3.880 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^{11}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{a^2 (a B+3 A b)}{9 x^9}-\frac{a^3 A}{10 x^{10}}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{6 x^6}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{8 x^8}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{5 x^5}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{7 x^7}-\frac{c^2 (A c+3 b B)}{4 x^4}-\frac{B c^3}{3 x^3} \]

[Out]

-(a^3*A)/(10*x^10) - (a^2*(3*A*b + a*B))/(9*x^9) - (3*a*(a*b*B + A*(b^2 + a*c)))/(8*x^8) - (3*a*B*(b^2 + a*c)
+ A*(b^3 + 6*a*b*c))/(7*x^7) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/(6*x^6) - (3*c*(b^2*B + A*b*c + a*B
*c))/(5*x^5) - (c^2*(3*b*B + A*c))/(4*x^4) - (B*c^3)/(3*x^3)

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Rubi [A]  time = 0.103633, antiderivative size = 166, 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} \[ -\frac{a^2 (a B+3 A b)}{9 x^9}-\frac{a^3 A}{10 x^{10}}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{6 x^6}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{8 x^8}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{5 x^5}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{7 x^7}-\frac{c^2 (A c+3 b B)}{4 x^4}-\frac{B c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*A)/(10*x^10) - (a^2*(3*A*b + a*B))/(9*x^9) - (3*a*(a*b*B + A*(b^2 + a*c)))/(8*x^8) - (3*a*B*(b^2 + a*c)
+ A*(b^3 + 6*a*b*c))/(7*x^7) - (b^3*B + 3*A*b^2*c + 6*a*b*B*c + 3*a*A*c^2)/(6*x^6) - (3*c*(b^2*B + A*b*c + a*B
*c))/(5*x^5) - (c^2*(3*b*B + A*c))/(4*x^4) - (B*c^3)/(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 )^3}{x^{11}} \, dx &=\int \left (\frac{a^3 A}{x^{11}}+\frac{a^2 (3 A b+a B)}{x^{10}}+\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{x^9}+\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{x^8}+\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{x^7}+\frac{3 c \left (b^2 B+A b c+a B c\right )}{x^6}+\frac{c^2 (3 b B+A c)}{x^5}+\frac{B c^3}{x^4}\right ) \, dx\\ &=-\frac{a^3 A}{10 x^{10}}-\frac{a^2 (3 A b+a B)}{9 x^9}-\frac{3 a \left (a b B+A \left (b^2+a c\right )\right )}{8 x^8}-\frac{3 a B \left (b^2+a c\right )+A \left (b^3+6 a b c\right )}{7 x^7}-\frac{b^3 B+3 A b^2 c+6 a b B c+3 a A c^2}{6 x^6}-\frac{3 c \left (b^2 B+A b c+a B c\right )}{5 x^5}-\frac{c^2 (3 b B+A c)}{4 x^4}-\frac{B c^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0637301, size = 176, normalized size = 1.06 \[ -\frac{15 a^2 x (7 A (8 b+9 c x)+9 B x (7 b+8 c x))+28 a^3 (9 A+10 B x)+9 a x^2 \left (5 A \left (21 b^2+48 b c x+28 c^2 x^2\right )+8 B x \left (15 b^2+35 b c x+21 c^2 x^2\right )\right )+6 x^3 \left (3 A \left (70 b^2 c x+20 b^3+84 b c^2 x^2+35 c^3 x^3\right )+7 B x \left (36 b^2 c x+10 b^3+45 b c^2 x^2+20 c^3 x^3\right )\right )}{2520 x^{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(28*a^3*(9*A + 10*B*x) + 15*a^2*x*(9*B*x*(7*b + 8*c*x) + 7*A*(8*b + 9*c*x)) + 9*a*x^2*(8*B*x*(15*b^2 + 35*b*c
*x + 21*c^2*x^2) + 5*A*(21*b^2 + 48*b*c*x + 28*c^2*x^2)) + 6*x^3*(7*B*x*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 +
20*c^3*x^3) + 3*A*(20*b^3 + 70*b^2*c*x + 84*b*c^2*x^2 + 35*c^3*x^3)))/(2520*x^10)

