3.1329 \(\int \frac{(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=334 \[ \frac{c \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{7 e^8 (d+e x)^7}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{6 e^8 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{9 e^8 (d+e x)^9}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{10 e^8 (d+e x)^{10}}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8 (d+e x)^8}+\frac{c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac{B c^3}{3 e^8 (d+e x)^3} \]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(10*e^8*(d + e*x)^10) - ((c*d^2 + a*e^2)^2*(7*B*
c*d^2 - 6*A*c*d*e + a*B*e^2))/(9*e^8*(d + e*x)^9) + (3*c*(c*d^2 + a*e^2)*(7*B*c*
d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(8*e^8*(d + e*x)^8) + (c*(4*A*c*d*e*
(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)))/(7*e^8*(d +
e*x)^7) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(6*e^8*(d
+ e*x)^6) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(5*e^8*(d + e*x)^5) + (c^
3*(7*B*d - A*e))/(4*e^8*(d + e*x)^4) - (B*c^3)/(3*e^8*(d + e*x)^3)

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Rubi [A]  time = 0.964739, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{c \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{7 e^8 (d+e x)^7}-\frac{3 c^2 \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac{c^2 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{6 e^8 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{9 e^8 (d+e x)^9}+\frac{\left (a e^2+c d^2\right )^3 (B d-A e)}{10 e^8 (d+e x)^{10}}+\frac{3 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8 (d+e x)^8}+\frac{c^3 (7 B d-A e)}{4 e^8 (d+e x)^4}-\frac{B c^3}{3 e^8 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^11,x]

[Out]

((B*d - A*e)*(c*d^2 + a*e^2)^3)/(10*e^8*(d + e*x)^10) - ((c*d^2 + a*e^2)^2*(7*B*
c*d^2 - 6*A*c*d*e + a*B*e^2))/(9*e^8*(d + e*x)^9) + (3*c*(c*d^2 + a*e^2)*(7*B*c*
d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(8*e^8*(d + e*x)^8) - (c*(35*B*c^2*d
^4 - 20*A*c^2*d^3*e + 30*a*B*c*d^2*e^2 - 12*a*A*c*d*e^3 + 3*a^2*B*e^4))/(7*e^8*(
d + e*x)^7) + (c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3))/(6*e^
8*(d + e*x)^6) - (3*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2))/(5*e^8*(d + e*x)^5) +
(c^3*(7*B*d - A*e))/(4*e^8*(d + e*x)^4) - (B*c^3)/(3*e^8*(d + e*x)^3)

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Rubi in Sympy [A]  time = 154.702, size = 345, normalized size = 1.03 \[ - \frac{B c^{3}}{3 e^{8} \left (d + e x\right )^{3}} - \frac{c^{3} \left (A e - 7 B d\right )}{4 e^{8} \left (d + e x\right )^{4}} - \frac{3 c^{2} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right )}{5 e^{8} \left (d + e x\right )^{5}} - \frac{c^{2} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{6 e^{8} \left (d + e x\right )^{6}} - \frac{c \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{7 e^{8} \left (d + e x\right )^{7}} - \frac{3 c \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{8 e^{8} \left (d + e x\right )^{8}} - \frac{\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{9 e^{8} \left (d + e x\right )^{9}} - \frac{\left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{10 e^{8} \left (d + e x\right )^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**11,x)

[Out]

-B*c**3/(3*e**8*(d + e*x)**3) - c**3*(A*e - 7*B*d)/(4*e**8*(d + e*x)**4) - 3*c**
2*(-2*A*c*d*e + B*a*e**2 + 7*B*c*d**2)/(5*e**8*(d + e*x)**5) - c**2*(3*A*a*e**3
+ 15*A*c*d**2*e - 15*B*a*d*e**2 - 35*B*c*d**3)/(6*e**8*(d + e*x)**6) - c*(-12*A*
a*c*d*e**3 - 20*A*c**2*d**3*e + 3*B*a**2*e**4 + 30*B*a*c*d**2*e**2 + 35*B*c**2*d
**4)/(7*e**8*(d + e*x)**7) - 3*c*(a*e**2 + c*d**2)*(A*a*e**3 + 5*A*c*d**2*e - 3*
B*a*d*e**2 - 7*B*c*d**3)/(8*e**8*(d + e*x)**8) - (a*e**2 + c*d**2)**2*(-6*A*c*d*
e + B*a*e**2 + 7*B*c*d**2)/(9*e**8*(d + e*x)**9) - (A*e - B*d)*(a*e**2 + c*d**2)
**3/(10*e**8*(d + e*x)**10)

