3.878 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^9} \, dx\)

Optimal. Leaf size=162 \[ -\frac{a^3 A}{8 x^8}-\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac{c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac{c^2 (A c+3 b B)}{2 x^2}-\frac{B c^3}{x} \]

[Out]

-(a^3*A)/(8*x^8) - (a^2*(3*A*b + a*B))/(7*x^7) - (a*(a*b*B + A*(b^2 + a*c)))/(2*
x^6) - (3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))/(5*x^5) - (b^3*B + 3*A*b^2*c + 6*
a*b*B*c + 3*a*A*c^2)/(4*x^4) - (c*(b^2*B + A*b*c + a*B*c))/x^3 - (c^2*(3*b*B + A
*c))/(2*x^2) - (B*c^3)/x

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Rubi [A]  time = 0.28791, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ -\frac{a^3 A}{8 x^8}-\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac{c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac{c^2 (A c+3 b B)}{2 x^2}-\frac{B c^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^9,x]

[Out]

-(a^3*A)/(8*x^8) - (a^2*(3*A*b + a*B))/(7*x^7) - (a*(a*b*B + A*(b^2 + a*c)))/(2*
x^6) - (3*a*B*(b^2 + a*c) + A*(b^3 + 6*a*b*c))/(5*x^5) - (b^3*B + 3*A*b^2*c + 6*
a*b*B*c + 3*a*A*c^2)/(4*x^4) - (c*(b^2*B + A*b*c + a*B*c))/x^3 - (c^2*(3*b*B + A
*c))/(2*x^2) - (B*c^3)/x

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Rubi in Sympy [A]  time = 40.7503, size = 175, normalized size = 1.08 \[ - \frac{A a^{3}}{8 x^{8}} - \frac{B c^{3}}{x} - \frac{a^{2} \left (3 A b + B a\right )}{7 x^{7}} - \frac{a \left (A a c + A b^{2} + B a b\right )}{2 x^{6}} - \frac{c^{2} \left (A c + 3 B b\right )}{2 x^{2}} - \frac{c \left (A b c + B a c + B b^{2}\right )}{x^{3}} - \frac{\frac{3 A a c^{2}}{4} + \frac{3 A b^{2} c}{4} + \frac{3 B a b c}{2} + \frac{B b^{3}}{4}}{x^{4}} - \frac{\frac{6 A a b c}{5} + \frac{A b^{3}}{5} + \frac{3 B a^{2} c}{5} + \frac{3 B a b^{2}}{5}}{x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**3/x**9,x)

[Out]

-A*a**3/(8*x**8) - B*c**3/x - a**2*(3*A*b + B*a)/(7*x**7) - a*(A*a*c + A*b**2 +
B*a*b)/(2*x**6) - c**2*(A*c + 3*B*b)/(2*x**2) - c*(A*b*c + B*a*c + B*b**2)/x**3
- (3*A*a*c**2/4 + 3*A*b**2*c/4 + 3*B*a*b*c/2 + B*b**3/4)/x**4 - (6*A*a*b*c/5 + A
*b**3/5 + 3*B*a**2*c/5 + 3*B*a*b**2/5)/x**5

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Mathematica [A]  time = 0.127194, size = 172, normalized size = 1.06 \[ -\frac{5 a^3 (7 A+8 B x)+4 a^2 x (5 A (6 b+7 c x)+7 B x (5 b+6 c x))+14 a x^2 \left (A \left (10 b^2+24 b c x+15 c^2 x^2\right )+2 B x \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+14 x^3 \left (A \left (4 b^3+15 b^2 c x+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (b^3+4 b^2 c x+6 b c^2 x^2+4 c^3 x^3\right )\right )}{280 x^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^9,x]

[Out]

-(5*a^3*(7*A + 8*B*x) + 4*a^2*x*(7*B*x*(5*b + 6*c*x) + 5*A*(6*b + 7*c*x)) + 14*a
*x^2*(2*B*x*(6*b^2 + 15*b*c*x + 10*c^2*x^2) + A*(10*b^2 + 24*b*c*x + 15*c^2*x^2)
) + 14*x^3*(5*B*x*(b^3 + 4*b^2*c*x + 6*b*c^2*x^2 + 4*c^3*x^3) + A*(4*b^3 + 15*b^
2*c*x + 20*b*c^2*x^2 + 10*c^3*x^3)))/(280*x^8)

