Optimal. Leaf size=389 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac{5 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{c^2 x \left (-c e (35 b d-6 a e)+9 b^2 e^2+30 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8 (d+e x)^4}-\frac{c^3 x^2 (10 c d-7 b e)}{2 e^6}+\frac{2 c^4 x^3}{3 e^5} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 1.37255, antiderivative size = 389, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ -\frac{6 c^2 e^2 \left (a^2 e^2-10 a b d e+15 b^2 d^2\right )-4 b^2 c e^3 (5 b d-3 a e)-20 c^3 d^2 e (7 b d-3 a e)+b^4 e^4+70 c^4 d^4}{e^8 (d+e x)}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{2 e^8 (d+e x)^2}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-2 c e (7 b d-a e)+3 b^2 e^2+14 c^2 d^2\right )}{3 e^8 (d+e x)^3}-\frac{5 c (2 c d-b e) \log (d+e x) \left (-c e (7 b d-3 a e)+b^2 e^2+7 c^2 d^2\right )}{e^8}+\frac{c^2 x \left (-c e (35 b d-6 a e)+9 b^2 e^2+30 c^2 d^2\right )}{e^7}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8 (d+e x)^4}-\frac{c^3 x^2 (10 c d-7 b e)}{2 e^6}+\frac{2 c^4 x^3}{3 e^5} \]
Antiderivative was successfully verified.
[In] Int[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^5,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**5,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.659105, size = 614, normalized size = 1.58 \[ -\frac{3 c^2 e^2 \left (6 a^2 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 a b d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+3 b^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )+c e^3 \left (2 a^3 e^3 (d+4 e x)+9 a^2 b e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+36 a b^2 e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-5 b^3 d \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+3 b e^4 \left (a^3 e^3+a^2 b e^2 (d+4 e x)+a b^2 e \left (d^2+4 d e x+6 e^2 x^2\right )+b^3 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )\right )+60 c (d+e x)^4 (2 c d-b e) \log (d+e x) \left (c e (3 a e-7 b d)+b^2 e^2+7 c^2 d^2\right )-3 c^3 e \left (2 a e \left (-77 d^5-248 d^4 e x-252 d^3 e^2 x^2-48 d^2 e^3 x^3+48 d e^4 x^4+12 e^5 x^5\right )+7 b \left (57 d^6+168 d^5 e x+132 d^4 e^2 x^2-32 d^3 e^3 x^3-68 d^2 e^4 x^4-12 d e^5 x^5+2 e^6 x^6\right )\right )+2 c^4 \left (319 d^7+856 d^6 e x+444 d^5 e^2 x^2-544 d^4 e^3 x^3-556 d^3 e^4 x^4-84 d^2 e^5 x^5+14 d e^6 x^6-4 e^7 x^7\right )}{12 e^8 (d+e x)^4} \]
Antiderivative was successfully verified.
[In] Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^3)/(d + e*x)^5,x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.021, size = 1056, normalized size = 2.7 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2*c*x+b)*(c*x^2+b*x+a)^3/(e*x+d)^5,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.728971, size = 917, normalized size = 2.36 \[ -\frac{638 \, c^{4} d^{7} - 1197 \, b c^{3} d^{6} e + 3 \, a^{3} b e^{7} + 231 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{5} e^{2} - 125 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{4} e^{3} + 3 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{3} e^{4} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d^{2} e^{5} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} d e^{6} + 12 \,{\left (70 \, c^{4} d^{4} e^{3} - 140 \, b c^{3} d^{3} e^{4} + 30 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{2} e^{5} - 20 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d e^{6} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} e^{7}\right )} x^{3} + 18 \,{\left (126 \, c^{4} d^{5} e^{2} - 245 \, b c^{3} d^{4} e^{3} + 50 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{3} e^{4} - 30 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{2} e^{5} +{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d e^{6} +{\left (a b^{3} + 3 \, a^{2} b c\right )} e^{7}\right )} x^{2} + 4 \,{\left (518 \, c^{4} d^{6} e - 987 \, b c^{3} d^{5} e^{2} + 195 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} e^{3} - 110 \,{\left (b^{3} c + 3 \, a b c^{2}\right )} d^{3} e^{4} + 3 \,{\left (b^{4} + 12 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} d^{2} e^{5} + 3 \,{\left (a b^{3} + 3 \, a^{2} b c\right )} d e^{6} +{\left (3 \, a^{2} b^{2} + 2 \, a^{3} c\right )} e^{7}\right )} x}{12 \,{\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} + \frac{4 \, c^{4} e^{2} x^{3} - 3 \,{\left (10 \, c^{4} d e - 7 \, b c^{3} e^{2}\right )} x^{2} + 6 \,{\left (30 \, c^{4} d^{2} - 35 \, b c^{3} d e + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} e^{2}\right )} x}{6 \, e^{7}} - \frac{5 \,{\left (14 \, c^{4} d^{3} - 21 \, b c^{3} d^{2} e + 3 \,{\left (3 \, b^{2} c^{2} + 2 \, a c^{3}\right )} d e^{2} -{\left (b^{3} c + 3 \, a b c^{2}\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(2*c*x + b)/(e*x + d)^5,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.268258, size = 1373, normalized size = 3.53 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(2*c*x + b)/(e*x + d)^5,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*c*x+b)*(c*x**2+b*x+a)**3/(e*x+d)**5,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.283017, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3*(2*c*x + b)/(e*x + d)^5,x, algorithm="giac")
[Out]