3.258 \(\int \frac{\left (b x+c x^2\right )^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=234 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac{c^3}{3 e^7 (d+e x)^3} \]

[Out]

-(d^3*(c*d - b*e)^3)/(9*e^7*(d + e*x)^9) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(
8*e^7*(d + e*x)^8) - (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(7*e^7*
(d + e*x)^7) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2))/(6*e^7*(d + e
*x)^6) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(5*e^7*(d + e*x)^5) + (3*c^2*(2
*c*d - b*e))/(4*e^7*(d + e*x)^4) - c^3/(3*e^7*(d + e*x)^3)

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Rubi [A]  time = 0.523506, antiderivative size = 234, 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 \[ -\frac{3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{5 e^7 (d+e x)^5}+\frac{(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )}{6 e^7 (d+e x)^6}-\frac{3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )}{7 e^7 (d+e x)^7}+\frac{3 c^2 (2 c d-b e)}{4 e^7 (d+e x)^4}-\frac{d^3 (c d-b e)^3}{9 e^7 (d+e x)^9}+\frac{3 d^2 (c d-b e)^2 (2 c d-b e)}{8 e^7 (d+e x)^8}-\frac{c^3}{3 e^7 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

-(d^3*(c*d - b*e)^3)/(9*e^7*(d + e*x)^9) + (3*d^2*(c*d - b*e)^2*(2*c*d - b*e))/(
8*e^7*(d + e*x)^8) - (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(7*e^7*
(d + e*x)^7) + ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2))/(6*e^7*(d + e
*x)^6) - (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))/(5*e^7*(d + e*x)^5) + (3*c^2*(2
*c*d - b*e))/(4*e^7*(d + e*x)^4) - c^3/(3*e^7*(d + e*x)^3)

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Rubi in Sympy [A]  time = 72.8789, size = 226, normalized size = 0.97 \[ - \frac{c^{3}}{3 e^{7} \left (d + e x\right )^{3}} - \frac{3 c^{2} \left (b e - 2 c d\right )}{4 e^{7} \left (d + e x\right )^{4}} - \frac{3 c \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{5 e^{7} \left (d + e x\right )^{5}} + \frac{d^{3} \left (b e - c d\right )^{3}}{9 e^{7} \left (d + e x\right )^{9}} - \frac{3 d^{2} \left (b e - 2 c d\right ) \left (b e - c d\right )^{2}}{8 e^{7} \left (d + e x\right )^{8}} + \frac{3 d \left (b e - c d\right ) \left (b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{7 e^{7} \left (d + e x\right )^{7}} - \frac{\left (b e - 2 c d\right ) \left (b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{6 e^{7} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**3/(e*x+d)**10,x)

[Out]

-c**3/(3*e**7*(d + e*x)**3) - 3*c**2*(b*e - 2*c*d)/(4*e**7*(d + e*x)**4) - 3*c*(
b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(5*e**7*(d + e*x)**5) + d**3*(b*e - c*d)**3
/(9*e**7*(d + e*x)**9) - 3*d**2*(b*e - 2*c*d)*(b*e - c*d)**2/(8*e**7*(d + e*x)**
8) + 3*d*(b*e - c*d)*(b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(7*e**7*(d + e*x)**7)
- (b*e - 2*c*d)*(b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)/(6*e**7*(d + e*x)**6)

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Mathematica [A]  time = 0.136876, size = 222, normalized size = 0.95 \[ -\frac{5 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+12 b^2 c e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+15 b c^2 e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+10 c^3 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{2520 e^7 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x + c*x^2)^3/(d + e*x)^10,x]

[Out]

-(5*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 12*b^2*c*e^2*(d^4 +
9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 15*b*c^2*e*(d^5 + 9*d
^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^5) + 10*c^3
*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 + 126*d*e^
5*x^5 + 84*e^6*x^6))/(2520*e^7*(d + e*x)^9)