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Mathematica [A]  time = 0.491852, size = 357, normalized size = 1.07 \[ -\frac{3 A e \left (84 a^3 e^6+7 a^2 c e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+2 a c^2 e^2 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )+c^3 \left (d^6+10 d^5 e x+45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+252 d e^5 x^5+210 e^6 x^6\right )\right )+B \left (28 a^3 e^6 (d+10 e x)+9 a^2 c e^4 \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+6 a c^2 e^2 \left (d^5+10 d^4 e x+45 d^3 e^2 x^2+120 d^2 e^3 x^3+210 d e^4 x^4+252 e^5 x^5\right )+7 c^3 \left (d^7+10 d^6 e x+45 d^5 e^2 x^2+120 d^4 e^3 x^3+210 d^3 e^4 x^4+252 d^2 e^5 x^5+210 d e^6 x^6+120 e^7 x^7\right )\right )}{2520 e^8 (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^11,x]

[Out]

-(3*A*e*(84*a^3*e^6 + 7*a^2*c*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 2*a*c^2*e^2*(d
^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4) + c^3*(d^6 + 10*
d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 2
10*e^6*x^6)) + B*(28*a^3*e^6*(d + 10*e*x) + 9*a^2*c*e^4*(d^3 + 10*d^2*e*x + 45*d
*e^2*x^2 + 120*e^3*x^3) + 6*a*c^2*e^2*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d
^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5) + 7*c^3*(d^7 + 10*d^6*e*x + 45*d^5*e^2
*x^2 + 120*d^4*e^3*x^3 + 210*d^3*e^4*x^4 + 252*d^2*e^5*x^5 + 210*d*e^6*x^6 + 120
*e^7*x^7)))/(2520*e^8*(d + e*x)^10)

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Maple [A]  time = 0.013, size = 449, normalized size = 1.3 \[ -{\frac{{c}^{2} \left ( 3\,aA{e}^{3}+15\,Ac{d}^{2}e-15\,aBd{e}^{2}-35\,Bc{d}^{3} \right ) }{6\,{e}^{8} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{3} \left ( Ae-7\,Bd \right ) }{4\,{e}^{8} \left ( ex+d \right ) ^{4}}}-{\frac{B{c}^{3}}{3\,{e}^{8} \left ( ex+d \right ) ^{3}}}+{\frac{3\,{c}^{2} \left ( 2\,Acde-aB{e}^{2}-7\,Bc{d}^{2} \right ) }{5\,{e}^{8} \left ( ex+d \right ) ^{5}}}-{\frac{-6\,Ad{a}^{2}c{e}^{5}-12\,A{d}^{3}a{c}^{2}{e}^{3}-6\,A{d}^{5}{c}^{3}e+B{a}^{3}{e}^{6}+9\,B{a}^{2}c{d}^{2}{e}^{4}+15\,Ba{c}^{2}{d}^{4}{e}^{2}+7\,B{c}^{3}{d}^{6}}{9\,{e}^{8} \left ( ex+d \right ) ^{9}}}-{\frac{A{a}^{3}{e}^{7}+3\,A{d}^{2}{a}^{2}c{e}^{5}+3\,A{d}^{4}a{c}^{2}{e}^{3}+A{d}^{6}{c}^{3}e-B{a}^{3}d{e}^{6}-3\,B{a}^{2}c{d}^{3}{e}^{4}-3\,Ba{c}^{2}{d}^{5}{e}^{2}-B{c}^{3}{d}^{7}}{10\,{e}^{8} \left ( ex+d \right ) ^{10}}}-{\frac{3\,c \left ( A{a}^{2}{e}^{5}+6\,A{d}^{2}ac{e}^{3}+5\,A{d}^{4}{c}^{2}e-3\,Bd{a}^{2}{e}^{4}-10\,aBc{d}^{3}{e}^{2}-7\,B{c}^{2}{d}^{5} \right ) }{8\,{e}^{8} \left ( ex+d \right ) ^{8}}}+{\frac{c \left ( 12\,Aacd{e}^{3}+20\,A{c}^{2}{d}^{3}e-3\,B{e}^{4}{a}^{2}-30\,Bac{d}^{2}{e}^{2}-35\,B{c}^{2}{d}^{4} \right ) }{7\,{e}^{8} \left ( ex+d \right ) ^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^3/(e*x+d)^11,x)

[Out]

