\(\int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx\) [134]

Optimal result

Integrand size = 16, antiderivative size = 115 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {a^5 A}{9 x^9}-\frac {a^4 (5 A b+a B)}{8 x^8}-\frac {5 a^3 b (2 A b+a B)}{7 x^7}-\frac {5 a^2 b^2 (A b+a B)}{3 x^6}-\frac {a b^3 (A b+2 a B)}{x^5}-\frac {b^4 (A b+5 a B)}{4 x^4}-\frac {b^5 B}{3 x^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {77} \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {a^5 A}{9 x^9}-\frac {a^4 (a B+5 A b)}{8 x^8}-\frac {5 a^3 b (a B+2 A b)}{7 x^7}-\frac {5 a^2 b^2 (a B+A b)}{3 x^6}-\frac {b^4 (5 a B+A b)}{4 x^4}-\frac {a b^3 (2 a B+A b)}{x^5}-\frac {b^5 B}{3 x^3} \]

[In]

Int[((a + b*x)^5*(A + B*x))/x^10,x]

[Out]

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

Rule 77

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {42 b^5 x^5 (3 A+4 B x)+126 a b^4 x^4 (4 A+5 B x)+168 a^2 b^3 x^3 (5 A+6 B x)+120 a^3 b^2 x^2 (6 A+7 B x)+45 a^4 b x (7 A+8 B x)+7 a^5 (8 A+9 B x)}{504 x^9} \]

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^10,x]

[Out]

-1/504*(42*b^5*x^5*(3*A + 4*B*x) + 126*a*b^4*x^4*(4*A + 5*B*x) + 168*a^2*b^3*x^3*(5*A + 6*B*x) + 120*a^3*b^2*x
^2*(6*A + 7*B*x) + 45*a^4*b*x*(7*A + 8*B*x) + 7*a^5*(8*A + 9*B*x))/x^9

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.90

method	result	size
default	\(-\frac {a^{5} A}{9 x^{9}}-\frac {a^{4} \left (5 A b +B a \right )}{8 x^{8}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{7 x^{7}}-\frac {5 a^{2} b^{2} \left (A b +B a \right )}{3 x^{6}}-\frac {a \,b^{3} \left (A b +2 B a \right )}{x^{5}}-\frac {b^{4} \left (A b +5 B a \right )}{4 x^{4}}-\frac {b^{5} B}{3 x^{3}}\)	\(104\)
norman	\(\frac {-\frac {b^{5} B \,x^{6}}{3}+\left (-\frac {1}{4} b^{5} A -\frac {5}{4} a \,b^{4} B \right ) x^{5}+\left (-a \,b^{4} A -2 a^{2} b^{3} B \right ) x^{4}+\left (-\frac {5}{3} a^{2} b^{3} A -\frac {5}{3} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {10}{7} a^{3} b^{2} A -\frac {5}{7} a^{4} b B \right ) x^{2}+\left (-\frac {5}{8} a^{4} b A -\frac {1}{8} a^{5} B \right ) x -\frac {a^{5} A}{9}}{x^{9}}\)	\(120\)
risch	\(\frac {-\frac {b^{5} B \,x^{6}}{3}+\left (-\frac {1}{4} b^{5} A -\frac {5}{4} a \,b^{4} B \right ) x^{5}+\left (-a \,b^{4} A -2 a^{2} b^{3} B \right ) x^{4}+\left (-\frac {5}{3} a^{2} b^{3} A -\frac {5}{3} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {10}{7} a^{3} b^{2} A -\frac {5}{7} a^{4} b B \right ) x^{2}+\left (-\frac {5}{8} a^{4} b A -\frac {1}{8} a^{5} B \right ) x -\frac {a^{5} A}{9}}{x^{9}}\)	\(120\)
gosper	\(-\frac {168 b^{5} B \,x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 a A \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 a^{2} A \,b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 a^{3} A \,b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 a^{4} A b x +63 a^{5} B x +56 a^{5} A}{504 x^{9}}\)	\(124\)
parallelrisch	\(-\frac {168 b^{5} B \,x^{6}+126 A \,b^{5} x^{5}+630 B a \,b^{4} x^{5}+504 a A \,b^{4} x^{4}+1008 B \,a^{2} b^{3} x^{4}+840 a^{2} A \,b^{3} x^{3}+840 B \,a^{3} b^{2} x^{3}+720 a^{3} A \,b^{2} x^{2}+360 B \,a^{4} b \,x^{2}+315 a^{4} A b x +63 a^{5} B x +56 a^{5} A}{504 x^{9}}\)	\(124\)

