\(\int \frac {x}{(a+b x)^{10}} \, dx\) [233]

Optimal result

Integrand size = 9, antiderivative size = 30 \[ \int \frac {x}{(a+b x)^{10}} \, dx=\frac {a}{9 b^2 (a+b x)^9}-\frac {1}{8 b^2 (a+b x)^8} \]

[Out]

1/9*a/b^2/(b*x+a)^9-1/8/b^2/(b*x+a)^8

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x}{(a+b x)^{10}} \, dx=\frac {a}{9 b^2 (a+b x)^9}-\frac {1}{8 b^2 (a+b x)^8} \]

[In]

Int[x/(a + b*x)^10,x]

[Out]

a/(9*b^2*(a + b*x)^9) - 1/(8*b^2*(a + b*x)^8)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a}{b (a+b x)^{10}}+\frac {1}{b (a+b x)^9}\right ) \, dx \\ & = \frac {a}{9 b^2 (a+b x)^9}-\frac {1}{8 b^2 (a+b x)^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67 \[ \int \frac {x}{(a+b x)^{10}} \, dx=-\frac {a+9 b x}{72 b^2 (a+b x)^9} \]

[In]

Integrate[x/(a + b*x)^10,x]

[Out]

-1/72*(a + 9*b*x)/(b^2*(a + b*x)^9)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63

method	result	size
gosper	\(-\frac {9 b x +a}{72 \left (b x +a \right )^{9} b^{2}}\)	\(19\)
norman	\(\frac {-\frac {x}{8 b}-\frac {a}{72 b^{2}}}{\left (b x +a \right )^{9}}\)	\(22\)
risch	\(\frac {-\frac {x}{8 b}-\frac {a}{72 b^{2}}}{\left (b x +a \right )^{9}}\)	\(22\)
parallelrisch	\(\frac {-9 b^{8} x -a \,b^{7}}{72 b^{9} \left (b x +a \right )^{9}}\)	\(26\)
default	\(\frac {a}{9 b^{2} \left (b x +a \right )^{9}}-\frac {1}{8 b^{2} \left (b x +a \right )^{8}}\)	\(27\)

[In]

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

[Out]

-1/72*(9*b*x+a)/(b*x+a)^9/b^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (26) = 52\).

Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.63 \[ \int \frac {x}{(a+b x)^{10}} \, dx=-\frac {9 \, b x + a}{72 \, {\left (b^{11} x^{9} + 9 \, a b^{10} x^{8} + 36 \, a^{2} b^{9} x^{7} + 84 \, a^{3} b^{8} x^{6} + 126 \, a^{4} b^{7} x^{5} + 126 \, a^{5} b^{6} x^{4} + 84 \, a^{6} b^{5} x^{3} + 36 \, a^{7} b^{4} x^{2} + 9 \, a^{8} b^{3} x + a^{9} b^{2}\right )}} \]

[In]

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

[Out]

-1/72*(9*b*x + a)/(b^11*x^9 + 9*a*b^10*x^8 + 36*a^2*b^9*x^7 + 84*a^3*b^8*x^6 + 126*a^4*b^7*x^5 + 126*a^5*b^6*x
^4 + 84*a^6*b^5*x^3 + 36*a^7*b^4*x^2 + 9*a^8*b^3*x + a^9*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (26) = 52\).

Time = 0.32 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.87 \[ \int \frac {x}{(a+b x)^{10}} \, dx=\frac {- a - 9 b x}{72 a^{9} b^{2} + 648 a^{8} b^{3} x + 2592 a^{7} b^{4} x^{2} + 6048 a^{6} b^{5} x^{3} + 9072 a^{5} b^{6} x^{4} + 9072 a^{4} b^{7} x^{5} + 6048 a^{3} b^{8} x^{6} + 2592 a^{2} b^{9} x^{7} + 648 a b^{10} x^{8} + 72 b^{11} x^{9}} \]

[In]

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

[Out]

(-a - 9*b*x)/(72*a**9*b**2 + 648*a**8*b**3*x + 2592*a**7*b**4*x**2 + 6048*a**6*b**5*x**3 + 9072*a**5*b**6*x**4
+ 9072*a**4*b**7*x**5 + 6048*a**3*b**8*x**6 + 2592*a**2*b**9*x**7 + 648*a*b**10*x**8 + 72*b**11*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.63 \[ \int \frac {x}{(a+b x)^{10}} \, dx=-\frac {9 \, b x + a}{72 \, {\left (b^{11} x^{9} + 9 \, a b^{10} x^{8} + 36 \, a^{2} b^{9} x^{7} + 84 \, a^{3} b^{8} x^{6} + 126 \, a^{4} b^{7} x^{5} + 126 \, a^{5} b^{6} x^{4} + 84 \, a^{6} b^{5} x^{3} + 36 \, a^{7} b^{4} x^{2} + 9 \, a^{8} b^{3} x + a^{9} b^{2}\right )}} \]

[In]

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

[Out]

-1/72*(9*b*x + a)/(b^11*x^9 + 9*a*b^10*x^8 + 36*a^2*b^9*x^7 + 84*a^3*b^8*x^6 + 126*a^4*b^7*x^5 + 126*a^5*b^6*x
^4 + 84*a^6*b^5*x^3 + 36*a^7*b^4*x^2 + 9*a^8*b^3*x + a^9*b^2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60 \[ \int \frac {x}{(a+b x)^{10}} \, dx=-\frac {9 \, b x + a}{72 \, {\left (b x + a\right )}^{9} b^{2}} \]

[In]

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

[Out]

-1/72*(9*b*x + a)/((b*x + a)^9*b^2)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.60 \[ \int \frac {x}{(a+b x)^{10}} \, dx=-\frac {a+9\,b\,x}{72\,b^2\,{\left (a+b\,x\right )}^9} \]

[In]

int(x/(a + b*x)^10,x)

[Out]

-(a + 9*b*x)/(72*b^2*(a + b*x)^9)