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Maple [A]  time = 0.011, size = 274, normalized size = 1.2 \[ -{\frac{3\,{d}^{2} \left ({b}^{3}{e}^{3}-4\,{b}^{2}cd{e}^{2}+5\,b{c}^{2}{d}^{2}e-2\,{c}^{3}{d}^{3} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}+{\frac{3\,d \left ({b}^{3}{e}^{3}-6\,{b}^{2}cd{e}^{2}+10\,b{c}^{2}{d}^{2}e-5\,{c}^{3}{d}^{3} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}+{\frac{{d}^{3} \left ({b}^{3}{e}^{3}-3\,{b}^{2}cd{e}^{2}+3\,b{c}^{2}{d}^{2}e-{c}^{3}{d}^{3} \right ) }{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{3\,c \left ({b}^{2}{e}^{2}-5\,bcde+5\,{c}^{2}{d}^{2} \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{3}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3}{e}^{3}-12\,{b}^{2}cd{e}^{2}+30\,b{c}^{2}{d}^{2}e-20\,{c}^{3}{d}^{3}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{c}^{2} \left ( be-2\,cd \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^3/(e*x+d)^10,x)

[Out]

-3/8*d^2*(b^3*e^3-4*b^2*c*d*e^2+5*b*c^2*d^2*e-2*c^3*d^3)/e^7/(e*x+d)^8+3/7*d*(b^
3*e^3-6*b^2*c*d*e^2+10*b*c^2*d^2*e-5*c^3*d^3)/e^7/(e*x+d)^7+1/9*d^3*(b^3*e^3-3*b
^2*c*d*e^2+3*b*c^2*d^2*e-c^3*d^3)/e^7/(e*x+d)^9-3/5*c*(b^2*e^2-5*b*c*d*e+5*c^2*d
^2)/e^7/(e*x+d)^5-1/3*c^3/e^7/(e*x+d)^3-1/6*(b^3*e^3-12*b^2*c*d*e^2+30*b*c^2*d^2
*e-20*c^3*d^3)/e^7/(e*x+d)^6-3/4*c^2*(b*e-2*c*d)/e^7/(e*x+d)^4

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Maxima [A]  time = 0.723234, size = 487, normalized size = 2.08 \[ -\frac{840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \,{\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \,{\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \,{\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \,{\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \,{\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3/(e*x + d)^10,x, algorithm="maxima")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 5*b^
3*d^3*e^3 + 630*(2*c^3*d*e^5 + 3*b*c^2*e^6)*x^5 + 126*(10*c^3*d^2*e^4 + 15*b*c^2
*d*e^5 + 12*b^2*c*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d^2*e^4 + 12*b^2*c*d*
e^5 + 5*b^3*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4
+ 5*b^3*d*e^5)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 5*b
^3*d^2*e^4)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 12
6*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8
*x + d^9*e^7)

