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

Optimal. Leaf size=101 \[ -\frac{a^2 A}{8 x^8}-\frac{2 a B c+2 A b c+b^2 B}{5 x^5}-\frac{A \left (2 a c+b^2\right )+2 a b B}{6 x^6}-\frac{a (a B+2 A b)}{7 x^7}-\frac{c (A c+2 b B)}{4 x^4}-\frac{B c^2}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0523615, antiderivative size = 101, 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} \[ -\frac{a^2 A}{8 x^8}-\frac{2 a B c+2 A b c+b^2 B}{5 x^5}-\frac{A \left (2 a c+b^2\right )+2 a b B}{6 x^6}-\frac{a (a B+2 A b)}{7 x^7}-\frac{c (A c+2 b B)}{4 x^4}-\frac{B c^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0377546, size = 99, normalized size = 0.98 \[ -\frac{15 a^2 (7 A+8 B x)+8 a x (5 A (6 b+7 c x)+7 B x (5 b+6 c x))+14 x^2 \left (A \left (10 b^2+24 b c x+15 c^2 x^2\right )+2 B x \left (6 b^2+15 b c x+10 c^2 x^2\right )\right )}{840 x^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(15*a^2*(7*A + 8*B*x) + 8*a*x*(7*B*x*(5*b + 6*c*x) + 5*A*(6*b + 7*c*x)) + 14*x^2*(2*B*x*(6*b^2 + 15*b*c*x + 1
0*c^2*x^2) + A*(10*b^2 + 24*b*c*x + 15*c^2*x^2)))/(840*x^8)

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{B{c}^{2}}{3\,{x}^{3}}}-{\frac{A{a}^{2}}{8\,{x}^{8}}}-{\frac{a \left ( 2\,Ab+aB \right ) }{7\,{x}^{7}}}-{\frac{2\,Abc+2\,aBc+{b}^{2}B}{5\,{x}^{5}}}-{\frac{c \left ( Ac+2\,bB \right ) }{4\,{x}^{4}}}-{\frac{2\,aAc+A{b}^{2}+2\,abB}{6\,{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)^2/x^9,x)

[Out]

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

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Maxima [A]  time = 1.15238, size = 126, normalized size = 1.25 \begin{align*} -\frac{280 \, B c^{2} x^{5} + 210 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 168 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 105 \, A a^{2} + 140 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 120 \,{\left (B a^{2} + 2 \, A a b\right )} x}{840 \, 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)^2/x^9,x, algorithm="maxima")

[Out]

-1/840*(280*B*c^2*x^5 + 210*(2*B*b*c + A*c^2)*x^4 + 168*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 105*A*a^2 + 140*(2*B*a
*b + A*b^2 + 2*A*a*c)*x^2 + 120*(B*a^2 + 2*A*a*b)*x)/x^8

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Fricas [A]  time = 1.18294, size = 227, normalized size = 2.25 \begin{align*} -\frac{280 \, B c^{2} x^{5} + 210 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 168 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} x^{3} + 105 \, A a^{2} + 140 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 120 \,{\left (B a^{2} + 2 \, A a b\right )} x}{840 \, 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)^2/x^9,x, algorithm="fricas")

[Out]

-1/840*(280*B*c^2*x^5 + 210*(2*B*b*c + A*c^2)*x^4 + 168*(B*b^2 + 2*(B*a + A*b)*c)*x^3 + 105*A*a^2 + 140*(2*B*a
*b + A*b^2 + 2*A*a*c)*x^2 + 120*(B*a^2 + 2*A*a*b)*x)/x^8

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Sympy [A]  time = 48.5055, size = 102, normalized size = 1.01 \begin{align*} - \frac{105 A a^{2} + 280 B c^{2} x^{5} + x^{4} \left (210 A c^{2} + 420 B b c\right ) + x^{3} \left (336 A b c + 336 B a c + 168 B b^{2}\right ) + x^{2} \left (280 A a c + 140 A b^{2} + 280 B a b\right ) + x \left (240 A a b + 120 B a^{2}\right )}{840 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)**2/x**9,x)

[Out]

-(105*A*a**2 + 280*B*c**2*x**5 + x**4*(210*A*c**2 + 420*B*b*c) + x**3*(336*A*b*c + 336*B*a*c + 168*B*b**2) + x
**2*(280*A*a*c + 140*A*b**2 + 280*B*a*b) + x*(240*A*a*b + 120*B*a**2))/(840*x**8)

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Giac [A]  time = 1.26743, size = 136, normalized size = 1.35 \begin{align*} -\frac{280 \, B c^{2} x^{5} + 420 \, B b c x^{4} + 210 \, A c^{2} x^{4} + 168 \, B b^{2} x^{3} + 336 \, B a c x^{3} + 336 \, A b c x^{3} + 280 \, B a b x^{2} + 140 \, A b^{2} x^{2} + 280 \, A a c x^{2} + 120 \, B a^{2} x + 240 \, A a b x + 105 \, A a^{2}}{840 \, 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)^2/x^9,x, algorithm="giac")

[Out]

-1/840*(280*B*c^2*x^5 + 420*B*b*c*x^4 + 210*A*c^2*x^4 + 168*B*b^2*x^3 + 336*B*a*c*x^3 + 336*A*b*c*x^3 + 280*B*
a*b*x^2 + 140*A*b^2*x^2 + 280*A*a*c*x^2 + 120*B*a^2*x + 240*A*a*b*x + 105*A*a^2)/x^8