Optimal. Leaf size=69 \[ -2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} A (a+b x)^{3/2}+2 a A \sqrt{a+b x}+\frac{2 B (a+b x)^{5/2}}{5 b} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0874744, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )+\frac{2}{3} A (a+b x)^{3/2}+2 a A \sqrt{a+b x}+\frac{2 B (a+b x)^{5/2}}{5 b} \]
Antiderivative was successfully verified.
[In] Int[((a + b*x)^(3/2)*(A + B*x))/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.35232, size = 65, normalized size = 0.94 \[ - 2 A a^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 2 A a \sqrt{a + b x} + \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**(3/2)*(B*x+A)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.135795, size = 73, normalized size = 1.06 \[ \frac{2 \sqrt{a+b x} \left (3 a^2 B+a (20 A b+6 b B x)+b^2 x (5 A+3 B x)\right )}{15 b}-2 a^{3/2} A \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((a + b*x)^(3/2)*(A + B*x))/x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 58, normalized size = 0.8 \[ 2\,{\frac{1}{b} \left ( 1/5\,B \left ( bx+a \right ) ^{5/2}+1/3\,Ab \left ( bx+a \right ) ^{3/2}+abA\sqrt{bx+a}-A{a}^{3/2}b{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^(3/2)*(B*x+A)/x,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,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.227443, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, A a^{\frac{3}{2}} b \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (3 \, B b^{2} x^{2} + 3 \, B a^{2} + 20 \, A a b +{\left (6 \, B a b + 5 \, A b^{2}\right )} x\right )} \sqrt{b x + a}}{15 \, b}, -\frac{2 \,{\left (15 \, A \sqrt{-a} a b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (3 \, B b^{2} x^{2} + 3 \, B a^{2} + 20 \, A a b +{\left (6 \, B a b + 5 \, A b^{2}\right )} x\right )} \sqrt{b x + a}\right )}}{15 \, b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 11.2265, size = 128, normalized size = 1.86 \[ - 2 A a^{2} \left (\begin{cases} - \frac{\operatorname{atan}{\left (\frac{\sqrt{a + b x}}{\sqrt{- a}} \right )}}{\sqrt{- a}} & \text{for}\: - a > 0 \\\frac{\operatorname{acoth}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: - a < 0 \wedge a < a + b x \\\frac{\operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} & \text{for}\: a > a + b x \wedge - a < 0 \end{cases}\right ) + 2 A a \sqrt{a + b x} + \frac{2 A \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{2 B \left (a + b x\right )^{\frac{5}{2}}}{5 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**(3/2)*(B*x+A)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.210209, size = 97, normalized size = 1.41 \[ \frac{2 \, A a^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \frac{2 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} B b^{4} + 5 \,{\left (b x + a\right )}^{\frac{3}{2}} A b^{5} + 15 \, \sqrt{b x + a} A a b^{5}\right )}}{15 \, b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^(3/2)/x,x, algorithm="giac")
[Out]