Optimal. Leaf size=80 \[ \frac{16 b^2 \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{8 b \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.09171, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{16 b^2 \sqrt{b x+c x^2}}{15 c^3 \sqrt{x}}-\frac{8 b \sqrt{x} \sqrt{b x+c x^2}}{15 c^2}+\frac{2 x^{3/2} \sqrt{b x+c x^2}}{5 c} \]
Antiderivative was successfully verified.
[In] Int[x^(5/2)/Sqrt[b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.4552, size = 73, normalized size = 0.91 \[ \frac{16 b^{2} \sqrt{b x + c x^{2}}}{15 c^{3} \sqrt{x}} - \frac{8 b \sqrt{x} \sqrt{b x + c x^{2}}}{15 c^{2}} + \frac{2 x^{\frac{3}{2}} \sqrt{b x + c x^{2}}}{5 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(5/2)/(c*x**2+b*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0310636, size = 42, normalized size = 0.52 \[ \frac{2 \sqrt{x (b+c x)} \left (8 b^2-4 b c x+3 c^2 x^2\right )}{15 c^3 \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(5/2)/Sqrt[b*x + c*x^2],x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.007, size = 44, normalized size = 0.6 \[{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3\,{c}^{2}{x}^{2}-4\,bcx+8\,{b}^{2} \right ) }{15\,{c}^{3}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(5/2)/(c*x^2+b*x)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.707365, size = 57, normalized size = 0.71 \[ \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )}}{15 \, \sqrt{c x + b} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.220536, size = 70, normalized size = 0.88 \[ \frac{2 \,{\left (3 \, c^{3} x^{4} - b c^{2} x^{3} + 4 \, b^{2} c x^{2} + 8 \, b^{3} x\right )}}{15 \, \sqrt{c x^{2} + b x} c^{3} \sqrt{x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{\frac{5}{2}}}{\sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(5/2)/(c*x**2+b*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.210165, size = 62, normalized size = 0.78 \[ -\frac{16 \, b^{\frac{5}{2}}}{15 \, c^{3}} + \frac{2 \,{\left (3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 10 \,{\left (c x + b\right )}^{\frac{3}{2}} b + 15 \, \sqrt{c x + b} b^{2}\right )}}{15 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(5/2)/sqrt(c*x^2 + b*x),x, algorithm="giac")
[Out]