\(\int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx\) [133]

Optimal result

Integrand size = 16, antiderivative size = 70 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {A (a+b x)^6}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^6}{28 a^2 x^7}-\frac {b (A b-4 a B) (a+b x)^6}{168 a^3 x^6} \]

[Out]

-1/8*A*(b*x+a)^6/a/x^8+1/28*(A*b-4*B*a)*(b*x+a)^6/a^2/x^7-1/168*b*(A*b-4*B*a)*(b*x+a)^6/a^3/x^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {79, 47, 37} \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {b (a+b x)^6 (A b-4 a B)}{168 a^3 x^6}+\frac {(a+b x)^6 (A b-4 a B)}{28 a^2 x^7}-\frac {A (a+b x)^6}{8 a x^8} \]

[In]

Int[((a + b*x)^5*(A + B*x))/x^9,x]

[Out]

-1/8*(A*(a + b*x)^6)/(a*x^8) + ((A*b - 4*a*B)*(a + b*x)^6)/(28*a^2*x^7) - (b*(A*b - 4*a*B)*(a + b*x)^6)/(168*a
^3*x^6)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^6}{8 a x^8}+\frac {(-2 A b+8 a B) \int \frac {(a+b x)^5}{x^8} \, dx}{8 a} \\ & = -\frac {A (a+b x)^6}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^6}{28 a^2 x^7}+\frac {(b (A b-4 a B)) \int \frac {(a+b x)^5}{x^7} \, dx}{28 a^2} \\ & = -\frac {A (a+b x)^6}{8 a x^8}+\frac {(A b-4 a B) (a+b x)^6}{28 a^2 x^7}-\frac {b (A b-4 a B) (a+b x)^6}{168 a^3 x^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {28 b^5 x^5 (2 A+3 B x)+70 a b^4 x^4 (3 A+4 B x)+84 a^2 b^3 x^3 (4 A+5 B x)+56 a^3 b^2 x^2 (5 A+6 B x)+20 a^4 b x (6 A+7 B x)+3 a^5 (7 A+8 B x)}{168 x^8} \]

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^9,x]

[Out]

-1/168*(28*b^5*x^5*(2*A + 3*B*x) + 70*a*b^4*x^4*(3*A + 4*B*x) + 84*a^2*b^3*x^3*(4*A + 5*B*x) + 56*a^3*b^2*x^2*
(5*A + 6*B*x) + 20*a^4*b*x*(6*A + 7*B*x) + 3*a^5*(7*A + 8*B*x))/x^8

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.49

method	result	size
default	\(-\frac {5 a^{3} b \left (2 A b +B a \right )}{6 x^{6}}-\frac {a^{4} \left (5 A b +B a \right )}{7 x^{7}}-\frac {a^{5} A}{8 x^{8}}-\frac {b^{4} \left (A b +5 B a \right )}{3 x^{3}}-\frac {b^{5} B}{2 x^{2}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{4 x^{4}}-\frac {2 a^{2} b^{2} \left (A b +B a \right )}{x^{5}}\)	\(104\)
norman	\(\frac {-\frac {b^{5} B \,x^{6}}{2}+\left (-\frac {1}{3} b^{5} A -\frac {5}{3} a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{4} a \,b^{4} A -\frac {5}{2} a^{2} b^{3} B \right ) x^{4}+\left (-2 a^{2} b^{3} A -2 a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{3} a^{3} b^{2} A -\frac {5}{6} a^{4} b B \right ) x^{2}+\left (-\frac {5}{7} a^{4} b A -\frac {1}{7} a^{5} B \right ) x -\frac {a^{5} A}{8}}{x^{8}}\)	\(120\)
risch	\(\frac {-\frac {b^{5} B \,x^{6}}{2}+\left (-\frac {1}{3} b^{5} A -\frac {5}{3} a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{4} a \,b^{4} A -\frac {5}{2} a^{2} b^{3} B \right ) x^{4}+\left (-2 a^{2} b^{3} A -2 a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{3} a^{3} b^{2} A -\frac {5}{6} a^{4} b B \right ) x^{2}+\left (-\frac {5}{7} a^{4} b A -\frac {1}{7} a^{5} B \right ) x -\frac {a^{5} A}{8}}{x^{8}}\)	\(120\)
gosper	\(-\frac {84 b^{5} B \,x^{6}+56 A \,b^{5} x^{5}+280 B a \,b^{4} x^{5}+210 a A \,b^{4} x^{4}+420 B \,a^{2} b^{3} x^{4}+336 a^{2} A \,b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+280 a^{3} A \,b^{2} x^{2}+140 B \,a^{4} b \,x^{2}+120 a^{4} A b x +24 a^{5} B x +21 a^{5} A}{168 x^{8}}\)	\(124\)
parallelrisch	\(-\frac {84 b^{5} B \,x^{6}+56 A \,b^{5} x^{5}+280 B a \,b^{4} x^{5}+210 a A \,b^{4} x^{4}+420 B \,a^{2} b^{3} x^{4}+336 a^{2} A \,b^{3} x^{3}+336 B \,a^{3} b^{2} x^{3}+280 a^{3} A \,b^{2} x^{2}+140 B \,a^{4} b \,x^{2}+120 a^{4} A b x +24 a^{5} B x +21 a^{5} A}{168 x^{8}}\)	\(124\)

