Optimal. Leaf size=117 \[ -\frac{2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt{x}}+\frac{b \sqrt{x} \sqrt{a+b x} (3 a B+2 A b)}{a}+\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.143367, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{2 (a+b x)^{3/2} (3 a B+2 A b)}{3 a \sqrt{x}}+\frac{b \sqrt{x} \sqrt{a+b x} (3 a B+2 A b)}{a}+\sqrt{b} (3 a B+2 A b) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )-\frac{2 A (a+b x)^{5/2}}{3 a x^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/x^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.8721, size = 112, normalized size = 0.96 \[ - \frac{2 A \left (a + b x\right )^{\frac{5}{2}}}{3 a x^{\frac{3}{2}}} + 2 \sqrt{b} \left (A b + \frac{3 B a}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a + b x}} \right )} + \frac{b \sqrt{x} \sqrt{a + b x} \left (2 A b + 3 B a\right )}{a} - \frac{4 \left (a + b x\right )^{\frac{3}{2}} \left (A b + \frac{3 B a}{2}\right )}{3 a \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.114963, size = 79, normalized size = 0.68 \[ \sqrt{b} (3 a B+2 A b) \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )-\frac{\sqrt{a+b x} (2 a (A+3 B x)+b x (8 A-3 B x))}{3 x^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/x^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.019, size = 151, normalized size = 1.3 \[{\frac{1}{6}\sqrt{bx+a} \left ( 6\,A\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ){x}^{2}{b}^{3/2}+9\,B\sqrt{b}\ln \left ( 1/2\,{\frac{2\,\sqrt{x \left ( bx+a \right ) }\sqrt{b}+2\,bx+a}{\sqrt{b}}} \right ) a{x}^{2}+6\,B{x}^{2}b\sqrt{x \left ( bx+a \right ) }-16\,Axb\sqrt{x \left ( bx+a \right ) }-12\,Bxa\sqrt{x \left ( bx+a \right ) }-4\,Aa\sqrt{x \left ( bx+a \right ) } \right ){x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x \left ( bx+a \right ) }}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/x^(5/2),x)
[Out]
_______________________________________________________________________________________
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)/x^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.248888, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{b} x^{2} \log \left (2 \, b x + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x} + a\right ) + 2 \,{\left (3 \, B b x^{2} - 2 \, A a - 2 \,{\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{6 \, x^{2}}, \frac{3 \,{\left (3 \, B a + 2 \, A b\right )} \sqrt{-b} x^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-b} \sqrt{x}}\right ) +{\left (3 \, B b x^{2} - 2 \, A a - 2 \,{\left (3 \, B a + 4 \, A b\right )} x\right )} \sqrt{b x + a} \sqrt{x}}{3 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 170.571, size = 168, normalized size = 1.44 \[ A \left (- \frac{2 a \sqrt{b} \sqrt{\frac{a}{b x} + 1}}{3 x} - \frac{8 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}}{3} - b^{\frac{3}{2}} \log{\left (\frac{a}{b x} \right )} + 2 b^{\frac{3}{2}} \log{\left (\sqrt{\frac{a}{b x} + 1} + 1 \right )}\right ) + B \left (- \frac{2 a^{\frac{3}{2}}}{\sqrt{x} \sqrt{1 + \frac{b x}{a}}} - \frac{\sqrt{a} b \sqrt{x}}{\sqrt{1 + \frac{b x}{a}}} + 3 a \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )} + \frac{b^{2} x^{\frac{3}{2}}}{\sqrt{a} \sqrt{1 + \frac{b x}{a}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)*(B*x+A)/x**(5/2),x)
[Out]
_______________________________________________________________________________________
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)/x^(5/2),x, algorithm="giac")
[Out]