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Fricas [A]  time = 0.21428, size = 487, normalized size = 2.08 \[ -\frac{840 \, c^{3} e^{6} x^{6} + 10 \, c^{3} d^{6} + 15 \, b c^{2} d^{5} e + 12 \, b^{2} c d^{4} e^{2} + 5 \, b^{3} d^{3} e^{3} + 630 \,{\left (2 \, c^{3} d e^{5} + 3 \, b c^{2} e^{6}\right )} x^{5} + 126 \,{\left (10 \, c^{3} d^{2} e^{4} + 15 \, b c^{2} d e^{5} + 12 \, b^{2} c e^{6}\right )} x^{4} + 84 \,{\left (10 \, c^{3} d^{3} e^{3} + 15 \, b c^{2} d^{2} e^{4} + 12 \, b^{2} c d e^{5} + 5 \, b^{3} e^{6}\right )} x^{3} + 36 \,{\left (10 \, c^{3} d^{4} e^{2} + 15 \, b c^{2} d^{3} e^{3} + 12 \, b^{2} c d^{2} e^{4} + 5 \, b^{3} d e^{5}\right )} x^{2} + 9 \,{\left (10 \, c^{3} d^{5} e + 15 \, b c^{2} d^{4} e^{2} + 12 \, b^{2} c d^{3} e^{3} + 5 \, b^{3} d^{2} e^{4}\right )} x}{2520 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/2520*(840*c^3*e^6*x^6 + 10*c^3*d^6 + 15*b*c^2*d^5*e + 12*b^2*c*d^4*e^2 + 5*b^
3*d^3*e^3 + 630*(2*c^3*d*e^5 + 3*b*c^2*e^6)*x^5 + 126*(10*c^3*d^2*e^4 + 15*b*c^2
*d*e^5 + 12*b^2*c*e^6)*x^4 + 84*(10*c^3*d^3*e^3 + 15*b*c^2*d^2*e^4 + 12*b^2*c*d*
e^5 + 5*b^3*e^6)*x^3 + 36*(10*c^3*d^4*e^2 + 15*b*c^2*d^3*e^3 + 12*b^2*c*d^2*e^4
+ 5*b^3*d*e^5)*x^2 + 9*(10*c^3*d^5*e + 15*b*c^2*d^4*e^2 + 12*b^2*c*d^3*e^3 + 5*b
^3*d^2*e^4)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7 + 84*d^3*e^13*x^6 + 12
6*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*d^7*e^9*x^2 + 9*d^8*e^8
*x + d^9*e^7)

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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)**3/(e*x+d)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.207598, size = 362, normalized size = 1.55 \[ -\frac{{\left (840 \, c^{3} x^{6} e^{6} + 1260 \, c^{3} d x^{5} e^{5} + 1260 \, c^{3} d^{2} x^{4} e^{4} + 840 \, c^{3} d^{3} x^{3} e^{3} + 360 \, c^{3} d^{4} x^{2} e^{2} + 90 \, c^{3} d^{5} x e + 10 \, c^{3} d^{6} + 1890 \, b c^{2} x^{5} e^{6} + 1890 \, b c^{2} d x^{4} e^{5} + 1260 \, b c^{2} d^{2} x^{3} e^{4} + 540 \, b c^{2} d^{3} x^{2} e^{3} + 135 \, b c^{2} d^{4} x e^{2} + 15 \, b c^{2} d^{5} e + 1512 \, b^{2} c x^{4} e^{6} + 1008 \, b^{2} c d x^{3} e^{5} + 432 \, b^{2} c d^{2} x^{2} e^{4} + 108 \, b^{2} c d^{3} x e^{3} + 12 \, b^{2} c d^{4} e^{2} + 420 \, b^{3} x^{3} e^{6} + 180 \, b^{3} d x^{2} e^{5} + 45 \, b^{3} d^{2} x e^{4} + 5 \, b^{3} d^{3} e^{3}\right )} e^{\left (-7\right )}}{2520 \,{\left (x e + d\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^3/(e*x + d)^10,x, algorithm="giac")

[Out]

-1/2520*(840*c^3*x^6*e^6 + 1260*c^3*d*x^5*e^5 + 1260*c^3*d^2*x^4*e^4 + 840*c^3*d
^3*x^3*e^3 + 360*c^3*d^4*x^2*e^2 + 90*c^3*d^5*x*e + 10*c^3*d^6 + 1890*b*c^2*x^5*
e^6 + 1890*b*c^2*d*x^4*e^5 + 1260*b*c^2*d^2*x^3*e^4 + 540*b*c^2*d^3*x^2*e^3 + 13
5*b*c^2*d^4*x*e^2 + 15*b*c^2*d^5*e + 1512*b^2*c*x^4*e^6 + 1008*b^2*c*d*x^3*e^5 +
432*b^2*c*d^2*x^2*e^4 + 108*b^2*c*d^3*x*e^3 + 12*b^2*c*d^4*e^2 + 420*b^3*x^3*e^
6 + 180*b^3*d*x^2*e^5 + 45*b^3*d^2*x*e^4 + 5*b^3*d^3*e^3)*e^(-7)/(x*e + d)^9