[In]

int((b*x+a)^5*(B*x+A)/x^9,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.70 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^9,x, algorithm="fricas")

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 336*(B*a^3*b^
2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (63) = 126\).

Time = 2.73 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.90 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=\frac {- 21 A a^{5} - 84 B b^{5} x^{6} + x^{5} \left (- 56 A b^{5} - 280 B a b^{4}\right ) + x^{4} \left (- 210 A a b^{4} - 420 B a^{2} b^{3}\right ) + x^{3} \left (- 336 A a^{2} b^{3} - 336 B a^{3} b^{2}\right ) + x^{2} \left (- 280 A a^{3} b^{2} - 140 B a^{4} b\right ) + x \left (- 120 A a^{4} b - 24 B a^{5}\right )}{168 x^{8}} \]

[In]

integrate((b*x+a)**5*(B*x+A)/x**9,x)

[Out]

(-21*A*a**5 - 84*B*b**5*x**6 + x**5*(-56*A*b**5 - 280*B*a*b**4) + x**4*(-210*A*a*b**4 - 420*B*a**2*b**3) + x**
3*(-336*A*a**2*b**3 - 336*B*a**3*b**2) + x**2*(-280*A*a**3*b**2 - 140*B*a**4*b) + x*(-120*A*a**4*b - 24*B*a**5
))/(168*x**8)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.70 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {84 \, B b^{5} x^{6} + 21 \, A a^{5} + 56 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 210 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 336 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 140 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 24 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{168 \, x^{8}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^9,x, algorithm="maxima")

[Out]

-1/168*(84*B*b^5*x^6 + 21*A*a^5 + 56*(5*B*a*b^4 + A*b^5)*x^5 + 210*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 336*(B*a^3*b^
2 + A*a^2*b^3)*x^3 + 140*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 24*(B*a^5 + 5*A*a^4*b)*x)/x^8

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.76 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {84 \, B b^{5} x^{6} + 280 \, B a b^{4} x^{5} + 56 \, A b^{5} x^{5} + 420 \, B a^{2} b^{3} x^{4} + 210 \, A a b^{4} x^{4} + 336 \, B a^{3} b^{2} x^{3} + 336 \, A a^{2} b^{3} x^{3} + 140 \, B a^{4} b x^{2} + 280 \, A a^{3} b^{2} x^{2} + 24 \, B a^{5} x + 120 \, A a^{4} b x + 21 \, A a^{5}}{168 \, x^{8}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^9,x, algorithm="giac")

[Out]

-1/168*(84*B*b^5*x^6 + 280*B*a*b^4*x^5 + 56*A*b^5*x^5 + 420*B*a^2*b^3*x^4 + 210*A*a*b^4*x^4 + 336*B*a^3*b^2*x^
3 + 336*A*a^2*b^3*x^3 + 140*B*a^4*b*x^2 + 280*A*a^3*b^2*x^2 + 24*B*a^5*x + 120*A*a^4*b*x + 21*A*a^5)/x^8

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.71 \[ \int \frac {(a+b x)^5 (A+B x)}{x^9} \, dx=-\frac {x\,\left (\frac {B\,a^5}{7}+\frac {5\,A\,b\,a^4}{7}\right )+\frac {A\,a^5}{8}+x^4\,\left (\frac {5\,B\,a^2\,b^3}{2}+\frac {5\,A\,a\,b^4}{4}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{6}+\frac {5\,A\,a^3\,b^2}{3}\right )+x^5\,\left (\frac {A\,b^5}{3}+\frac {5\,B\,a\,b^4}{3}\right )+x^3\,\left (2\,B\,a^3\,b^2+2\,A\,a^2\,b^3\right )+\frac {B\,b^5\,x^6}{2}}{x^8} \]

[In]

int(((A + B*x)*(a + b*x)^5)/x^9,x)

[Out]

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