Optimal. Leaf size=63 \[ -\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac{2 B \sqrt{a+b x}}{b^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0816219, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{2 (A b-2 a B)}{b^3 \sqrt{a+b x}}+\frac{2 a (A b-a B)}{3 b^3 (a+b x)^{3/2}}+\frac{2 B \sqrt{a+b x}}{b^3} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x))/(a + b*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.9819, size = 60, normalized size = 0.95 \[ \frac{2 B \sqrt{a + b x}}{b^{3}} + \frac{2 a \left (A b - B a\right )}{3 b^{3} \left (a + b x\right )^{\frac{3}{2}}} - \frac{2 \left (A b - 2 B a\right )}{b^{3} \sqrt{a + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)/(b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0534743, size = 46, normalized size = 0.73 \[ \frac{16 a^2 B-4 a b (A-6 B x)+6 b^2 x (B x-A)}{3 b^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x))/(a + b*x)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 47, normalized size = 0.8 \[ -{\frac{-6\,{b}^{2}B{x}^{2}+6\,Ax{b}^{2}-24\,Bxab+4\,Aab-16\,B{a}^{2}}{3\,{b}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)/(b*x+a)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.33996, size = 78, normalized size = 1.24 \[ \frac{2 \,{\left (\frac{3 \, \sqrt{b x + a} B}{b} - \frac{B a^{2} - A a b - 3 \,{\left (2 \, B a - A b\right )}{\left (b x + a\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b}\right )}}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(b*x + a)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.214182, size = 78, normalized size = 1.24 \[ \frac{2 \,{\left (3 \, B b^{2} x^{2} + 8 \, B a^{2} - 2 \, A a b + 3 \,{\left (4 \, B a b - A b^{2}\right )} x\right )}}{3 \,{\left (b^{4} x + a b^{3}\right )} \sqrt{b x + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(b*x + a)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 4.07984, size = 211, normalized size = 3.35 \[ \begin{cases} - \frac{4 A a b}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} - \frac{6 A b^{2} x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{16 B a^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{24 B a b x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{6 B b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{\frac{A x^{2}}{2} + \frac{B x^{3}}{3}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)/(b*x+a)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.213045, size = 74, normalized size = 1.17 \[ \frac{2 \, \sqrt{b x + a} B}{b^{3}} + \frac{2 \,{\left (6 \,{\left (b x + a\right )} B a - B a^{2} - 3 \,{\left (b x + a\right )} A b + A a b\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(b*x + a)^(5/2),x, algorithm="giac")
[Out]