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

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.104902, 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, Rules used = {765} \[ -\frac{a^2 (a B+3 A b)}{7 x^7}-\frac{a^3 A}{8 x^8}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{4 x^4}-\frac{a \left (A \left (a c+b^2\right )+a b B\right )}{2 x^6}-\frac{c \left (a B c+A b c+b^2 B\right )}{x^3}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{5 x^5}-\frac{c^2 (A c+3 b B)}{2 x^2}-\frac{B c^3}{x} \]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{(A+B x) \left (a+b x+c x^2\right )^3}{x^9} \, dx &=\int \left (\frac{a^3 A}{x^9}+\frac{a^2 (3 A b+a B)}{x^8}+\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^7}+\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^6}+\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^5}+\frac{3 c \left (b^2 B+A b c+a B c\right )}{x^4}+\frac{c^2 (3 b B+A c)}{x^3}+\frac{B c^3}{x^2}\right ) \, dx\\ &=-\frac{a^3 A}{8 x^8}-\frac{a^2 (3 A b+a B)}{7 x^7}-\frac{a \left (a b B+A \left (b^2+a c\right )\right )}{2 x^6}-\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{5 x^5}-\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{4 x^4}-\frac{c \left (b^2 B+A b c+a B c\right )}{x^3}-\frac{c^2 (3 b B+A c)}{2 x^2}-\frac{B c^3}{x}\\ \end{align*}

Mathematica [A]  time = 0.0607239, size = 172, normalized size = 1.06 \[ -\frac{4 a^2 x (5 A (6 b+7 c x)+7 B x (5 b+6 c x))+5 a^3 (7 A+8 B x)+14 a x^2 \left (A \left (10 b^2+24 b c x+15 c^2 x^2\right )+2 B x \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )+14 x^3 \left (A \left (15 b^2 c x+4 b^3+20 b c^2 x^2+10 c^3 x^3\right )+5 B x \left (4 b^2 c x+b^3+6 b c^2 x^2+4 c^3 x^3\right )\right )}{280 x^8} \]

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)

________________________________________________________________________________________

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

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]

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

________________________________________________________________________________________

Maxima [A]  time = 1.12904, size = 224, normalized size = 1.38 \begin{align*} -\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}} \end{align*}

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, 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

________________________________________________________________________________________

Fricas [A]  time = 1.45746, size = 378, normalized size = 2.33 \begin{align*} -\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}} \end{align*}

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, 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

________________________________________________________________________________________

Sympy [A]  time = 171.838, size = 187, normalized size = 1.15 \begin{align*} - \frac{35 A a^{3} + 280 B c^{3} x^{7} + x^{6} \left (140 A c^{3} + 420 B b c^{2}\right ) + x^{5} \left (280 A b c^{2} + 280 B a c^{2} + 280 B b^{2} c\right ) + x^{4} \left (210 A a c^{2} + 210 A b^{2} c + 420 B a b c + 70 B b^{3}\right ) + x^{3} \left (336 A a b c + 56 A b^{3} + 168 B a^{2} c + 168 B a b^{2}\right ) + x^{2} \left (140 A a^{2} c + 140 A a b^{2} + 140 B a^{2} b\right ) + x \left (120 A a^{2} b + 40 B a^{3}\right )}{280 x^{8}} \end{align*}

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]

-(35*A*a**3 + 280*B*c**3*x**7 + x**6*(140*A*c**3 + 420*B*b*c**2) + x**5*(280*A*b*c**2 + 280*B*a*c**2 + 280*B*b
**2*c) + x**4*(210*A*a*c**2 + 210*A*b**2*c + 420*B*a*b*c + 70*B*b**3) + x**3*(336*A*a*b*c + 56*A*b**3 + 168*B*
a**2*c + 168*B*a*b**2) + x**2*(140*A*a**2*c + 140*A*a*b**2 + 140*B*a**2*b) + x*(120*A*a**2*b + 40*B*a**3))/(28
0*x**8)

________________________________________________________________________________________

Giac [A]  time = 1.2879, size = 258, normalized size = 1.59 \begin{align*} -\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}} \end{align*}

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, 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