Optimal. Leaf size=144 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{\sqrt{x} (a+b x)^{5/2} (a B+6 A b)}{3 a}+\frac{5}{12} \sqrt{x} (a+b x)^{3/2} (a B+6 A b)+\frac{5}{8} a \sqrt{x} \sqrt{a+b x} (a B+6 A b)-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}} \]
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Rubi [A] time = 0.176545, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 a^2 (a B+6 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{8 \sqrt{b}}+\frac{\sqrt{x} (a+b x)^{5/2} (a B+6 A b)}{3 a}+\frac{5}{12} \sqrt{x} (a+b x)^{3/2} (a B+6 A b)+\frac{5}{8} a \sqrt{x} \sqrt{a+b x} (a B+6 A b)-\frac{2 A (a+b x)^{7/2}}{a \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(5/2)*(A + B*x))/x^(3/2),x]
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Rubi in Sympy [A] time = 14.8557, size = 138, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{7}{2}}}{a \sqrt{x}} + \frac{5 a^{2} \left (6 A b + B a\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )}}{8 \sqrt{b}} + \frac{5 a \sqrt{x} \sqrt{a + b x} \left (6 A b + B a\right )}{8} + \sqrt{x} \left (a + b x\right )^{\frac{3}{2}} \left (\frac{5 A b}{2} + \frac{5 B a}{12}\right ) + \frac{\sqrt{x} \left (a + b x\right )^{\frac{5}{2}} \left (6 A b + B a\right )}{3 a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(5/2)*(B*x+A)/x**(3/2),x)
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Mathematica [A] time = 0.17596, size = 105, normalized size = 0.73 \[ \frac{\sqrt{a+b x} \left (a^2 (33 B x-48 A)+2 a b x (27 A+13 B x)+4 b^2 x^2 (3 A+2 B x)\right )}{24 \sqrt{x}}+\frac{5 a^2 (a B+6 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{8 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(5/2)*(A + B*x))/x^(3/2),x]
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Maple [A] time = 0.021, size = 202, normalized size = 1.4 \[{\frac{1}{48}\sqrt{bx+a} \left ( 16\,B{x}^{3}{b}^{5/2}\sqrt{x \left ( bx+a \right ) }+24\,{b}^{5/2}A\sqrt{x \left ( bx+a \right ) }{x}^{2}+52\,B{b}^{3/2}a\sqrt{x \left ( bx+a \right ) }{x}^{2}+90\,bA{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x+108\,{b}^{3/2}A\sqrt{x \left ( bx+a \right ) }ax+15\,B{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) x+66\,B{a}^{2}\sqrt{x \left ( bx+a \right ) }x\sqrt{b}-96\,A{a}^{2}\sqrt{x \left ( bx+a \right ) }\sqrt{b} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(5/2)*(B*x+A)/x^(3/2),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)^(5/2)/x^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.244613, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \,{\left (8 \, B b^{2} x^{3} - 48 \, A a^{2} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{b} \sqrt{x}}{48 \, \sqrt{b} x}, \frac{15 \,{\left (B a^{3} + 6 \, A a^{2} b\right )} x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) +{\left (8 \, B b^{2} x^{3} - 48 \, A a^{2} + 2 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} x^{2} + 3 \,{\left (11 \, B a^{2} + 18 \, A a b\right )} x\right )} \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(5/2)/x^(3/2),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)**(5/2)*(B*x+A)/x**(3/2),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((B*x + A)*(b*x + a)^(5/2)/x^(3/2),x, algorithm="giac")
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