-1/6*c^2*(3*A*a*e^3+15*A*c*d^2*e-15*B*a*d*e^2-35*B*c*d^3)/e^8/(e*x+d)^6-1/4*c^3*
(A*e-7*B*d)/e^8/(e*x+d)^4-1/3*B*c^3/e^8/(e*x+d)^3+3/5*c^2*(2*A*c*d*e-B*a*e^2-7*B
*c*d^2)/e^8/(e*x+d)^5-1/9*(-6*A*a^2*c*d*e^5-12*A*a*c^2*d^3*e^3-6*A*c^3*d^5*e+B*a
^3*e^6+9*B*a^2*c*d^2*e^4+15*B*a*c^2*d^4*e^2+7*B*c^3*d^6)/e^8/(e*x+d)^9-1/10*(A*a
^3*e^7+3*A*a^2*c*d^2*e^5+3*A*a*c^2*d^4*e^3+A*c^3*d^6*e-B*a^3*d*e^6-3*B*a^2*c*d^3
*e^4-3*B*a*c^2*d^5*e^2-B*c^3*d^7)/e^8/(e*x+d)^10-3/8*c*(A*a^2*e^5+6*A*a*c*d^2*e^
3+5*A*c^2*d^4*e-3*B*a^2*d*e^4-10*B*a*c*d^3*e^2-7*B*c^2*d^5)/e^8/(e*x+d)^8+1/7*c*
(12*A*a*c*d*e^3+20*A*c^2*d^3*e-3*B*a^2*e^4-30*B*a*c*d^2*e^2-35*B*c^2*d^4)/e^8/(e
*x+d)^7

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Maxima [A]  time = 0.729228, size = 752, normalized size = 2.25 \[ -\frac{840 \, B c^{3} e^{7} x^{7} + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7} + 210 \,{\left (7 \, B c^{3} d e^{6} + 3 \, A c^{3} e^{7}\right )} x^{6} + 252 \,{\left (7 \, B c^{3} d^{2} e^{5} + 3 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} + 210 \,{\left (7 \, B c^{3} d^{3} e^{4} + 3 \, A c^{3} d^{2} e^{5} + 6 \, B a c^{2} d e^{6} + 6 \, A a c^{2} e^{7}\right )} x^{4} + 120 \,{\left (7 \, B c^{3} d^{4} e^{3} + 3 \, A c^{3} d^{3} e^{4} + 6 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} + 45 \,{\left (7 \, B c^{3} d^{5} e^{2} + 3 \, A c^{3} d^{4} e^{3} + 6 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 21 \, A a^{2} c e^{7}\right )} x^{2} + 10 \,{\left (7 \, B c^{3} d^{6} e + 3 \, A c^{3} d^{5} e^{2} + 6 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 21 \, A a^{2} c d e^{6} + 28 \, B a^{3} e^{7}\right )} x}{2520 \,{\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^11,x, algorithm="maxima")

[Out]

-1/2520*(840*B*c^3*e^7*x^7 + 7*B*c^3*d^7 + 3*A*c^3*d^6*e + 6*B*a*c^2*d^5*e^2 + 6
*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 21*A*a^2*c*d^2*e^5 + 28*B*a^3*d*e^6 + 252
*A*a^3*e^7 + 210*(7*B*c^3*d*e^6 + 3*A*c^3*e^7)*x^6 + 252*(7*B*c^3*d^2*e^5 + 3*A*
c^3*d*e^6 + 6*B*a*c^2*e^7)*x^5 + 210*(7*B*c^3*d^3*e^4 + 3*A*c^3*d^2*e^5 + 6*B*a*
c^2*d*e^6 + 6*A*a*c^2*e^7)*x^4 + 120*(7*B*c^3*d^4*e^3 + 3*A*c^3*d^3*e^4 + 6*B*a*
c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 9*B*a^2*c*e^7)*x^3 + 45*(7*B*c^3*d^5*e^2 + 3*A*c
^3*d^4*e^3 + 6*B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5 + 9*B*a^2*c*d*e^6 + 21*A*a^2*
c*e^7)*x^2 + 10*(7*B*c^3*d^6*e + 3*A*c^3*d^5*e^2 + 6*B*a*c^2*d^4*e^3 + 6*A*a*c^2
*d^3*e^4 + 9*B*a^2*c*d^2*e^5 + 21*A*a^2*c*d*e^6 + 28*B*a^3*e^7)*x)/(e^18*x^10 +
10*d*e^17*x^9 + 45*d^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4*e^14*x^6 + 252*d^5*
e^13*x^5 + 210*d^6*e^12*x^4 + 120*d^7*e^11*x^3 + 45*d^8*e^10*x^2 + 10*d^9*e^9*x
+ d^10*e^8)

