\(\int \frac {(a+b x)^{10}}{x^{12}} \, dx\) [146]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {(a+b x)^{11}}{11 a x^{11}} \]

[Out]

-1/11*(b*x+a)^11/a/x^11

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {(a+b x)^{11}}{11 a x^{11}} \]

[In]

Int[(a + b*x)^10/x^12,x]

[Out]

-1/11*(a + b*x)^11/(a*x^11)

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{11}}{11 a x^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(114\) vs. \(2(17)=34\).

Time = 0.01 (sec) , antiderivative size = 114, normalized size of antiderivative = 6.71 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {a^{10}}{11 x^{11}}-\frac {a^9 b}{x^{10}}-\frac {5 a^8 b^2}{x^9}-\frac {15 a^7 b^3}{x^8}-\frac {30 a^6 b^4}{x^7}-\frac {42 a^5 b^5}{x^6}-\frac {42 a^4 b^6}{x^5}-\frac {30 a^3 b^7}{x^4}-\frac {15 a^2 b^8}{x^3}-\frac {5 a b^9}{x^2}-\frac {b^{10}}{x} \]

[In]

Integrate[(a + b*x)^10/x^12,x]

[Out]

-1/11*a^10/x^11 - (a^9*b)/x^10 - (5*a^8*b^2)/x^9 - (15*a^7*b^3)/x^8 - (30*a^6*b^4)/x^7 - (42*a^5*b^5)/x^6 - (4
2*a^4*b^6)/x^5 - (30*a^3*b^7)/x^4 - (15*a^2*b^8)/x^3 - (5*a*b^9)/x^2 - b^10/x

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs. \(2(15)=30\).

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 6.53

method	result	size
gosper	\(-\frac {11 b^{10} x^{10}+55 a \,b^{9} x^{9}+165 a^{2} b^{8} x^{8}+330 a^{3} b^{7} x^{7}+462 a^{4} b^{6} x^{6}+462 a^{5} b^{5} x^{5}+330 a^{6} b^{4} x^{4}+165 a^{7} b^{3} x^{3}+55 a^{8} b^{2} x^{2}+11 a^{9} b x +a^{10}}{11 x^{11}}\)	\(111\)
norman	\(\frac {-b^{10} x^{10}-5 a \,b^{9} x^{9}-15 a^{2} b^{8} x^{8}-30 a^{3} b^{7} x^{7}-42 a^{4} b^{6} x^{6}-42 a^{5} b^{5} x^{5}-30 a^{6} b^{4} x^{4}-15 a^{7} b^{3} x^{3}-5 a^{8} b^{2} x^{2}-a^{9} b x -\frac {1}{11} a^{10}}{x^{11}}\)	\(112\)
risch	\(\frac {-b^{10} x^{10}-5 a \,b^{9} x^{9}-15 a^{2} b^{8} x^{8}-30 a^{3} b^{7} x^{7}-42 a^{4} b^{6} x^{6}-42 a^{5} b^{5} x^{5}-30 a^{6} b^{4} x^{4}-15 a^{7} b^{3} x^{3}-5 a^{8} b^{2} x^{2}-a^{9} b x -\frac {1}{11} a^{10}}{x^{11}}\)	\(112\)
default	\(-\frac {a^{9} b}{x^{10}}-\frac {42 a^{5} b^{5}}{x^{6}}-\frac {30 a^{6} b^{4}}{x^{7}}-\frac {5 a^{8} b^{2}}{x^{9}}-\frac {15 a^{2} b^{8}}{x^{3}}-\frac {a^{10}}{11 x^{11}}-\frac {b^{10}}{x}-\frac {5 a \,b^{9}}{x^{2}}-\frac {30 a^{3} b^{7}}{x^{4}}-\frac {42 a^{4} b^{6}}{x^{5}}-\frac {15 a^{7} b^{3}}{x^{8}}\)	\(113\)
parallelrisch	\(\frac {-11 b^{10} x^{10}-55 a \,b^{9} x^{9}-165 a^{2} b^{8} x^{8}-330 a^{3} b^{7} x^{7}-462 a^{4} b^{6} x^{6}-462 a^{5} b^{5} x^{5}-330 a^{6} b^{4} x^{4}-165 a^{7} b^{3} x^{3}-55 a^{8} b^{2} x^{2}-11 a^{9} b x -a^{10}}{11 x^{11}}\)	\(113\)

[In]

int((b*x+a)^10/x^12,x,method=_RETURNVERBOSE)

[Out]

-1/11*(11*b^10*x^10+55*a*b^9*x^9+165*a^2*b^8*x^8+330*a^3*b^7*x^7+462*a^4*b^6*x^6+462*a^5*b^5*x^5+330*a^6*b^4*x
^4+165*a^7*b^3*x^3+55*a^8*b^2*x^2+11*a^9*b*x+a^10)/x^11

