\(\int \frac {x^8}{(a+b x)^{10}} \, dx\) [226]

Optimal result

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

[Out]

1/9*x^9/a/(b*x+a)^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} \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=\frac {x^9}{9 a (a+b x)^9} \]

[In]

Int[x^8/(a + b*x)^10,x]

[Out]

x^9/(9*a*(a + b*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 {x^9}{9 a (a+b x)^9} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(17)=34\).

Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 5.71 \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=-\frac {a^8+9 a^7 b x+36 a^6 b^2 x^2+84 a^5 b^3 x^3+126 a^4 b^4 x^4+126 a^3 b^5 x^5+84 a^2 b^6 x^6+36 a b^7 x^7+9 b^8 x^8}{9 b^9 (a+b x)^9} \]

[In]

Integrate[x^8/(a + b*x)^10,x]

[Out]

-1/9*(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)/(b^9*(a + b*x)^9)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(95\) vs. \(2(15)=30\).

Time = 0.04 (sec) , antiderivative size = 96, normalized size of antiderivative = 5.65

method	result	size
gosper	\(-\frac {9 b^{8} x^{8}+36 a \,x^{7} b^{7}+84 a^{2} x^{6} b^{6}+126 a^{3} x^{5} b^{5}+126 a^{4} x^{4} b^{4}+84 a^{5} b^{3} x^{3}+36 a^{6} x^{2} b^{2}+9 a^{7} x b +a^{8}}{9 \left (b x +a \right )^{9} b^{9}}\)	\(96\)
parallelrisch	\(\frac {-9 b^{8} x^{8}-36 a \,x^{7} b^{7}-84 a^{2} x^{6} b^{6}-126 a^{3} x^{5} b^{5}-126 a^{4} x^{4} b^{4}-84 a^{5} b^{3} x^{3}-36 a^{6} x^{2} b^{2}-9 a^{7} x b -a^{8}}{9 b^{9} \left (b x +a \right )^{9}}\)	\(98\)
norman	\(\frac {-\frac {x^{8}}{b}-\frac {4 a \,x^{7}}{b^{2}}-\frac {28 a^{2} x^{6}}{3 b^{3}}-\frac {14 a^{3} x^{5}}{b^{4}}-\frac {14 a^{4} x^{4}}{b^{5}}-\frac {28 a^{5} x^{3}}{3 b^{6}}-\frac {4 a^{6} x^{2}}{b^{7}}-\frac {a^{7} x}{b^{8}}-\frac {a^{8}}{9 b^{9}}}{\left (b x +a \right )^{9}}\)	\(99\)
risch	\(\frac {-\frac {x^{8}}{b}-\frac {4 a \,x^{7}}{b^{2}}-\frac {28 a^{2} x^{6}}{3 b^{3}}-\frac {14 a^{3} x^{5}}{b^{4}}-\frac {14 a^{4} x^{4}}{b^{5}}-\frac {28 a^{5} x^{3}}{3 b^{6}}-\frac {4 a^{6} x^{2}}{b^{7}}-\frac {a^{7} x}{b^{8}}-\frac {a^{8}}{9 b^{9}}}{\left (b x +a \right )^{9}}\)	\(99\)
default	\(\frac {a^{7}}{b^{9} \left (b x +a \right )^{8}}-\frac {a^{8}}{9 b^{9} \left (b x +a \right )^{9}}+\frac {28 a^{5}}{3 b^{9} \left (b x +a \right )^{6}}+\frac {14 a^{3}}{b^{9} \left (b x +a \right )^{4}}-\frac {28 a^{2}}{3 b^{9} \left (b x +a \right )^{3}}-\frac {14 a^{4}}{b^{9} \left (b x +a \right )^{5}}+\frac {4 a}{b^{9} \left (b x +a \right )^{2}}-\frac {1}{\left (b x +a \right ) b^{9}}-\frac {4 a^{6}}{b^{9} \left (b x +a \right )^{7}}\)	\(131\)

[In]

int(x^8/(b*x+a)^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)/(b*x+a)^9/b^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 186, normalized size of antiderivative = 10.94 \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=-\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, {\left (b^{18} x^{9} + 9 \, a b^{17} x^{8} + 36 \, a^{2} b^{16} x^{7} + 84 \, a^{3} b^{15} x^{6} + 126 \, a^{4} b^{14} x^{5} + 126 \, a^{5} b^{13} x^{4} + 84 \, a^{6} b^{12} x^{3} + 36 \, a^{7} b^{11} x^{2} + 9 \, a^{8} b^{10} x + a^{9} b^{9}\right )}} \]

[In]

integrate(x^8/(b*x+a)^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)/(b^18*x^9 + 9*a*b^17*x^8 + 36*a^2*b^16*x^7 + 84*a^3*b^15*x^6 + 126*a^4*b^14*x^5 + 1
26*a^5*b^13*x^4 + 84*a^6*b^12*x^3 + 36*a^7*b^11*x^2 + 9*a^8*b^10*x + a^9*b^9)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (12) = 24\).