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Fricas [A]  time = 0.285175, size = 752, normalized size = 2.25 \[ -\frac{840 \, B c^{3} e^{7} x^{7} + 7 \, B c^{3} d^{7} + 3 \, A c^{3} d^{6} e + 6 \, B a c^{2} d^{5} e^{2} + 6 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} + 21 \, A a^{2} c d^{2} e^{5} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7} + 210 \,{\left (7 \, B c^{3} d e^{6} + 3 \, A c^{3} e^{7}\right )} x^{6} + 252 \,{\left (7 \, B c^{3} d^{2} e^{5} + 3 \, A c^{3} d e^{6} + 6 \, B a c^{2} e^{7}\right )} x^{5} + 210 \,{\left (7 \, B c^{3} d^{3} e^{4} + 3 \, A c^{3} d^{2} e^{5} + 6 \, B a c^{2} d e^{6} + 6 \, A a c^{2} e^{7}\right )} x^{4} + 120 \,{\left (7 \, B c^{3} d^{4} e^{3} + 3 \, A c^{3} d^{3} e^{4} + 6 \, B a c^{2} d^{2} e^{5} + 6 \, A a c^{2} d e^{6} + 9 \, B a^{2} c e^{7}\right )} x^{3} + 45 \,{\left (7 \, B c^{3} d^{5} e^{2} + 3 \, A c^{3} d^{4} e^{3} + 6 \, B a c^{2} d^{3} e^{4} + 6 \, A a c^{2} d^{2} e^{5} + 9 \, B a^{2} c d e^{6} + 21 \, A a^{2} c e^{7}\right )} x^{2} + 10 \,{\left (7 \, B c^{3} d^{6} e + 3 \, A c^{3} d^{5} e^{2} + 6 \, B a c^{2} d^{4} e^{3} + 6 \, A a c^{2} d^{3} e^{4} + 9 \, B a^{2} c d^{2} e^{5} + 21 \, A a^{2} c d e^{6} + 28 \, B a^{3} e^{7}\right )} x}{2520 \,{\left (e^{18} x^{10} + 10 \, d e^{17} x^{9} + 45 \, d^{2} e^{16} x^{8} + 120 \, d^{3} e^{15} x^{7} + 210 \, d^{4} e^{14} x^{6} + 252 \, d^{5} e^{13} x^{5} + 210 \, d^{6} e^{12} x^{4} + 120 \, d^{7} e^{11} x^{3} + 45 \, d^{8} e^{10} x^{2} + 10 \, d^{9} e^{9} x + d^{10} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^11,x, algorithm="fricas")

[Out]

-1/2520*(840*B*c^3*e^7*x^7 + 7*B*c^3*d^7 + 3*A*c^3*d^6*e + 6*B*a*c^2*d^5*e^2 + 6
*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 + 21*A*a^2*c*d^2*e^5 + 28*B*a^3*d*e^6 + 252
*A*a^3*e^7 + 210*(7*B*c^3*d*e^6 + 3*A*c^3*e^7)*x^6 + 252*(7*B*c^3*d^2*e^5 + 3*A*
c^3*d*e^6 + 6*B*a*c^2*e^7)*x^5 + 210*(7*B*c^3*d^3*e^4 + 3*A*c^3*d^2*e^5 + 6*B*a*
c^2*d*e^6 + 6*A*a*c^2*e^7)*x^4 + 120*(7*B*c^3*d^4*e^3 + 3*A*c^3*d^3*e^4 + 6*B*a*
c^2*d^2*e^5 + 6*A*a*c^2*d*e^6 + 9*B*a^2*c*e^7)*x^3 + 45*(7*B*c^3*d^5*e^2 + 3*A*c
^3*d^4*e^3 + 6*B*a*c^2*d^3*e^4 + 6*A*a*c^2*d^2*e^5 + 9*B*a^2*c*d*e^6 + 21*A*a^2*
c*e^7)*x^2 + 10*(7*B*c^3*d^6*e + 3*A*c^3*d^5*e^2 + 6*B*a*c^2*d^4*e^3 + 6*A*a*c^2
*d^3*e^4 + 9*B*a^2*c*d^2*e^5 + 21*A*a^2*c*d*e^6 + 28*B*a^3*e^7)*x)/(e^18*x^10 +
10*d*e^17*x^9 + 45*d^2*e^16*x^8 + 120*d^3*e^15*x^7 + 210*d^4*e^14*x^6 + 252*d^5*
e^13*x^5 + 210*d^6*e^12*x^4 + 120*d^7*e^11*x^3 + 45*d^8*e^10*x^2 + 10*d^9*e^9*x
+ d^10*e^8)

