Optimal. Leaf size=71 \[ -\frac{A b-a B}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{B}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0641102, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{A b-a B}{4 b^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}-\frac{B}{3 b^2 \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.90373, size = 68, normalized size = 0.96 \[ - \frac{B}{3 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 a + 2 b x\right ) \left (A b - B a\right )}{8 b^{2} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0312899, size = 39, normalized size = 0.55 \[ \frac{-B (a+4 b x)-3 A b}{12 b^2 (a+b x)^3 \sqrt{(a+b x)^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 33, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( 4\,xBb+3\,Ab+Ba \right ) }{12\,{b}^{2}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.723556, size = 85, normalized size = 1.2 \[ -\frac{B}{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac{3}{2}} b^{2}} - \frac{A}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}}{\left (x + \frac{a}{b}\right )}^{4}} + \frac{B a}{4 \,{\left (b^{2}\right )}^{\frac{5}{2}} b{\left (x + \frac{a}{b}\right )}^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.277078, size = 82, normalized size = 1.15 \[ -\frac{4 \, B b x + B a + 3 \, A b}{12 \,{\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{\left (\left (a + b x\right )^{2}\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.649812, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(b^2*x^2 + 2*a*b*x + a^2)^(5/2),x, algorithm="giac")
[Out]