\(\int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx\) [135]

Optimal result

Integrand size = 16, antiderivative size = 117 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {a^5 A}{10 x^{10}}-\frac {a^4 (5 A b+a B)}{9 x^9}-\frac {5 a^3 b (2 A b+a B)}{8 x^8}-\frac {10 a^2 b^2 (A b+a B)}{7 x^7}-\frac {5 a b^3 (A b+2 a B)}{6 x^6}-\frac {b^4 (A b+5 a B)}{5 x^5}-\frac {b^5 B}{4 x^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 117, 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} \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {a^5 A}{10 x^{10}}-\frac {a^4 (a B+5 A b)}{9 x^9}-\frac {5 a^3 b (a B+2 A b)}{8 x^8}-\frac {10 a^2 b^2 (a B+A b)}{7 x^7}-\frac {b^4 (5 a B+A b)}{5 x^5}-\frac {5 a b^3 (2 a B+A b)}{6 x^6}-\frac {b^5 B}{4 x^4} \]

[In]

Int[((a + b*x)^5*(A + B*x))/x^11,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {126 b^5 x^5 (4 A+5 B x)+420 a b^4 x^4 (5 A+6 B x)+600 a^2 b^3 x^3 (6 A+7 B x)+450 a^3 b^2 x^2 (7 A+8 B x)+175 a^4 b x (8 A+9 B x)+28 a^5 (9 A+10 B x)}{2520 x^{10}} \]

[In]

Integrate[((a + b*x)^5*(A + B*x))/x^11,x]

[Out]

-1/2520*(126*b^5*x^5*(4*A + 5*B*x) + 420*a*b^4*x^4*(5*A + 6*B*x) + 600*a^2*b^3*x^3*(6*A + 7*B*x) + 450*a^3*b^2
*x^2*(7*A + 8*B*x) + 175*a^4*b*x*(8*A + 9*B*x) + 28*a^5*(9*A + 10*B*x))/x^10

Maple [A] (verified)

Time = 0.39 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.89

method	result	size
default	\(-\frac {a^{5} A}{10 x^{10}}-\frac {a^{4} \left (5 A b +B a \right )}{9 x^{9}}-\frac {5 a^{3} b \left (2 A b +B a \right )}{8 x^{8}}-\frac {10 a^{2} b^{2} \left (A b +B a \right )}{7 x^{7}}-\frac {5 a \,b^{3} \left (A b +2 B a \right )}{6 x^{6}}-\frac {b^{4} \left (A b +5 B a \right )}{5 x^{5}}-\frac {b^{5} B}{4 x^{4}}\)	\(104\)
norman	\(\frac {-\frac {b^{5} B \,x^{6}}{4}+\left (-\frac {1}{5} b^{5} A -a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{6} a \,b^{4} A -\frac {5}{3} a^{2} b^{3} B \right ) x^{4}+\left (-\frac {10}{7} a^{2} b^{3} A -\frac {10}{7} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{4} a^{3} b^{2} A -\frac {5}{8} a^{4} b B \right ) x^{2}+\left (-\frac {5}{9} a^{4} b A -\frac {1}{9} a^{5} B \right ) x -\frac {a^{5} A}{10}}{x^{10}}\)	\(120\)
risch	\(\frac {-\frac {b^{5} B \,x^{6}}{4}+\left (-\frac {1}{5} b^{5} A -a \,b^{4} B \right ) x^{5}+\left (-\frac {5}{6} a \,b^{4} A -\frac {5}{3} a^{2} b^{3} B \right ) x^{4}+\left (-\frac {10}{7} a^{2} b^{3} A -\frac {10}{7} a^{3} b^{2} B \right ) x^{3}+\left (-\frac {5}{4} a^{3} b^{2} A -\frac {5}{8} a^{4} b B \right ) x^{2}+\left (-\frac {5}{9} a^{4} b A -\frac {1}{9} a^{5} B \right ) x -\frac {a^{5} A}{10}}{x^{10}}\)	\(120\)
gosper	\(-\frac {630 b^{5} B \,x^{6}+504 A \,b^{5} x^{5}+2520 B a \,b^{4} x^{5}+2100 a A \,b^{4} x^{4}+4200 B \,a^{2} b^{3} x^{4}+3600 a^{2} A \,b^{3} x^{3}+3600 B \,a^{3} b^{2} x^{3}+3150 a^{3} A \,b^{2} x^{2}+1575 B \,a^{4} b \,x^{2}+1400 a^{4} A b x +280 a^{5} B x +252 a^{5} A}{2520 x^{10}}\)	\(124\)
parallelrisch	\(-\frac {630 b^{5} B \,x^{6}+504 A \,b^{5} x^{5}+2520 B a \,b^{4} x^{5}+2100 a A \,b^{4} x^{4}+4200 B \,a^{2} b^{3} x^{4}+3600 a^{2} A \,b^{3} x^{3}+3600 B \,a^{3} b^{2} x^{3}+3150 a^{3} A \,b^{2} x^{2}+1575 B \,a^{4} b \,x^{2}+1400 a^{4} A b x +280 a^{5} B x +252 a^{5} A}{2520 x^{10}}\)	\(124\)

