Optimal. Leaf size=132 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac{d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}+\frac{d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{e^5 (d+e x)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.252742, antiderivative size = 132, 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 \[ -\frac{b^2 e^2-6 b c d e+6 c^2 d^2}{3 e^5 (d+e x)^3}-\frac{d^2 (c d-b e)^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}+\frac{d (c d-b e) (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{e^5 (d+e x)} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^2/(d + e*x)^6,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 39.8921, size = 119, normalized size = 0.9 \[ - \frac{c^{2}}{e^{5} \left (d + e x\right )} - \frac{c \left (b e - 2 c d\right )}{e^{5} \left (d + e x\right )^{2}} - \frac{d^{2} \left (b e - c d\right )^{2}}{5 e^{5} \left (d + e x\right )^{5}} + \frac{d \left (b e - 2 c d\right ) \left (b e - c d\right )}{2 e^{5} \left (d + e x\right )^{4}} - \frac{b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{3 e^{5} \left (d + e x\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**2/(e*x+d)**6,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0854447, size = 116, normalized size = 0.88 \[ -\frac{b^2 e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 b c e \left (d^3+5 d^2 e x+10 d e^2 x^2+10 e^3 x^3\right )+6 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{30 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^2/(d + e*x)^6,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 143, normalized size = 1.1 \[ -{\frac{{d}^{2} \left ({b}^{2}{e}^{2}-2\,bcde+{c}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{c \left ( be-2\,cd \right ) }{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{d \left ({b}^{2}{e}^{2}-3\,bcde+2\,{c}^{2}{d}^{2} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^2/(e*x+d)^6,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.717025, size = 244, normalized size = 1.85 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2/(e*x + d)^6,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.219721, size = 244, normalized size = 1.85 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + b^{2} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2/(e*x + d)^6,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 11.1081, size = 192, normalized size = 1.45 \[ - \frac{b^{2} d^{2} e^{2} + 3 b c d^{3} e + 6 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (30 b c e^{4} + 60 c^{2} d e^{3}\right ) + x^{2} \left (10 b^{2} e^{4} + 30 b c d e^{3} + 60 c^{2} d^{2} e^{2}\right ) + x \left (5 b^{2} d e^{3} + 15 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+b*x)**2/(e*x+d)**6,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.211482, size = 178, normalized size = 1.35 \[ -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 5 \, b^{2} d x e^{3} + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^2/(e*x + d)^6,x, algorithm="giac")
[Out]