\(\int \frac {x^5}{(a+b x)^{10}} \, dx\) [229]

Optimal result

Integrand size = 11, antiderivative size = 69 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=\frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {x^6}{504 a^4 (a+b x)^6} \]

[Out]

1/9*x^6/a/(b*x+a)^9+1/24*x^6/a^2/(b*x+a)^8+1/84*x^6/a^3/(b*x+a)^7+1/504*x^6/a^4/(b*x+a)^6

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=\frac {x^6}{504 a^4 (a+b x)^6}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{9 a (a+b x)^9} \]

[In]

Int[x^5/(a + b*x)^10,x]

[Out]

x^6/(9*a*(a + b*x)^9) + x^6/(24*a^2*(a + b*x)^8) + x^6/(84*a^3*(a + b*x)^7) + x^6/(504*a^4*(a + b*x)^6)

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]

Rule 47

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] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \frac {x^6}{9 a (a+b x)^9}+\frac {\int \frac {x^5}{(a+b x)^9} \, dx}{3 a} \\ & = \frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {\int \frac {x^5}{(a+b x)^8} \, dx}{12 a^2} \\ & = \frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {\int \frac {x^5}{(a+b x)^7} \, dx}{84 a^3} \\ & = \frac {x^6}{9 a (a+b x)^9}+\frac {x^6}{24 a^2 (a+b x)^8}+\frac {x^6}{84 a^3 (a+b x)^7}+\frac {x^6}{504 a^4 (a+b x)^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=-\frac {a^5+9 a^4 b x+36 a^3 b^2 x^2+84 a^2 b^3 x^3+126 a b^4 x^4+126 b^5 x^5}{504 b^6 (a+b x)^9} \]

[In]

Integrate[x^5/(a + b*x)^10,x]

[Out]

-1/504*(a^5 + 9*a^4*b*x + 36*a^3*b^2*x^2 + 84*a^2*b^3*x^3 + 126*a*b^4*x^4 + 126*b^5*x^5)/(b^6*(a + b*x)^9)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91

method	result	size
gosper	\(-\frac {126 b^{5} x^{5}+126 a \,b^{4} x^{4}+84 a^{2} b^{3} x^{3}+36 a^{3} b^{2} x^{2}+9 a^{4} b x +a^{5}}{504 \left (b x +a \right )^{9} b^{6}}\)	\(63\)
norman	\(\frac {-\frac {x^{5}}{4 b}-\frac {a \,x^{4}}{4 b^{2}}-\frac {a^{2} x^{3}}{6 b^{3}}-\frac {a^{3} x^{2}}{14 b^{4}}-\frac {a^{4} x}{56 b^{5}}-\frac {a^{5}}{504 b^{6}}}{\left (b x +a \right )^{9}}\)	\(66\)
risch	\(\frac {-\frac {x^{5}}{4 b}-\frac {a \,x^{4}}{4 b^{2}}-\frac {a^{2} x^{3}}{6 b^{3}}-\frac {a^{3} x^{2}}{14 b^{4}}-\frac {a^{4} x}{56 b^{5}}-\frac {a^{5}}{504 b^{6}}}{\left (b x +a \right )^{9}}\)	\(66\)
parallelrisch	\(\frac {-126 b^{8} x^{5}-126 a \,b^{7} x^{4}-84 a^{2} b^{6} x^{3}-36 a^{3} b^{5} x^{2}-9 a^{4} b^{4} x -a^{5} b^{3}}{504 b^{9} \left (b x +a \right )^{9}}\)	\(70\)
default	\(\frac {a^{5}}{9 b^{6} \left (b x +a \right )^{9}}-\frac {5 a^{2}}{3 b^{6} \left (b x +a \right )^{6}}-\frac {1}{4 b^{6} \left (b x +a \right )^{4}}-\frac {5 a^{4}}{8 b^{6} \left (b x +a \right )^{8}}+\frac {10 a^{3}}{7 b^{6} \left (b x +a \right )^{7}}+\frac {a}{b^{6} \left (b x +a \right )^{5}}\)	\(86\)

[In]

int(x^5/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

-1/504*(126*b^5*x^5+126*a*b^4*x^4+84*a^2*b^3*x^3+36*a^3*b^2*x^2+9*a^4*b*x+a^5)/(b*x+a)^9/b^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (61) = 122\).

