Optimal. Leaf size=126 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{\sqrt{x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-a B)}{8 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b} \]
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Rubi [A] time = 0.139065, 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 \[ \frac{a^2 (6 A b-a B) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 b^{3/2}}+\frac{\sqrt{x} (a+b x)^{3/2} (6 A b-a B)}{12 b}+\frac{a \sqrt{x} \sqrt{a+b x} (6 A b-a B)}{8 b}+\frac{B \sqrt{x} (a+b x)^{5/2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/Sqrt[x],x]
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Rubi in Sympy [A] time = 11.7932, size = 109, normalized size = 0.87 \[ \frac{B \sqrt{x} \left (a + b x\right )^{\frac{5}{2}}}{3 b} + \frac{a^{2} \left (6 A b - B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} + \frac{a \sqrt{x} \sqrt{a + b x} \left (6 A b - B a\right )}{8 b} + \frac{\sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (6 A b - B a\right )}{12 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(1/2),x)
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Mathematica [A] time = 0.0990434, size = 100, normalized size = 0.79 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{a+b x} \left (3 a^2 B+2 a b (15 A+7 B x)+4 b^2 x (3 A+2 B x)\right )-3 a^2 (a B-6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{24 b^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/Sqrt[x],x]
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Maple [A] time = 0.017, size = 176, normalized size = 1.4 \[{\frac{1}{48}\sqrt{bx+a}\sqrt{x} \left ( 16\,B{x}^{2}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,A\sqrt{x \left ( bx+a \right ) }x{b}^{5/2}+28\,Ba\sqrt{x \left ( bx+a \right ) }x{b}^{3/2}+18\,A{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) b+60\,A\sqrt{x \left ( bx+a \right ) }a{b}^{3/2}-3\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) +6\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/x^(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((B*x + A)*(b*x + a)^(3/2)/sqrt(x),x, algorithm="maxima")
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Fricas [A] time = 0.247875, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (8 \, B b^{2} x^{2} + 3 \, B a^{2} + 30 \, A a b + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x} - 3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right )}{48 \, b^{\frac{3}{2}}}, \frac{{\left (8 \, B b^{2} x^{2} + 3 \, B a^{2} + 30 \, A a b + 2 \,{\left (7 \, B a b + 6 \, A b^{2}\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x} - 3 \,{\left (B a^{3} - 6 \, A a^{2} b\right )} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right )}{24 \, \sqrt{-b} b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/sqrt(x),x, algorithm="fricas")
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Sympy [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)*(B*x+A)/x**(1/2),x)
[Out]
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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((B*x + A)*(b*x + a)^(3/2)/sqrt(x),x, algorithm="giac")
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