Optimal. Leaf size=70 \[ \frac{2 (b+2 c x) (b B-4 A c)}{3 b^3 c \sqrt{b x+c x^2}}-\frac{2 x (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.107399, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 (b+2 c x) (b B-4 A c)}{3 b^3 c \sqrt{b x+c x^2}}-\frac{2 x (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(x*(A + B*x))/(b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.12253, size = 61, normalized size = 0.87 \[ \frac{2 x \left (A c - B b\right )}{3 b c \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{\left (2 b + 4 c x\right ) \left (4 A c - B b\right )}{3 b^{3} c \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(B*x+A)/(c*x**2+b*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0796265, size = 55, normalized size = 0.79 \[ \frac{x \left (2 b B x (3 b+2 c x)-2 A \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )}{3 b^3 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x*(A + B*x))/(b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 62, normalized size = 0.9 \[ -{\frac{2\,{x}^{2} \left ( cx+b \right ) \left ( 8\,A{c}^{2}{x}^{2}-2\,B{x}^{2}bc+12\,Abcx-3\,{b}^{2}Bx+3\,{b}^{2}A \right ) }{3\,{b}^{3}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(B*x+A)/(c*x^2+b*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.685086, size = 150, normalized size = 2.14 \[ \frac{4 \, B x}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, A x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{2 \, B x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{16 \, A c x}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \, A}{3 \, \sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, B}{3 \, \sqrt{c x^{2} + b x} b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.303694, size = 84, normalized size = 1.2 \[ -\frac{2 \,{\left (3 \, A b^{2} - 2 \,{\left (B b c - 4 \, A c^{2}\right )} x^{2} - 3 \,{\left (B b^{2} - 4 \, A b c\right )} x\right )}}{3 \,{\left (b^{3} c x + b^{4}\right )} \sqrt{c x^{2} + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(B*x+A)/(c*x**2+b*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x + A\right )} x}{{\left (c x^{2} + b x\right )}^{\frac{5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]