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.47 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {11 \, b^{10} x^{10} + 55 \, a b^{9} x^{9} + 165 \, a^{2} b^{8} x^{8} + 330 \, a^{3} b^{7} x^{7} + 462 \, a^{4} b^{6} x^{6} + 462 \, a^{5} b^{5} x^{5} + 330 \, a^{6} b^{4} x^{4} + 165 \, a^{7} b^{3} x^{3} + 55 \, a^{8} b^{2} x^{2} + 11 \, a^{9} b x + a^{10}}{11 \, x^{11}} \]

[In]

integrate((b*x+a)^10/x^12,x, algorithm="fricas")

[Out]

-1/11*(11*b^10*x^10 + 55*a*b^9*x^9 + 165*a^2*b^8*x^8 + 330*a^3*b^7*x^7 + 462*a^4*b^6*x^6 + 462*a^5*b^5*x^5 + 3
30*a^6*b^4*x^4 + 165*a^7*b^3*x^3 + 55*a^8*b^2*x^2 + 11*a^9*b*x + a^10)/x^11

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (14) = 28\).

Time = 0.53 (sec) , antiderivative size = 119, normalized size of antiderivative = 7.00 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=\frac {- a^{10} - 11 a^{9} b x - 55 a^{8} b^{2} x^{2} - 165 a^{7} b^{3} x^{3} - 330 a^{6} b^{4} x^{4} - 462 a^{5} b^{5} x^{5} - 462 a^{4} b^{6} x^{6} - 330 a^{3} b^{7} x^{7} - 165 a^{2} b^{8} x^{8} - 55 a b^{9} x^{9} - 11 b^{10} x^{10}}{11 x^{11}} \]

[In]

integrate((b*x+a)**10/x**12,x)

[Out]

(-a**10 - 11*a**9*b*x - 55*a**8*b**2*x**2 - 165*a**7*b**3*x**3 - 330*a**6*b**4*x**4 - 462*a**5*b**5*x**5 - 462
*a**4*b**6*x**6 - 330*a**3*b**7*x**7 - 165*a**2*b**8*x**8 - 55*a*b**9*x**9 - 11*b**10*x**10)/(11*x**11)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.47 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {11 \, b^{10} x^{10} + 55 \, a b^{9} x^{9} + 165 \, a^{2} b^{8} x^{8} + 330 \, a^{3} b^{7} x^{7} + 462 \, a^{4} b^{6} x^{6} + 462 \, a^{5} b^{5} x^{5} + 330 \, a^{6} b^{4} x^{4} + 165 \, a^{7} b^{3} x^{3} + 55 \, a^{8} b^{2} x^{2} + 11 \, a^{9} b x + a^{10}}{11 \, x^{11}} \]

[In]

integrate((b*x+a)^10/x^12,x, algorithm="maxima")

[Out]

-1/11*(11*b^10*x^10 + 55*a*b^9*x^9 + 165*a^2*b^8*x^8 + 330*a^3*b^7*x^7 + 462*a^4*b^6*x^6 + 462*a^5*b^5*x^5 + 3
30*a^6*b^4*x^4 + 165*a^7*b^3*x^3 + 55*a^8*b^2*x^2 + 11*a^9*b*x + a^10)/x^11

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.47 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {11 \, b^{10} x^{10} + 55 \, a b^{9} x^{9} + 165 \, a^{2} b^{8} x^{8} + 330 \, a^{3} b^{7} x^{7} + 462 \, a^{4} b^{6} x^{6} + 462 \, a^{5} b^{5} x^{5} + 330 \, a^{6} b^{4} x^{4} + 165 \, a^{7} b^{3} x^{3} + 55 \, a^{8} b^{2} x^{2} + 11 \, a^{9} b x + a^{10}}{11 \, x^{11}} \]

[In]

integrate((b*x+a)^10/x^12,x, algorithm="giac")

[Out]

-1/11*(11*b^10*x^10 + 55*a*b^9*x^9 + 165*a^2*b^8*x^8 + 330*a^3*b^7*x^7 + 462*a^4*b^6*x^6 + 462*a^5*b^5*x^5 + 3
30*a^6*b^4*x^4 + 165*a^7*b^3*x^3 + 55*a^8*b^2*x^2 + 11*a^9*b*x + a^10)/x^11

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 110, normalized size of antiderivative = 6.47 \[ \int \frac {(a+b x)^{10}}{x^{12}} \, dx=-\frac {\frac {a^{10}}{11}+a^9\,b\,x+5\,a^8\,b^2\,x^2+15\,a^7\,b^3\,x^3+30\,a^6\,b^4\,x^4+42\,a^5\,b^5\,x^5+42\,a^4\,b^6\,x^6+30\,a^3\,b^7\,x^7+15\,a^2\,b^8\,x^8+5\,a\,b^9\,x^9+b^{10}\,x^{10}}{x^{11}} \]

[In]

int((a + b*x)^10/x^12,x)

[Out]

-(a^10/11 + b^10*x^10 + 5*a*b^9*x^9 + 5*a^8*b^2*x^2 + 15*a^7*b^3*x^3 + 30*a^6*b^4*x^4 + 42*a^5*b^5*x^5 + 42*a^
4*b^6*x^6 + 30*a^3*b^7*x^7 + 15*a^2*b^8*x^8 + a^9*b*x)/x^11