Optimal. Leaf size=269 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^3}{8 e^7 (d+e x)^8}+\frac{c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{c^3}{2 e^7 (d+e x)^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.760465, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{3 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^7 (d+e x)^4}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{5 e^7 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^7 (d+e x)^6}+\frac{3 (2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{7 e^7 (d+e x)^7}-\frac{\left (a e^2-b d e+c d^2\right )^3}{8 e^7 (d+e x)^8}+\frac{c^2 (2 c d-b e)}{e^7 (d+e x)^3}-\frac{c^3}{2 e^7 (d+e x)^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x + c*x^2)^3/(d + e*x)^9,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 99.8395, size = 262, normalized size = 0.97 \[ - \frac{c^{3}}{2 e^{7} \left (d + e x\right )^{2}} - \frac{c^{2} \left (b e - 2 c d\right )}{e^{7} \left (d + e x\right )^{3}} - \frac{3 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{4 e^{7} \left (d + e x\right )^{4}} - \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{5 e^{7} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{2 e^{7} \left (d + e x\right )^{6}} - \frac{3 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{7 e^{7} \left (d + e x\right )^{7}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{3}}{8 e^{7} \left (d + e x\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x+a)**3/(e*x+d)**9,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.412007, size = 375, normalized size = 1.39 \[ -\frac{c e^2 \left (5 a^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a b e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+e^3 \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+c^2 e \left (3 a e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )}{280 e^7 (d+e x)^8} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x + c*x^2)^3/(d + e*x)^9,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.013, size = 461, normalized size = 1.7 \[ -{\frac{3\,{a}^{2}b{e}^{5}-6\,{a}^{2}cd{e}^{4}-6\,a{b}^{2}d{e}^{4}+18\,abc{d}^{2}{e}^{3}-12\,a{c}^{2}{d}^{3}{e}^{2}+3\,{b}^{3}{d}^{2}{e}^{3}-12\,{d}^{3}{b}^{2}c{e}^{2}+15\,{d}^{4}b{c}^{2}e-6\,{c}^{3}{d}^{5}}{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{{c}^{3}}{2\,{e}^{7} \left ( ex+d \right ) ^{2}}}-{\frac{{a}^{3}{e}^{6}-3\,{a}^{2}bd{e}^{5}+3\,{a}^{2}c{d}^{2}{e}^{4}+3\,a{b}^{2}{d}^{2}{e}^{4}-6\,{d}^{3}acb{e}^{3}+3\,{c}^{2}{d}^{4}a{e}^{2}-{b}^{3}{d}^{3}{e}^{3}+3\,{d}^{4}{b}^{2}c{e}^{2}-3\,b{c}^{2}{d}^{5}e+{c}^{3}{d}^{6}}{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{{c}^{2} \left ( be-2\,cd \right ) }{{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{6\,abc{e}^{3}-12\,a{c}^{2}{e}^{2}d+{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{3\,{a}^{2}c{e}^{4}+3\,a{b}^{2}{e}^{4}-18\,cabd{e}^{3}+18\,a{c}^{2}{d}^{2}{e}^{2}-3\,d{b}^{3}{e}^{3}+18\,c{b}^{2}{d}^{2}{e}^{2}-30\,{d}^{3}eb{c}^{2}+15\,{c}^{3}{d}^{4}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,c \left ( ac{e}^{2}+{b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x+a)^3/(e*x+d)^9,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.826762, size = 651, normalized size = 2.42 \[ -\frac{140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 5 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 280 \,{\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 5 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 15 \, a^{2} b e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 5 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^9,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.20369, size = 651, normalized size = 2.42 \[ -\frac{140 \, c^{3} e^{6} x^{6} + 5 \, c^{3} d^{6} + 5 \, b c^{2} d^{5} e + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 5 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4} + 280 \,{\left (c^{3} d e^{5} + b c^{2} e^{6}\right )} x^{5} + 70 \,{\left (5 \, c^{3} d^{2} e^{4} + 5 \, b c^{2} d e^{5} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6}\right )} x^{4} + 56 \,{\left (5 \, c^{3} d^{3} e^{3} + 5 \, b c^{2} d^{2} e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{5} +{\left (b^{3} + 6 \, a b c\right )} e^{6}\right )} x^{3} + 28 \,{\left (5 \, c^{3} d^{4} e^{2} + 5 \, b c^{2} d^{3} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} +{\left (b^{3} + 6 \, a b c\right )} d e^{5} + 5 \,{\left (a b^{2} + a^{2} c\right )} e^{6}\right )} x^{2} + 8 \,{\left (5 \, c^{3} d^{5} e + 5 \, b c^{2} d^{4} e^{2} + 15 \, a^{2} b e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 5 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} x}{280 \,{\left (e^{15} x^{8} + 8 \, d e^{14} x^{7} + 28 \, d^{2} e^{13} x^{6} + 56 \, d^{3} e^{12} x^{5} + 70 \, d^{4} e^{11} x^{4} + 56 \, d^{5} e^{10} x^{3} + 28 \, d^{6} e^{9} x^{2} + 8 \, d^{7} e^{8} x + d^{8} e^{7}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^9,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((c*x**2+b*x+a)**3/(e*x+d)**9,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.203912, size = 618, normalized size = 2.3 \[ -\frac{{\left (140 \, c^{3} x^{6} e^{6} + 280 \, c^{3} d x^{5} e^{5} + 350 \, c^{3} d^{2} x^{4} e^{4} + 280 \, c^{3} d^{3} x^{3} e^{3} + 140 \, c^{3} d^{4} x^{2} e^{2} + 40 \, c^{3} d^{5} x e + 5 \, c^{3} d^{6} + 280 \, b c^{2} x^{5} e^{6} + 350 \, b c^{2} d x^{4} e^{5} + 280 \, b c^{2} d^{2} x^{3} e^{4} + 140 \, b c^{2} d^{3} x^{2} e^{3} + 40 \, b c^{2} d^{4} x e^{2} + 5 \, b c^{2} d^{5} e + 210 \, b^{2} c x^{4} e^{6} + 210 \, a c^{2} x^{4} e^{6} + 168 \, b^{2} c d x^{3} e^{5} + 168 \, a c^{2} d x^{3} e^{5} + 84 \, b^{2} c d^{2} x^{2} e^{4} + 84 \, a c^{2} d^{2} x^{2} e^{4} + 24 \, b^{2} c d^{3} x e^{3} + 24 \, a c^{2} d^{3} x e^{3} + 3 \, b^{2} c d^{4} e^{2} + 3 \, a c^{2} d^{4} e^{2} + 56 \, b^{3} x^{3} e^{6} + 336 \, a b c x^{3} e^{6} + 28 \, b^{3} d x^{2} e^{5} + 168 \, a b c d x^{2} e^{5} + 8 \, b^{3} d^{2} x e^{4} + 48 \, a b c d^{2} x e^{4} + b^{3} d^{3} e^{3} + 6 \, a b c d^{3} e^{3} + 140 \, a b^{2} x^{2} e^{6} + 140 \, a^{2} c x^{2} e^{6} + 40 \, a b^{2} d x e^{5} + 40 \, a^{2} c d x e^{5} + 5 \, a b^{2} d^{2} e^{4} + 5 \, a^{2} c d^{2} e^{4} + 120 \, a^{2} b x e^{6} + 15 \, a^{2} b d e^{5} + 35 \, a^{3} e^{6}\right )} e^{\left (-7\right )}}{280 \,{\left (x e + d\right )}^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x + a)^3/(e*x + d)^9,x, algorithm="giac")
[Out]