[In]

int((b*x+a)^5*(B*x+A)/x^10,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^10,x, algorithm="fricas")

[Out]

-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 840*(B*a^3*
b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9

Sympy [A] (verification not implemented)

Time = 4.00 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=\frac {- 56 A a^{5} - 168 B b^{5} x^{6} + x^{5} \left (- 126 A b^{5} - 630 B a b^{4}\right ) + x^{4} \left (- 504 A a b^{4} - 1008 B a^{2} b^{3}\right ) + x^{3} \left (- 840 A a^{2} b^{3} - 840 B a^{3} b^{2}\right ) + x^{2} \left (- 720 A a^{3} b^{2} - 360 B a^{4} b\right ) + x \left (- 315 A a^{4} b - 63 B a^{5}\right )}{504 x^{9}} \]

[In]

integrate((b*x+a)**5*(B*x+A)/x**10,x)

[Out]

(-56*A*a**5 - 168*B*b**5*x**6 + x**5*(-126*A*b**5 - 630*B*a*b**4) + x**4*(-504*A*a*b**4 - 1008*B*a**2*b**3) +
x**3*(-840*A*a**2*b**3 - 840*B*a**3*b**2) + x**2*(-720*A*a**3*b**2 - 360*B*a**4*b) + x*(-315*A*a**4*b - 63*B*a
**5))/(504*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {168 \, B b^{5} x^{6} + 56 \, A a^{5} + 126 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 504 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 840 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 360 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 63 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{504 \, x^{9}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^10,x, algorithm="maxima")

[Out]

-1/504*(168*B*b^5*x^6 + 56*A*a^5 + 126*(5*B*a*b^4 + A*b^5)*x^5 + 504*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 840*(B*a^3*
b^2 + A*a^2*b^3)*x^3 + 360*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 63*(B*a^5 + 5*A*a^4*b)*x)/x^9

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {168 \, B b^{5} x^{6} + 630 \, B a b^{4} x^{5} + 126 \, A b^{5} x^{5} + 1008 \, B a^{2} b^{3} x^{4} + 504 \, A a b^{4} x^{4} + 840 \, B a^{3} b^{2} x^{3} + 840 \, A a^{2} b^{3} x^{3} + 360 \, B a^{4} b x^{2} + 720 \, A a^{3} b^{2} x^{2} + 63 \, B a^{5} x + 315 \, A a^{4} b x + 56 \, A a^{5}}{504 \, x^{9}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^10,x, algorithm="giac")

[Out]

-1/504*(168*B*b^5*x^6 + 630*B*a*b^4*x^5 + 126*A*b^5*x^5 + 1008*B*a^2*b^3*x^4 + 504*A*a*b^4*x^4 + 840*B*a^3*b^2
*x^3 + 840*A*a^2*b^3*x^3 + 360*B*a^4*b*x^2 + 720*A*a^3*b^2*x^2 + 63*B*a^5*x + 315*A*a^4*b*x + 56*A*a^5)/x^9

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{10}} \, dx=-\frac {x\,\left (\frac {B\,a^5}{8}+\frac {5\,A\,b\,a^4}{8}\right )+\frac {A\,a^5}{9}+x^4\,\left (2\,B\,a^2\,b^3+A\,a\,b^4\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{7}+\frac {10\,A\,a^3\,b^2}{7}\right )+x^5\,\left (\frac {A\,b^5}{4}+\frac {5\,B\,a\,b^4}{4}\right )+x^3\,\left (\frac {5\,B\,a^3\,b^2}{3}+\frac {5\,A\,a^2\,b^3}{3}\right )+\frac {B\,b^5\,x^6}{3}}{x^9} \]

[In]

int(((A + B*x)*(a + b*x)^5)/x^10,x)

[Out]

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