Optimal. Leaf size=87 \[ \frac{16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.126246, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{16 d (c d-b e) (x (2 c d-b e)+b d)}{3 b^4 \sqrt{b x+c x^2}}-\frac{2 (d+e x)^2 (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^3/(b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.2367, size = 83, normalized size = 0.95 \[ - \frac{2 \left (d + e x\right )^{2} \left (b d - x \left (b e - 2 c d\right )\right )}{3 b^{2} \left (b x + c x^{2}\right )^{\frac{3}{2}}} - \frac{8 d \left (2 b d - x \left (2 b e - 4 c d\right )\right ) \left (b e - c d\right )}{3 b^{4} \sqrt{b x + c x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**3/(c*x**2+b*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.143539, size = 105, normalized size = 1.21 \[ \frac{2 \left (b^3 \left (-d^3-9 d^2 e x+9 d e^2 x^2+e^3 x^3\right )+6 b^2 c d x \left (d^2-6 d e x+e^2 x^2\right )+24 b c^2 d^2 x^2 (d-e x)+16 c^3 d^3 x^3\right )}{3 b^4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^3/(b*x + c*x^2)^(5/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 136, normalized size = 1.6 \[ -{\frac{2\,x \left ( cx+b \right ) \left ( -{b}^{3}{e}^{3}{x}^{3}-6\,{b}^{2}cd{e}^{2}{x}^{3}+24\,b{c}^{2}{d}^{2}e{x}^{3}-16\,{c}^{3}{d}^{3}{x}^{3}-9\,{b}^{3}d{e}^{2}{x}^{2}+36\,{b}^{2}c{d}^{2}e{x}^{2}-24\,b{c}^{2}{d}^{3}{x}^{2}+9\,{b}^{3}{d}^{2}ex-6\,{b}^{2}c{d}^{3}x+{d}^{3}{b}^{3} \right ) }{3\,{b}^{4}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^3/(c*x^2+b*x)^(5/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.703837, size = 401, normalized size = 4.61 \[ -\frac{e^{3} x^{2}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{4 \, c d^{3} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b^{2}} + \frac{32 \, c^{2} d^{3} x}{3 \, \sqrt{c x^{2} + b x} b^{4}} + \frac{2 \, d^{2} e x}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} - \frac{16 \, c d^{2} e x}{\sqrt{c x^{2} + b x} b^{3}} + \frac{4 \, d e^{2} x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2 \, d e^{2} x}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} - \frac{b e^{3} x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c^{2}} + \frac{2 \, e^{3} x}{3 \, \sqrt{c x^{2} + b x} b c} - \frac{2 \, d^{3}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} b} + \frac{16 \, c d^{3}}{3 \, \sqrt{c x^{2} + b x} b^{3}} - \frac{8 \, d^{2} e}{\sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, d e^{2}}{\sqrt{c x^{2} + b x} b c} + \frac{e^{3}}{3 \, \sqrt{c x^{2} + b x} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x)^(5/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.214751, size = 181, normalized size = 2.08 \[ -\frac{2 \,{\left (b^{3} d^{3} -{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x^{3} - 3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )} x^{2} - 3 \,{\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )} x\right )}}{3 \,{\left (b^{4} c x^{2} + b^{5} x\right )} \sqrt{c x^{2} + b x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x)^(5/2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (d + e x\right )^{3}}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**3/(c*x**2+b*x)**(5/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.218898, size = 188, normalized size = 2.16 \[ \frac{{\left (x{\left (\frac{{\left (16 \, c^{3} d^{3} - 24 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} + b^{3} e^{3}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (8 \, b c^{2} d^{3} - 12 \, b^{2} c d^{2} e + 3 \, b^{3} d e^{2}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (2 \, b^{2} c d^{3} - 3 \, b^{3} d^{2} e\right )}}{b^{4} c^{2}}\right )} x - \frac{d^{3}}{b c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^3/(c*x^2 + b*x)^(5/2),x, algorithm="giac")
[Out]