[In]

int((b*x+a)^5*(B*x+A)/x^11,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2520 \, x^{10}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^11,x, algorithm="fricas")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 3600*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

Sympy [A] (verification not implemented)

Time = 7.07 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=\frac {- 252 A a^{5} - 630 B b^{5} x^{6} + x^{5} \left (- 504 A b^{5} - 2520 B a b^{4}\right ) + x^{4} \left (- 2100 A a b^{4} - 4200 B a^{2} b^{3}\right ) + x^{3} \left (- 3600 A a^{2} b^{3} - 3600 B a^{3} b^{2}\right ) + x^{2} \left (- 3150 A a^{3} b^{2} - 1575 B a^{4} b\right ) + x \left (- 1400 A a^{4} b - 280 B a^{5}\right )}{2520 x^{10}} \]

[In]

integrate((b*x+a)**5*(B*x+A)/x**11,x)

[Out]

(-252*A*a**5 - 630*B*b**5*x**6 + x**5*(-504*A*b**5 - 2520*B*a*b**4) + x**4*(-2100*A*a*b**4 - 4200*B*a**2*b**3)
+ x**3*(-3600*A*a**2*b**3 - 3600*B*a**3*b**2) + x**2*(-3150*A*a**3*b**2 - 1575*B*a**4*b) + x*(-1400*A*a**4*b
- 280*B*a**5))/(2520*x**10)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {630 \, B b^{5} x^{6} + 252 \, A a^{5} + 504 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 2100 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 3600 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 1575 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 280 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x}{2520 \, x^{10}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^11,x, algorithm="maxima")

[Out]

-1/2520*(630*B*b^5*x^6 + 252*A*a^5 + 504*(5*B*a*b^4 + A*b^5)*x^5 + 2100*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 3600*(B*
a^3*b^2 + A*a^2*b^3)*x^3 + 1575*(B*a^4*b + 2*A*a^3*b^2)*x^2 + 280*(B*a^5 + 5*A*a^4*b)*x)/x^10

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {630 \, B b^{5} x^{6} + 2520 \, B a b^{4} x^{5} + 504 \, A b^{5} x^{5} + 4200 \, B a^{2} b^{3} x^{4} + 2100 \, A a b^{4} x^{4} + 3600 \, B a^{3} b^{2} x^{3} + 3600 \, A a^{2} b^{3} x^{3} + 1575 \, B a^{4} b x^{2} + 3150 \, A a^{3} b^{2} x^{2} + 280 \, B a^{5} x + 1400 \, A a^{4} b x + 252 \, A a^{5}}{2520 \, x^{10}} \]

[In]

integrate((b*x+a)^5*(B*x+A)/x^11,x, algorithm="giac")

[Out]

-1/2520*(630*B*b^5*x^6 + 2520*B*a*b^4*x^5 + 504*A*b^5*x^5 + 4200*B*a^2*b^3*x^4 + 2100*A*a*b^4*x^4 + 3600*B*a^3
*b^2*x^3 + 3600*A*a^2*b^3*x^3 + 1575*B*a^4*b*x^2 + 3150*A*a^3*b^2*x^2 + 280*B*a^5*x + 1400*A*a^4*b*x + 252*A*a
^5)/x^10

Mupad [B] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.02 \[ \int \frac {(a+b x)^5 (A+B x)}{x^{11}} \, dx=-\frac {x\,\left (\frac {B\,a^5}{9}+\frac {5\,A\,b\,a^4}{9}\right )+\frac {A\,a^5}{10}+x^4\,\left (\frac {5\,B\,a^2\,b^3}{3}+\frac {5\,A\,a\,b^4}{6}\right )+x^2\,\left (\frac {5\,B\,a^4\,b}{8}+\frac {5\,A\,a^3\,b^2}{4}\right )+x^5\,\left (\frac {A\,b^5}{5}+B\,a\,b^4\right )+x^3\,\left (\frac {10\,B\,a^3\,b^2}{7}+\frac {10\,A\,a^2\,b^3}{7}\right )+\frac {B\,b^5\,x^6}{4}}{x^{10}} \]

[In]

int(((A + B*x)*(a + b*x)^5)/x^11,x)

[Out]

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