Optimal. Leaf size=110 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac{c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.189809, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac{c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^2/(d + e*x)^6,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 30.5937, size = 102, normalized size = 0.93 \[ \frac{2 c^{2} d}{e^{5} \left (d + e x\right )^{2}} - \frac{c^{2}}{e^{5} \left (d + e x\right )} + \frac{c d \left (a e^{2} + c d^{2}\right )}{e^{5} \left (d + e x\right )^{4}} - \frac{2 c \left (a e^{2} + 3 c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{3}} - \frac{\left (a e^{2} + c d^{2}\right )^{2}}{5 e^{5} \left (d + e x\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**2/(e*x+d)**6,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0657837, size = 90, normalized size = 0.82 \[ -\frac{3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{15 e^5 (d+e x)^5} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^2/(d + e*x)^6,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.008, size = 119, normalized size = 1.1 \[ -{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}+2\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{2\,c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{cd \left ( a{e}^{2}+c{d}^{2} \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^2/(e*x+d)^6,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.722902, size = 204, normalized size = 1.85 \[ -\frac{15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^6,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.198456, size = 204, normalized size = 1.85 \[ -\frac{15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^6,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 8.00162, size = 160, normalized size = 1.45 \[ - \frac{3 a^{2} e^{4} + a c d^{2} e^{2} + 3 c^{2} d^{4} + 30 c^{2} d e^{3} x^{3} + 15 c^{2} e^{4} x^{4} + x^{2} \left (10 a c e^{4} + 30 c^{2} d^{2} e^{2}\right ) + x \left (5 a c d e^{3} + 15 c^{2} d^{3} e\right )}{15 d^{5} e^{5} + 75 d^{4} e^{6} x + 150 d^{3} e^{7} x^{2} + 150 d^{2} e^{8} x^{3} + 75 d e^{9} x^{4} + 15 e^{10} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**2/(e*x+d)**6,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.209593, size = 132, normalized size = 1.2 \[ -\frac{{\left (15 \, c^{2} x^{4} e^{4} + 30 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 15 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 10 \, a c x^{2} e^{4} + 5 \, a c d x e^{3} + a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^6,x, algorithm="giac")
[Out]