Optimal. Leaf size=48 \[ \frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0439084, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 x}{3 a \left (a x+b x^2\right )^{3/2}}-\frac{8 (a+2 b x)}{3 a^3 \sqrt{a x+b x^2}} \]
Antiderivative was successfully verified.
[In] Int[x/(a*x + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.68897, size = 42, normalized size = 0.88 \[ \frac{2 x}{3 a \left (a x + b x^{2}\right )^{\frac{3}{2}}} - \frac{4 \left (2 a + 4 b x\right )}{3 a^{3} \sqrt{a x + b x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(b*x**2+a*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0357549, size = 38, normalized size = 0.79 \[ -\frac{2 x \left (3 a^2+12 a b x+8 b^2 x^2\right )}{3 a^3 (x (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x/(a*x + b*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 44, normalized size = 0.9 \[ -{\frac{2\,{x}^{2} \left ( bx+a \right ) \left ( 8\,{b}^{2}{x}^{2}+12\,abx+3\,{a}^{2} \right ) }{3\,{a}^{3}} \left ( b{x}^{2}+ax \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(b*x^2+a*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.695022, size = 70, normalized size = 1.46 \[ \frac{2 \, x}{3 \,{\left (b x^{2} + a x\right )}^{\frac{3}{2}} a} - \frac{16 \, b x}{3 \, \sqrt{b x^{2} + a x} a^{3}} - \frac{8}{3 \, \sqrt{b x^{2} + a x} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^2 + a*x)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.218623, size = 59, normalized size = 1.23 \[ -\frac{2 \,{\left (8 \, b^{2} x^{2} + 12 \, a b x + 3 \, a^{2}\right )}}{3 \,{\left (a^{3} b x + a^{4}\right )} \sqrt{b x^{2} + a x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^2 + a*x)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (x \left (a + b x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x**2+a*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (b x^{2} + a x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(b*x^2 + a*x)^(5/2),x, algorithm="giac")
[Out]