Optimal. Leaf size=126 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}-\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-5 a B)}{8 b^3}+\frac{x^{3/2} \sqrt{a+b x} (6 A b-5 a B)}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b} \]
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Rubi [A] time = 0.142528, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{a^2 (6 A b-5 a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{7/2}}-\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-5 a B)}{8 b^3}+\frac{x^{3/2} \sqrt{a+b x} (6 A b-5 a B)}{12 b^2}+\frac{B x^{5/2} \sqrt{a+b x}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(x^(3/2)*(A + B*x))/Sqrt[a + b*x],x]
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Rubi in Sympy [A] time = 12.681, size = 117, normalized size = 0.93 \[ \frac{B x^{\frac{5}{2}} \sqrt{a + b x}}{3 b} + \frac{a^{2} \left (6 A b - 5 B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{8 b^{\frac{7}{2}}} - \frac{a \sqrt{x} \sqrt{a + b x} \left (6 A b - 5 B a\right )}{8 b^{3}} + \frac{x^{\frac{3}{2}} \sqrt{a + b x} \left (6 A b - 5 B a\right )}{12 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(3/2)*(B*x+A)/(b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.111185, size = 101, normalized size = 0.8 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (15 a^2 B-2 a b (9 A+5 B x)+4 b^2 x (3 A+2 B x)\right )-3 a^2 (5 a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{7/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(3/2)*(A + B*x))/Sqrt[a + b*x],x]
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Maple [A] time = 0.022, size = 176, normalized size = 1.4 \[{\frac{1}{48}\sqrt{x}\sqrt{bx+a} \left ( 16\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}-20\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+18\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b-36\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-15\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +30\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(3/2)*(B*x+A)/(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)*x^(3/2)/sqrt(b*x + a),x, algorithm="maxima")
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Fricas [A] time = 0.240406, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 18 \, A a b - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{48 \, b^{\frac{7}{2}}}, \frac{{\left (8 \, B b^{2} x^{2} + 15 \, B a^{2} - 18 \, A a b - 2 \,{\left (5 \, B a b - 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 3 \,{\left (5 \, B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, \sqrt{-b} b^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^(3/2)/sqrt(b*x + a),x, algorithm="fricas")
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Sympy [A] time = 99.9212, size = 245, normalized size = 1.94 \[ - \frac{3 A a^{\frac{3}{2}} \sqrt{x}}{4 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{A \sqrt{a} x^{\frac{3}{2}}}{4 b \sqrt{1 + \frac{b x}{a}}} + \frac{3 A a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{4 b^{\frac{5}{2}}} + \frac{A x^{\frac{5}{2}}}{2 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} + \frac{5 B a^{\frac{5}{2}} \sqrt{x}}{8 b^{3} \sqrt{1 + \frac{b x}{a}}} + \frac{5 B a^{\frac{3}{2}} x^{\frac{3}{2}}}{24 b^{2} \sqrt{1 + \frac{b x}{a}}} - \frac{B \sqrt{a} x^{\frac{5}{2}}}{12 b \sqrt{1 + \frac{b x}{a}}} - \frac{5 B a^{3} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{8 b^{\frac{7}{2}}} + \frac{B x^{\frac{7}{2}}}{3 \sqrt{a} \sqrt{1 + \frac{b x}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(3/2)*(B*x+A)/(b*x+a)**(1/2),x)
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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)*x^(3/2)/sqrt(b*x + a),x, algorithm="giac")
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