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Maple [A]  time = 0.009, size = 154, normalized size = 1. \[ -{\frac{a \left ( aAc+{b}^{2}A+abB \right ) }{2\,{x}^{6}}}-{\frac{3\,aA{c}^{2}+3\,A{b}^{2}c+6\,abBc+B{b}^{3}}{4\,{x}^{4}}}-{\frac{A{a}^{3}}{8\,{x}^{8}}}-{\frac{c \left ( Abc+aBc+{b}^{2}B \right ) }{{x}^{3}}}-{\frac{{c}^{2} \left ( Ac+3\,Bb \right ) }{2\,{x}^{2}}}-{\frac{6\,Aabc+A{b}^{3}+3\,B{a}^{2}c+3\,a{b}^{2}B}{5\,{x}^{5}}}-{\frac{B{c}^{3}}{x}}-{\frac{{a}^{2} \left ( 3\,Ab+Ba \right ) }{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^3/x^9,x)

[Out]

-1/2*a*(A*a*c+A*b^2+B*a*b)/x^6-1/4*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+B*b^3)/x^4-1/8
*a^3*A/x^8-c*(A*b*c+B*a*c+B*b^2)/x^3-1/2*c^2*(A*c+3*B*b)/x^2-1/5*(6*A*a*b*c+A*b^
3+3*B*a^2*c+3*B*a*b^2)/x^5-B*c^3/x-1/7*a^2*(3*A*b+B*a)/x^7

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Maxima [A]  time = 0.693203, size = 224, normalized size = 1.38 \[ -\frac{280 \, B c^{3} x^{7} + 140 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^9,x, algorithm="maxima")

[Out]

-1/280*(280*B*c^3*x^7 + 140*(3*B*b*c^2 + A*c^3)*x^6 + 280*(B*b^2*c + (B*a + A*b)
*c^2)*x^5 + 70*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 35*A*a^3 + 56*(
3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 140*(B*a^2*b + A*a*b^2 + A*a^2*
c)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^8

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Fricas [A]  time = 0.260259, size = 224, normalized size = 1.38 \[ -\frac{280 \, B c^{3} x^{7} + 140 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^9,x, algorithm="fricas")

[Out]

-1/280*(280*B*c^3*x^7 + 140*(3*B*b*c^2 + A*c^3)*x^6 + 280*(B*b^2*c + (B*a + A*b)
*c^2)*x^5 + 70*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 35*A*a^3 + 56*(
3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 140*(B*a^2*b + A*a*b^2 + A*a^2*
c)*x^2 + 40*(B*a^3 + 3*A*a^2*b)*x)/x^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+b*x+a)**3/x**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.268179, size = 258, normalized size = 1.59 \[ -\frac{280 \, B c^{3} x^{7} + 420 \, B b c^{2} x^{6} + 140 \, A c^{3} x^{6} + 280 \, B b^{2} c x^{5} + 280 \, B a c^{2} x^{5} + 280 \, A b c^{2} x^{5} + 70 \, B b^{3} x^{4} + 420 \, B a b c x^{4} + 210 \, A b^{2} c x^{4} + 210 \, A a c^{2} x^{4} + 168 \, B a b^{2} x^{3} + 56 \, A b^{3} x^{3} + 168 \, B a^{2} c x^{3} + 336 \, A a b c x^{3} + 140 \, B a^{2} b x^{2} + 140 \, A a b^{2} x^{2} + 140 \, A a^{2} c x^{2} + 40 \, B a^{3} x + 120 \, A a^{2} b x + 35 \, A a^{3}}{280 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^3*(B*x + A)/x^9,x, algorithm="giac")

[Out]

-1/280*(280*B*c^3*x^7 + 420*B*b*c^2*x^6 + 140*A*c^3*x^6 + 280*B*b^2*c*x^5 + 280*
B*a*c^2*x^5 + 280*A*b*c^2*x^5 + 70*B*b^3*x^4 + 420*B*a*b*c*x^4 + 210*A*b^2*c*x^4
+ 210*A*a*c^2*x^4 + 168*B*a*b^2*x^3 + 56*A*b^3*x^3 + 168*B*a^2*c*x^3 + 336*A*a*
b*c*x^3 + 140*B*a^2*b*x^2 + 140*A*a*b^2*x^2 + 140*A*a^2*c*x^2 + 40*B*a^3*x + 120
*A*a^2*b*x + 35*A*a^3)/x^8