\(\int \frac {(a+b x)^8}{x^{10}} \, dx\) [243]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {(a+b x)^9}{9 a x^9} \]

[Out]

-1/9*(b*x+a)^9/a/x^9

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} \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {(a+b x)^9}{9 a x^9} \]

[In]

Int[(a + b*x)^8/x^10,x]

[Out]

-1/9*(a + b*x)^9/(a*x^9)

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)^9}{9 a x^9} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(17)=34\).

Time = 0.01 (sec) , antiderivative size = 96, normalized size of antiderivative = 5.65 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {a^8}{9 x^9}-\frac {a^7 b}{x^8}-\frac {4 a^6 b^2}{x^7}-\frac {28 a^5 b^3}{3 x^6}-\frac {14 a^4 b^4}{x^5}-\frac {14 a^3 b^5}{x^4}-\frac {28 a^2 b^6}{3 x^3}-\frac {4 a b^7}{x^2}-\frac {b^8}{x} \]

[In]

Integrate[(a + b*x)^8/x^10,x]

[Out]

-1/9*a^8/x^9 - (a^7*b)/x^8 - (4*a^6*b^2)/x^7 - (28*a^5*b^3)/(3*x^6) - (14*a^4*b^4)/x^5 - (14*a^3*b^5)/x^4 - (2
8*a^2*b^6)/(3*x^3) - (4*a*b^7)/x^2 - b^8/x

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(15)=30\).

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 5.24

method	result	size
gosper	\(-\frac {9 b^{8} x^{8}+36 a \,x^{7} b^{7}+84 a^{2} x^{6} b^{6}+126 a^{3} x^{5} b^{5}+126 a^{4} x^{4} b^{4}+84 a^{5} b^{3} x^{3}+36 a^{6} x^{2} b^{2}+9 a^{7} x b +a^{8}}{9 x^{9}}\)	\(89\)
norman	\(\frac {-b^{8} x^{8}-4 a \,x^{7} b^{7}-\frac {28}{3} a^{2} x^{6} b^{6}-14 a^{3} x^{5} b^{5}-14 a^{4} x^{4} b^{4}-\frac {28}{3} a^{5} b^{3} x^{3}-4 a^{6} x^{2} b^{2}-a^{7} x b -\frac {1}{9} a^{8}}{x^{9}}\)	\(90\)
risch	\(\frac {-b^{8} x^{8}-4 a \,x^{7} b^{7}-\frac {28}{3} a^{2} x^{6} b^{6}-14 a^{3} x^{5} b^{5}-14 a^{4} x^{4} b^{4}-\frac {28}{3} a^{5} b^{3} x^{3}-4 a^{6} x^{2} b^{2}-a^{7} x b -\frac {1}{9} a^{8}}{x^{9}}\)	\(90\)
default	\(-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {4 b^{2} a^{6}}{x^{7}}-\frac {a^{8}}{9 x^{9}}-\frac {28 a^{2} b^{6}}{3 x^{3}}-\frac {b^{8}}{x}-\frac {4 a \,b^{7}}{x^{2}}-\frac {14 a^{3} b^{5}}{x^{4}}-\frac {14 a^{4} b^{4}}{x^{5}}-\frac {a^{7} b}{x^{8}}\)	\(91\)
parallelrisch	\(\frac {-9 b^{8} x^{8}-36 a \,x^{7} b^{7}-84 a^{2} x^{6} b^{6}-126 a^{3} x^{5} b^{5}-126 a^{4} x^{4} b^{4}-84 a^{5} b^{3} x^{3}-36 a^{6} x^{2} b^{2}-9 a^{7} x b -a^{8}}{9 x^{9}}\)	\(91\)

[In]

int((b*x+a)^8/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/9*(9*b^8*x^8+36*a*b^7*x^7+84*a^2*b^6*x^6+126*a^3*b^5*x^5+126*a^4*b^4*x^4+84*a^5*b^3*x^3+36*a^6*b^2*x^2+9*a^
7*b*x+a^8)/x^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \]

[In]

integrate((b*x+a)^8/x^10,x, algorithm="fricas")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (14) = 28\).

Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 5.59 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=\frac {- a^{8} - 9 a^{7} b x - 36 a^{6} b^{2} x^{2} - 84 a^{5} b^{3} x^{3} - 126 a^{4} b^{4} x^{4} - 126 a^{3} b^{5} x^{5} - 84 a^{2} b^{6} x^{6} - 36 a b^{7} x^{7} - 9 b^{8} x^{8}}{9 x^{9}} \]

[In]

integrate((b*x+a)**8/x**10,x)

[Out]

(-a**8 - 9*a**7*b*x - 36*a**6*b**2*x**2 - 84*a**5*b**3*x**3 - 126*a**4*b**4*x**4 - 126*a**3*b**5*x**5 - 84*a**
2*b**6*x**6 - 36*a*b**7*x**7 - 9*b**8*x**8)/(9*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).

Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \]

[In]

integrate((b*x+a)^8/x^10,x, algorithm="maxima")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, x^{9}} \]

[In]

integrate((b*x+a)^8/x^10,x, algorithm="giac")

[Out]

-1/9*(9*b^8*x^8 + 36*a*b^7*x^7 + 84*a^2*b^6*x^6 + 126*a^3*b^5*x^5 + 126*a^4*b^4*x^4 + 84*a^5*b^3*x^3 + 36*a^6*
b^2*x^2 + 9*a^7*b*x + a^8)/x^9

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 88, normalized size of antiderivative = 5.18 \[ \int \frac {(a+b x)^8}{x^{10}} \, dx=-\frac {\frac {a^8}{9}+a^7\,b\,x+4\,a^6\,b^2\,x^2+\frac {28\,a^5\,b^3\,x^3}{3}+14\,a^4\,b^4\,x^4+14\,a^3\,b^5\,x^5+\frac {28\,a^2\,b^6\,x^6}{3}+4\,a\,b^7\,x^7+b^8\,x^8}{x^9} \]

[In]

int((a + b*x)^8/x^10,x)

[Out]

-(a^8/9 + b^8*x^8 + 4*a*b^7*x^7 + 4*a^6*b^2*x^2 + (28*a^5*b^3*x^3)/3 + 14*a^4*b^4*x^4 + 14*a^3*b^5*x^5 + (28*a
^2*b^6*x^6)/3 + a^7*b*x)/x^9