Optimal. Leaf size=93 \[ -\frac{a (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}+\frac{\sqrt{x} \sqrt{a+b x} (4 A b-3 a B)}{4 b^2}+\frac{B x^{3/2} \sqrt{a+b x}}{2 b} \]
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Rubi [A] time = 0.103734, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a (4 A b-3 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{4 b^{5/2}}+\frac{\sqrt{x} \sqrt{a+b x} (4 A b-3 a B)}{4 b^2}+\frac{B x^{3/2} \sqrt{a+b x}}{2 b} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[x]*(A + B*x))/Sqrt[a + b*x],x]
[Out]
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Rubi in Sympy [A] time = 8.87174, size = 85, normalized size = 0.91 \[ \frac{B x^{\frac{3}{2}} \sqrt{a + b x}}{2 b} - \frac{a \left (4 A b - 3 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{4 b^{\frac{5}{2}}} + \frac{\sqrt{x} \sqrt{a + b x} \left (4 A b - 3 B a\right )}{4 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*x**(1/2)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0843091, size = 79, normalized size = 0.85 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} (-3 a B+4 A b+2 b B x)+a (3 a B-4 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{4 b^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[x]*(A + B*x))/Sqrt[a + b*x],x]
[Out]
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Maple [A] time = 0.017, size = 136, normalized size = 1.5 \[ -{\frac{1}{8}\sqrt{x}\sqrt{bx+a} \left ( -4\,Bx{b}^{3/2}\sqrt{x \left ( bx+a \right ) }+4\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) ab-8\,A{b}^{3/2}\sqrt{x \left ( bx+a \right ) }-3\,B\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){a}^{2}+6\,Ba\sqrt{b}\sqrt{x \left ( bx+a \right ) } \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*x^(1/2)/(b*x+a)^(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)*sqrt(x)/sqrt(b*x + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237557, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, B b x - 3 \, B a + 4 \, A b\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} -{\left (3 \, B a^{2} - 4 \, A a b\right )} \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{8 \, b^{\frac{5}{2}}}, \frac{{\left (2 \, B b x - 3 \, B a + 4 \, A b\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} +{\left (3 \, B a^{2} - 4 \, A a b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{4 \, \sqrt{-b} b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*sqrt(x)/sqrt(b*x + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 20.9745, size = 156, normalized size = 1.68 \[ \frac{A \sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{b} - \frac{A a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} - \frac{3 B a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{B \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 B a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{B x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*x**(1/2)/(b*x+a)**(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)*sqrt(x)/sqrt(b*x + a),x, algorithm="giac")
[Out]