Time = 0.22 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.22 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=-\frac {126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \, {\left (b^{15} x^{9} + 9 \, a b^{14} x^{8} + 36 \, a^{2} b^{13} x^{7} + 84 \, a^{3} b^{12} x^{6} + 126 \, a^{4} b^{11} x^{5} + 126 \, a^{5} b^{10} x^{4} + 84 \, a^{6} b^{9} x^{3} + 36 \, a^{7} b^{8} x^{2} + 9 \, a^{8} b^{7} x + a^{9} b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*x + a^5)/(b^15*x^9 + 9*a*b^14*
x^8 + 36*a^2*b^13*x^7 + 84*a^3*b^12*x^6 + 126*a^4*b^11*x^5 + 126*a^5*b^10*x^4 + 84*a^6*b^9*x^3 + 36*a^7*b^8*x^
2 + 9*a^8*b^7*x + a^9*b^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (58) = 116\).

Time = 0.54 (sec) , antiderivative size = 163, normalized size of antiderivative = 2.36 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=\frac {- a^{5} - 9 a^{4} b x - 36 a^{3} b^{2} x^{2} - 84 a^{2} b^{3} x^{3} - 126 a b^{4} x^{4} - 126 b^{5} x^{5}}{504 a^{9} b^{6} + 4536 a^{8} b^{7} x + 18144 a^{7} b^{8} x^{2} + 42336 a^{6} b^{9} x^{3} + 63504 a^{5} b^{10} x^{4} + 63504 a^{4} b^{11} x^{5} + 42336 a^{3} b^{12} x^{6} + 18144 a^{2} b^{13} x^{7} + 4536 a b^{14} x^{8} + 504 b^{15} x^{9}} \]

[In]

integrate(x**5/(b*x+a)**10,x)

[Out]

(-a**5 - 9*a**4*b*x - 36*a**3*b**2*x**2 - 84*a**2*b**3*x**3 - 126*a*b**4*x**4 - 126*b**5*x**5)/(504*a**9*b**6
+ 4536*a**8*b**7*x + 18144*a**7*b**8*x**2 + 42336*a**6*b**9*x**3 + 63504*a**5*b**10*x**4 + 63504*a**4*b**11*x*
*5 + 42336*a**3*b**12*x**6 + 18144*a**2*b**13*x**7 + 4536*a*b**14*x**8 + 504*b**15*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (61) = 122\).

Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.22 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=-\frac {126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \, {\left (b^{15} x^{9} + 9 \, a b^{14} x^{8} + 36 \, a^{2} b^{13} x^{7} + 84 \, a^{3} b^{12} x^{6} + 126 \, a^{4} b^{11} x^{5} + 126 \, a^{5} b^{10} x^{4} + 84 \, a^{6} b^{9} x^{3} + 36 \, a^{7} b^{8} x^{2} + 9 \, a^{8} b^{7} x + a^{9} b^{6}\right )}} \]

[In]

integrate(x^5/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*x + a^5)/(b^15*x^9 + 9*a*b^14*
x^8 + 36*a^2*b^13*x^7 + 84*a^3*b^12*x^6 + 126*a^4*b^11*x^5 + 126*a^5*b^10*x^4 + 84*a^6*b^9*x^3 + 36*a^7*b^8*x^
2 + 9*a^8*b^7*x + a^9*b^6)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=-\frac {126 \, b^{5} x^{5} + 126 \, a b^{4} x^{4} + 84 \, a^{2} b^{3} x^{3} + 36 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x + a^{5}}{504 \, {\left (b x + a\right )}^{9} b^{6}} \]

[In]

integrate(x^5/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^5 + 126*a*b^4*x^4 + 84*a^2*b^3*x^3 + 36*a^3*b^2*x^2 + 9*a^4*b*x + a^5)/((b*x + a)^9*b^6)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \frac {x^5}{(a+b x)^{10}} \, dx=\frac {\frac {a}{{\left (a+b\,x\right )}^5}-\frac {1}{4\,{\left (a+b\,x\right )}^4}-\frac {5\,a^2}{3\,{\left (a+b\,x\right )}^6}+\frac {10\,a^3}{7\,{\left (a+b\,x\right )}^7}-\frac {5\,a^4}{8\,{\left (a+b\,x\right )}^8}+\frac {a^5}{9\,{\left (a+b\,x\right )}^9}}{b^6} \]

[In]

int(x^5/(a + b*x)^10,x)

[Out]

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