Time = 0.49 (sec) , antiderivative size = 199, normalized size of antiderivative = 11.71 \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=\frac {- a^{8} - 9 a^{7} b x - 36 a^{6} b^{2} x^{2} - 84 a^{5} b^{3} x^{3} - 126 a^{4} b^{4} x^{4} - 126 a^{3} b^{5} x^{5} - 84 a^{2} b^{6} x^{6} - 36 a b^{7} x^{7} - 9 b^{8} x^{8}}{9 a^{9} b^{9} + 81 a^{8} b^{10} x + 324 a^{7} b^{11} x^{2} + 756 a^{6} b^{12} x^{3} + 1134 a^{5} b^{13} x^{4} + 1134 a^{4} b^{14} x^{5} + 756 a^{3} b^{15} x^{6} + 324 a^{2} b^{16} x^{7} + 81 a b^{17} x^{8} + 9 b^{18} x^{9}} \]

[In]

integrate(x**8/(b*x+a)**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*a**9*b**9 + 81*a**8*b**10*x + 324*a**7*b**11*x**2 + 756*a**6*b*
*12*x**3 + 1134*a**5*b**13*x**4 + 1134*a**4*b**14*x**5 + 756*a**3*b**15*x**6 + 324*a**2*b**16*x**7 + 81*a*b**1
7*x**8 + 9*b**18*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 186, normalized size of antiderivative = 10.94 \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=-\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, {\left (b^{18} x^{9} + 9 \, a b^{17} x^{8} + 36 \, a^{2} b^{16} x^{7} + 84 \, a^{3} b^{15} x^{6} + 126 \, a^{4} b^{14} x^{5} + 126 \, a^{5} b^{13} x^{4} + 84 \, a^{6} b^{12} x^{3} + 36 \, a^{7} b^{11} x^{2} + 9 \, a^{8} b^{10} x + a^{9} b^{9}\right )}} \]

[In]

integrate(x^8/(b*x+a)^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)/(b^18*x^9 + 9*a*b^17*x^8 + 36*a^2*b^16*x^7 + 84*a^3*b^15*x^6 + 126*a^4*b^14*x^5 + 1
26*a^5*b^13*x^4 + 84*a^6*b^12*x^3 + 36*a^7*b^11*x^2 + 9*a^8*b^10*x + a^9*b^9)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 95, normalized size of antiderivative = 5.59 \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=-\frac {9 \, b^{8} x^{8} + 36 \, a b^{7} x^{7} + 84 \, a^{2} b^{6} x^{6} + 126 \, a^{3} b^{5} x^{5} + 126 \, a^{4} b^{4} x^{4} + 84 \, a^{5} b^{3} x^{3} + 36 \, a^{6} b^{2} x^{2} + 9 \, a^{7} b x + a^{8}}{9 \, {\left (b x + a\right )}^{9} b^{9}} \]

[In]

integrate(x^8/(b*x+a)^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)/((b*x + a)^9*b^9)

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 107, normalized size of antiderivative = 6.29 \[ \int \frac {x^8}{(a+b x)^{10}} \, dx=-\frac {\frac {1}{a+b\,x}-\frac {4\,a}{{\left (a+b\,x\right )}^2}+\frac {28\,a^2}{3\,{\left (a+b\,x\right )}^3}-\frac {14\,a^3}{{\left (a+b\,x\right )}^4}+\frac {14\,a^4}{{\left (a+b\,x\right )}^5}-\frac {28\,a^5}{3\,{\left (a+b\,x\right )}^6}+\frac {4\,a^6}{{\left (a+b\,x\right )}^7}-\frac {a^7}{{\left (a+b\,x\right )}^8}+\frac {a^8}{9\,{\left (a+b\,x\right )}^9}}{b^9} \]

[In]

int(x^8/(a + b*x)^10,x)

[Out]

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