqrt{-b x + a}}{\sqrt{b} \sqrt{x}}\right ) -{\left (8 \, b^{2} x^{2} - 2 \, a b x - 3 \, a^{2}\right )} \sqrt{-b x + a} \sqrt{b} \sqrt{x}}{24 \, b^{\frac{5}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x + a)*x^(3/2),x, algorithm="fricas")
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Sympy [A] time = 27.1826, size = 260, normalized size = 2.55 \[ \begin{cases} \frac{i a^{\frac{5}{2}} \sqrt{x}}{8 b^{2} \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b \sqrt{-1 + \frac{b x}{a}}} - \frac{5 i \sqrt{a} x^{\frac{5}{2}}}{12 \sqrt{-1 + \frac{b x}{a}}} - \frac{i a^{3} \operatorname{acosh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} + \frac{i b x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \left |{\frac{b x}{a}}\right | > 1 \\- \frac{a^{\frac{5}{2}} \sqrt{x}}{8 b^{2} \sqrt{1 - \frac{b x}{a}}} + \frac{a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b \sqrt{1 - \frac{b x}{a}}} + \frac{5 \sqrt{a} x^{\frac{5}{2}}}{12 \sqrt{1 - \frac{b x}{a}}} + \frac{a^{3} \operatorname{asin}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{5}{2}}} - \frac{b x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(-b*x+a)**(1/2),x)
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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(sqrt(-b*x + a)*x^(3/2),x, algorithm="giac")
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