Optimal. Leaf size=114 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{5 e^5 (d+e x)^5}+\frac{2 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2}{7 e^5 (d+e x)^7}-\frac{c^2}{3 e^5 (d+e x)^3}+\frac{c^2 d}{e^5 (d+e x)^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.186919, antiderivative size = 114, 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 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{5 e^5 (d+e x)^5}+\frac{2 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2}{7 e^5 (d+e x)^7}-\frac{c^2}{3 e^5 (d+e x)^3}+\frac{c^2 d}{e^5 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)^2/(d + e*x)^8,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 31.3423, size = 107, normalized size = 0.94 \[ \frac{c^{2} d}{e^{5} \left (d + e x\right )^{4}} - \frac{c^{2}}{3 e^{5} \left (d + e x\right )^{3}} + \frac{2 c d \left (a e^{2} + c d^{2}\right )}{3 e^{5} \left (d + e x\right )^{6}} - \frac{2 c \left (a e^{2} + 3 c d^{2}\right )}{5 e^{5} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} + c d^{2}\right )^{2}}{7 e^{5} \left (d + e x\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)**2/(e*x+d)**8,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0654663, size = 90, normalized size = 0.79 \[ -\frac{15 a^2 e^4+2 a c e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)^2/(d + e*x)^8,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 119, normalized size = 1. \[ -{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}+{\frac{2\,cd \left ( a{e}^{2}+c{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{2\,c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)^2/(e*x+d)^8,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.707614, size = 234, normalized size = 2.05 \[ -\frac{35 \, c^{2} e^{4} x^{4} + 35 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + 21 \,{\left (c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} + 7 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^8,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.201483, size = 234, normalized size = 2.05 \[ -\frac{35 \, c^{2} e^{4} x^{4} + 35 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + 21 \,{\left (c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} + 7 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^8,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 11.9258, size = 184, normalized size = 1.61 \[ - \frac{15 a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4} + 35 c^{2} d e^{3} x^{3} + 35 c^{2} e^{4} x^{4} + x^{2} \left (42 a c e^{4} + 21 c^{2} d^{2} e^{2}\right ) + x \left (14 a c d e^{3} + 7 c^{2} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)**2/(e*x+d)**8,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.209851, size = 132, normalized size = 1.16 \[ -\frac{{\left (35 \, c^{2} x^{4} e^{4} + 35 \, c^{2} d x^{3} e^{3} + 21 \, c^{2} d^{2} x^{2} e^{2} + 7 \, c^{2} d^{3} x e + c^{2} d^{4} + 42 \, a c x^{2} e^{4} + 14 \, a c d x e^{3} + 2 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{105 \,{\left (x e + d\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)^2/(e*x + d)^8,x, algorithm="giac")
[Out]