Optimal. Leaf size=74 \[ -\frac{16 c^2 \sqrt{b x+c x^2}}{15 b^3 x}+\frac{8 c \sqrt{b x+c x^2}}{15 b^2 x^2}-\frac{2 \sqrt{b x+c x^2}}{5 b x^3} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0956, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{16 c^2 \sqrt{b x+c x^2}}{15 b^3 x}+\frac{8 c \sqrt{b x+c x^2}}{15 b^2 x^2}-\frac{2 \sqrt{b x+c x^2}}{5 b x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[b*x + c*x^2]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 9.85392, size = 66, normalized size = 0.89 \[ - \frac{2 \sqrt{b x + c x^{2}}}{5 b x^{3}} + \frac{8 c \sqrt{b x + c x^{2}}}{15 b^{2} x^{2}} - \frac{16 c^{2} \sqrt{b x + c x^{2}}}{15 b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(c*x**2+b*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.031278, size = 40, normalized size = 0.54 \[ -\frac{2 \sqrt{x (b+c x)} \left (3 b^2-4 b c x+8 c^2 x^2\right )}{15 b^3 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*Sqrt[b*x + c*x^2]),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 8\,{c}^{2}{x}^{2}-4\,bcx+3\,{b}^{2} \right ) }{15\,{x}^{2}{b}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(c*x^2+b*x)^(1/2),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(1/(sqrt(c*x^2 + b*x)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.223293, size = 51, normalized size = 0.69 \[ -\frac{2 \,{\left (8 \, c^{2} x^{2} - 4 \, b c x + 3 \, b^{2}\right )} \sqrt{c x^{2} + b x}}{15 \, b^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(c*x**2+b*x)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.219465, size = 105, normalized size = 1.42 \[ \frac{2 \,{\left (20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} c + 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} b \sqrt{c} + 3 \, b^{2}\right )}}{15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x)*x^3),x, algorithm="giac")
[Out]