3.879 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{x^{10}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.105163, 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} \[ -\frac{a^2 (a B+3 A b)}{8 x^8}-\frac{a^3 A}{9 x^9}-\frac{3 a A c^2+6 a b B c+3 A b^2 c+b^3 B}{5 x^5}-\frac{3 a \left (A \left (a c+b^2\right )+a b B\right )}{7 x^7}-\frac{3 c \left (a B c+A b c+b^2 B\right )}{4 x^4}-\frac{A \left (6 a b c+b^3\right )+3 a B \left (a c+b^2\right )}{6 x^6}-\frac{c^2 (A c+3 b B)}{3 x^3}-\frac{B c^3}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]

[Out]

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

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

Mathematica [A]  time = 0.0614397, size = 175, normalized size = 1.05 \[ -\frac{45 a^2 x (3 A (7 b+8 c x)+4 B x (6 b+7 c x))+35 a^3 (8 A+9 B x)+18 a x^2 \left (4 A \left (15 b^2+35 b c x+21 c^2 x^2\right )+7 B x \left (10 b^2+24 b c x+15 c^2 x^2\right )\right )+42 x^3 \left (A \left (36 b^2 c x+10 b^3+45 b c^2 x^2+20 c^3 x^3\right )+3 B x \left (15 b^2 c x+4 b^3+20 b c^2 x^2+10 c^3 x^3\right )\right )}{2520 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/x^10,x]

[Out]

-(35*a^3*(8*A + 9*B*x) + 45*a^2*x*(4*B*x*(6*b + 7*c*x) + 3*A*(7*b + 8*c*x)) + 18*a*x^2*(7*B*x*(10*b^2 + 24*b*c
*x + 15*c^2*x^2) + 4*A*(15*b^2 + 35*b*c*x + 21*c^2*x^2)) + 42*x^3*(3*B*x*(4*b^3 + 15*b^2*c*x + 20*b*c^2*x^2 +
10*c^3*x^3) + A*(10*b^3 + 36*b^2*c*x + 45*b*c^2*x^2 + 20*c^3*x^3)))/(2520*x^9)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.06706, size = 224, normalized size = 1.35 \begin{align*} -\frac{1260 \, B c^{3} x^{7} + 840 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1890 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 504 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \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^10,x, algorithm="maxima")

[Out]

-1/2520*(1260*B*c^3*x^7 + 840*(3*B*b*c^2 + A*c^3)*x^6 + 1890*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 504*(B*b^3 + 3*
A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 1080*
(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

________________________________________________________________________________________

Fricas [A]  time = 1.49143, size = 389, normalized size = 2.34 \begin{align*} -\frac{1260 \, B c^{3} x^{7} + 840 \,{\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 1890 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} x^{5} + 504 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 280 \, A a^{3} + 420 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 1080 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 315 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x}{2520 \, x^{9}} \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^10,x, algorithm="fricas")

[Out]

-1/2520*(1260*B*c^3*x^7 + 840*(3*B*b*c^2 + A*c^3)*x^6 + 1890*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 504*(B*b^3 + 3*
A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 + 280*A*a^3 + 420*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*x^3 + 1080*
(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 + 315*(B*a^3 + 3*A*a^2*b)*x)/x^9

________________________________________________________________________________________

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**10,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.25134, size = 258, normalized size = 1.55 \begin{align*} -\frac{1260 \, B c^{3} x^{7} + 2520 \, B b c^{2} x^{6} + 840 \, A c^{3} x^{6} + 1890 \, B b^{2} c x^{5} + 1890 \, B a c^{2} x^{5} + 1890 \, A b c^{2} x^{5} + 504 \, B b^{3} x^{4} + 3024 \, B a b c x^{4} + 1512 \, A b^{2} c x^{4} + 1512 \, A a c^{2} x^{4} + 1260 \, B a b^{2} x^{3} + 420 \, A b^{3} x^{3} + 1260 \, B a^{2} c x^{3} + 2520 \, A a b c x^{3} + 1080 \, B a^{2} b x^{2} + 1080 \, A a b^{2} x^{2} + 1080 \, A a^{2} c x^{2} + 315 \, B a^{3} x + 945 \, A a^{2} b x + 280 \, A a^{3}}{2520 \, x^{9}} \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^10,x, algorithm="giac")

[Out]

-1/2520*(1260*B*c^3*x^7 + 2520*B*b*c^2*x^6 + 840*A*c^3*x^6 + 1890*B*b^2*c*x^5 + 1890*B*a*c^2*x^5 + 1890*A*b*c^
2*x^5 + 504*B*b^3*x^4 + 3024*B*a*b*c*x^4 + 1512*A*b^2*c*x^4 + 1512*A*a*c^2*x^4 + 1260*B*a*b^2*x^3 + 420*A*b^3*
x^3 + 1260*B*a^2*c*x^3 + 2520*A*a*b*c*x^3 + 1080*B*a^2*b*x^2 + 1080*A*a*b^2*x^2 + 1080*A*a^2*c*x^2 + 315*B*a^3
*x + 945*A*a^2*b*x + 280*A*a^3)/x^9