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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((B*x+A)*(c*x**2+a)**3/(e*x+d)**11,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.285312, size = 617, normalized size = 1.85 \[ -\frac{{\left (840 \, B c^{3} x^{7} e^{7} + 1470 \, B c^{3} d x^{6} e^{6} + 1764 \, B c^{3} d^{2} x^{5} e^{5} + 1470 \, B c^{3} d^{3} x^{4} e^{4} + 840 \, B c^{3} d^{4} x^{3} e^{3} + 315 \, B c^{3} d^{5} x^{2} e^{2} + 70 \, B c^{3} d^{6} x e + 7 \, B c^{3} d^{7} + 630 \, A c^{3} x^{6} e^{7} + 756 \, A c^{3} d x^{5} e^{6} + 630 \, A c^{3} d^{2} x^{4} e^{5} + 360 \, A c^{3} d^{3} x^{3} e^{4} + 135 \, A c^{3} d^{4} x^{2} e^{3} + 30 \, A c^{3} d^{5} x e^{2} + 3 \, A c^{3} d^{6} e + 1512 \, B a c^{2} x^{5} e^{7} + 1260 \, B a c^{2} d x^{4} e^{6} + 720 \, B a c^{2} d^{2} x^{3} e^{5} + 270 \, B a c^{2} d^{3} x^{2} e^{4} + 60 \, B a c^{2} d^{4} x e^{3} + 6 \, B a c^{2} d^{5} e^{2} + 1260 \, A a c^{2} x^{4} e^{7} + 720 \, A a c^{2} d x^{3} e^{6} + 270 \, A a c^{2} d^{2} x^{2} e^{5} + 60 \, A a c^{2} d^{3} x e^{4} + 6 \, A a c^{2} d^{4} e^{3} + 1080 \, B a^{2} c x^{3} e^{7} + 405 \, B a^{2} c d x^{2} e^{6} + 90 \, B a^{2} c d^{2} x e^{5} + 9 \, B a^{2} c d^{3} e^{4} + 945 \, A a^{2} c x^{2} e^{7} + 210 \, A a^{2} c d x e^{6} + 21 \, A a^{2} c d^{2} e^{5} + 280 \, B a^{3} x e^{7} + 28 \, B a^{3} d e^{6} + 252 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{2520 \,{\left (x e + d\right )}^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)/(e*x + d)^11,x, algorithm="giac")

[Out]

-1/2520*(840*B*c^3*x^7*e^7 + 1470*B*c^3*d*x^6*e^6 + 1764*B*c^3*d^2*x^5*e^5 + 147
0*B*c^3*d^3*x^4*e^4 + 840*B*c^3*d^4*x^3*e^3 + 315*B*c^3*d^5*x^2*e^2 + 70*B*c^3*d
^6*x*e + 7*B*c^3*d^7 + 630*A*c^3*x^6*e^7 + 756*A*c^3*d*x^5*e^6 + 630*A*c^3*d^2*x
^4*e^5 + 360*A*c^3*d^3*x^3*e^4 + 135*A*c^3*d^4*x^2*e^3 + 30*A*c^3*d^5*x*e^2 + 3*
A*c^3*d^6*e + 1512*B*a*c^2*x^5*e^7 + 1260*B*a*c^2*d*x^4*e^6 + 720*B*a*c^2*d^2*x^
3*e^5 + 270*B*a*c^2*d^3*x^2*e^4 + 60*B*a*c^2*d^4*x*e^3 + 6*B*a*c^2*d^5*e^2 + 126
0*A*a*c^2*x^4*e^7 + 720*A*a*c^2*d*x^3*e^6 + 270*A*a*c^2*d^2*x^2*e^5 + 60*A*a*c^2
*d^3*x*e^4 + 6*A*a*c^2*d^4*e^3 + 1080*B*a^2*c*x^3*e^7 + 405*B*a^2*c*d*x^2*e^6 +
90*B*a^2*c*d^2*x*e^5 + 9*B*a^2*c*d^3*e^4 + 945*A*a^2*c*x^2*e^7 + 210*A*a^2*c*d*x
*e^6 + 21*A*a^2*c*d^2*e^5 + 280*B*a^3*x*e^7 + 28*B*a^3*d*e^6 + 252*A*a^3*e^7)*e^
(-8)/(x*e + d)^10