Optimal. Leaf size=116 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^3 \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a+b x}+\frac{5}{24} a x^{3/2} (a+b x)^{3/2}+\frac{1}{4} x^{3/2} (a+b x)^{5/2} \]
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Rubi [A] time = 0.0909257, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{64 b^{3/2}}+\frac{5 a^3 \sqrt{x} \sqrt{a+b x}}{64 b}+\frac{5}{32} a^2 x^{3/2} \sqrt{a+b x}+\frac{5}{24} a x^{3/2} (a+b x)^{3/2}+\frac{1}{4} x^{3/2} (a+b x)^{5/2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(a + b*x)^(5/2),x]
[Out]
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Rubi in Sympy [A] time = 14.1474, size = 110, normalized size = 0.95 \[ - \frac{5 a^{4} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{64 b^{\frac{3}{2}}} - \frac{5 a^{3} \sqrt{x} \sqrt{a + b x}}{64 b} - \frac{5 a^{2} \sqrt{x} \left (a + b x\right )^{\frac{3}{2}}}{96 b} - \frac{a \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{24 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{7}{2}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*x**(1/2),x)
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Mathematica [A] time = 0.0629352, size = 89, normalized size = 0.77 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^3+118 a^2 b x+136 a b^2 x^2+48 b^3 x^3\right )-15 a^4 \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{192 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(a + b*x)^(5/2),x]
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Maple [A] time = 0.008, size = 111, normalized size = 1. \[{\frac{1}{4}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a}{24}{x}^{{\frac{3}{2}}} \left ( bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}}{32}{x}^{{\frac{3}{2}}}\sqrt{bx+a}}+{\frac{5\,{a}^{3}}{64\,b}\sqrt{x}\sqrt{bx+a}}-{\frac{5\,{a}^{4}}{128}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(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 + a)^(5/2)*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.224735, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, a^{4} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{384 \, b^{\frac{3}{2}}}, -\frac{15 \, a^{4} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) -{\left (48 \, b^{3} x^{3} + 136 \, a b^{2} x^{2} + 118 \, a^{2} b x + 15 \, a^{3}\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{192 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*sqrt(x),x, algorithm="fricas")
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Sympy [A] time = 101.738, size = 155, normalized size = 1.34 \[ \frac{5 a^{\frac{7}{2}} \sqrt{x}}{64 b \sqrt{1 + \frac{b x}{a}}} + \frac{133 a^{\frac{5}{2}} x^{\frac{3}{2}}}{192 \sqrt{1 + \frac{b x}{a}}} + \frac{127 a^{\frac{3}{2}} b x^{\frac{5}{2}}}{96 \sqrt{1 + \frac{b x}{a}}} + \frac{23 \sqrt{a} b^{2} x^{\frac{7}{2}}}{24 \sqrt{1 + \frac{b x}{a}}} - \frac{5 a^{4} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{64 b^{\frac{3}{2}}} + \frac{b^{3} x^{\frac{9}{2}}}{4 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(5/2)*x**(1/2),x)
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GIAC/XCAS [A] time = 24.6269, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^(5/2)*sqrt(x),x, algorithm="giac")
[Out]