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Maple [A]  time = 0.007, size = 154, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2} \left ( 3\,Ab+aB \right ) }{9\,{x}^{9}}}-{\frac{B{c}^{3}}{3\,{x}^{3}}}-{\frac{3\,a \left ( aAc+A{b}^{2}+abB \right ) }{8\,{x}^{8}}}-{\frac{6\,Aabc+A{b}^{3}+3\,B{a}^{2}c+3\,Ba{b}^{2}}{7\,{x}^{7}}}-{\frac{3\,c \left ( Abc+aBc+{b}^{2}B \right ) }{5\,{x}^{5}}}-{\frac{A{a}^{3}}{10\,{x}^{10}}}-{\frac{{c}^{2} \left ( Ac+3\,bB \right ) }{4\,{x}^{4}}}-{\frac{3\,aA{c}^{2}+3\,A{b}^{2}c+6\,abBc+{b}^{3}B}{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)^3/x^11,x)

[Out]

-1/9*a^2*(3*A*b+B*a)/x^9-1/3*B*c^3/x^3-3/8*a*(A*a*c+A*b^2+B*a*b)/x^8-1/7*(6*A*a*b*c+A*b^3+3*B*a^2*c+3*B*a*b^2)
/x^7-3/5*c*(A*b*c+B*a*c+B*b^2)/x^5-1/10*a^3*A/x^10-1/4*c^2*(A*c+3*B*b)/x^4-1/6*(3*A*a*c^2+3*A*b^2*c+6*B*a*b*c+
B*b^3)/x^6

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Maxima [A]  time = 1.08073, size = 224, normalized size = 1.35 \begin{align*} -\frac{840 \, B c^{3} x^{7} + 630 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1512 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 420 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 252 \, A a^{3} + 360 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 945 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 280 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \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^11,x, algorithm="maxima")

[Out]

-1/2520*(840*B*c^3*x^7 + 630*(3*B*b*c^2 + A*c^3)*x^6 + 1512*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 420*(B*b^3 + 3*A
*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 252*A*a^3 + 360*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 945*(B
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10

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Fricas [A]  time = 1.42767, size = 387, normalized size = 2.33 \begin{align*} -\frac{840 \, B c^{3} x^{7} + 630 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1512 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 420 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 252 \, A a^{3} + 360 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 945 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 280 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{10}} \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^11,x, algorithm="fricas")

[Out]

-1/2520*(840*B*c^3*x^7 + 630*(3*B*b*c^2 + A*c^3)*x^6 + 1512*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 420*(B*b^3 + 3*A
*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 252*A*a^3 + 360*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 945*(B
*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 280*(B*a^3 + 3*A*a^2*b)*x)/x^10

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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**11,x)

[Out]

Timed out

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Giac [A]  time = 1.21754, size = 258, normalized size = 1.55 \begin{align*} -\frac{840 \, B c^{3} x^{7} + 1890 \, B b c^{2} x^{6} + 630 \, A c^{3} x^{6} + 1512 \, B b^{2} c x^{5} + 1512 \, B a c^{2} x^{5} + 1512 \, A b c^{2} x^{5} + 420 \, B b^{3} x^{4} + 2520 \, B a b c x^{4} + 1260 \, A b^{2} c x^{4} + 1260 \, A a c^{2} x^{4} + 1080 \, B a b^{2} x^{3} + 360 \, A b^{3} x^{3} + 1080 \, B a^{2} c x^{3} + 2160 \, A a b c x^{3} + 945 \, B a^{2} b x^{2} + 945 \, A a b^{2} x^{2} + 945 \, A a^{2} c x^{2} + 280 \, B a^{3} x + 840 \, A a^{2} b x + 252 \, A a^{3}}{2520 \, x^{10}} \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^11,x, algorithm="giac")

[Out]

-1/2520*(840*B*c^3*x^7 + 1890*B*b*c^2*x^6 + 630*A*c^3*x^6 + 1512*B*b^2*c*x^5 + 1512*B*a*c^2*x^5 + 1512*A*b*c^2
*x^5 + 420*B*b^3*x^4 + 2520*B*a*b*c*x^4 + 1260*A*b^2*c*x^4 + 1260*A*a*c^2*x^4 + 1080*B*a*b^2*x^3 + 360*A*b^3*x
^3 + 1080*B*a^2*c*x^3 + 2160*A*a*b*c*x^3 + 945*B*a^2*b*x^2 + 945*A*a*b^2*x^2 + 945*A*a^2*c*x^2 + 280*B*a^3*x +
840*A*a^2*b*x + 252*A*a^3)/x^10