\(\int \frac {x^6}{(a+b x)^{10}} \, dx\) [228]

Optimal result

Integrand size = 11, antiderivative size = 52 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=\frac {x^7}{9 a (a+b x)^9}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {x^7}{252 a^3 (a+b x)^7} \]

[Out]

1/9*x^7/a/(b*x+a)^9+1/36*x^7/a^2/(b*x+a)^8+1/252*x^7/a^3/(b*x+a)^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=\frac {x^7}{252 a^3 (a+b x)^7}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {x^7}{9 a (a+b x)^9} \]

[In]

Int[x^6/(a + b*x)^10,x]

[Out]

x^7/(9*a*(a + b*x)^9) + x^7/(36*a^2*(a + b*x)^8) + x^7/(252*a^3*(a + b*x)^7)

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^7}{9 a (a+b x)^9}+\frac {2 \int \frac {x^6}{(a+b x)^9} \, dx}{9 a} \\ & = \frac {x^7}{9 a (a+b x)^9}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {\int \frac {x^6}{(a+b x)^8} \, dx}{36 a^2} \\ & = \frac {x^7}{9 a (a+b x)^9}+\frac {x^7}{36 a^2 (a+b x)^8}+\frac {x^7}{252 a^3 (a+b x)^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.44 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=-\frac {a^6+9 a^5 b x+36 a^4 b^2 x^2+84 a^3 b^3 x^3+126 a^2 b^4 x^4+126 a b^5 x^5+84 b^6 x^6}{252 b^7 (a+b x)^9} \]

[In]

Integrate[x^6/(a + b*x)^10,x]

[Out]

-1/252*(a^6 + 9*a^5*b*x + 36*a^4*b^2*x^2 + 84*a^3*b^3*x^3 + 126*a^2*b^4*x^4 + 126*a*b^5*x^5 + 84*b^6*x^6)/(b^7
*(a + b*x)^9)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.42

method	result	size
gosper	\(-\frac {84 b^{6} x^{6}+126 a \,x^{5} b^{5}+126 a^{2} x^{4} b^{4}+84 a^{3} x^{3} b^{3}+36 a^{4} x^{2} b^{2}+9 a^{5} x b +a^{6}}{252 \left (b x +a \right )^{9} b^{7}}\)	\(74\)
norman	\(\frac {-\frac {x^{6}}{3 b}-\frac {a \,x^{5}}{2 b^{2}}-\frac {a^{2} x^{4}}{2 b^{3}}-\frac {a^{3} x^{3}}{3 b^{4}}-\frac {a^{4} x^{2}}{7 b^{5}}-\frac {a^{5} x}{28 b^{6}}-\frac {a^{6}}{252 b^{7}}}{\left (b x +a \right )^{9}}\)	\(77\)
risch	\(\frac {-\frac {x^{6}}{3 b}-\frac {a \,x^{5}}{2 b^{2}}-\frac {a^{2} x^{4}}{2 b^{3}}-\frac {a^{3} x^{3}}{3 b^{4}}-\frac {a^{4} x^{2}}{7 b^{5}}-\frac {a^{5} x}{28 b^{6}}-\frac {a^{6}}{252 b^{7}}}{\left (b x +a \right )^{9}}\)	\(77\)
parallelrisch	\(\frac {-84 b^{8} x^{6}-126 a \,b^{7} x^{5}-126 a^{2} b^{6} x^{4}-84 a^{3} b^{5} x^{3}-36 a^{4} b^{4} x^{2}-9 a^{5} b^{3} x -b^{2} a^{6}}{252 b^{9} \left (b x +a \right )^{9}}\)	\(81\)
default	\(-\frac {a^{6}}{9 b^{7} \left (b x +a \right )^{9}}-\frac {15 a^{4}}{7 b^{7} \left (b x +a \right )^{7}}+\frac {10 a^{3}}{3 b^{7} \left (b x +a \right )^{6}}-\frac {3 a^{2}}{b^{7} \left (b x +a \right )^{5}}+\frac {3 a}{2 b^{7} \left (b x +a \right )^{4}}-\frac {1}{3 b^{7} \left (b x +a \right )^{3}}+\frac {3 a^{5}}{4 b^{7} \left (b x +a \right )^{8}}\)	\(102\)

[In]

int(x^6/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

-1/252*(84*b^6*x^6+126*a*b^5*x^5+126*a^2*b^4*x^4+84*a^3*b^3*x^3+36*a^4*b^2*x^2+9*a^5*b*x+a^6)/(b*x+a)^9/b^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (46) = 92\).

Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 3.15 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=-\frac {84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \, {\left (b^{16} x^{9} + 9 \, a b^{15} x^{8} + 36 \, a^{2} b^{14} x^{7} + 84 \, a^{3} b^{13} x^{6} + 126 \, a^{4} b^{12} x^{5} + 126 \, a^{5} b^{11} x^{4} + 84 \, a^{6} b^{10} x^{3} + 36 \, a^{7} b^{9} x^{2} + 9 \, a^{8} b^{8} x + a^{9} b^{7}\right )}} \]

[In]

integrate(x^6/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*x^6 + 126*a*b^5*x^5 + 126*a^2*b^4*x^4 + 84*a^3*b^3*x^3 + 36*a^4*b^2*x^2 + 9*a^5*b*x + a^6)/(b^1
6*x^9 + 9*a*b^15*x^8 + 36*a^2*b^14*x^7 + 84*a^3*b^13*x^6 + 126*a^4*b^12*x^5 + 126*a^5*b^11*x^4 + 84*a^6*b^10*x
^3 + 36*a^7*b^9*x^2 + 9*a^8*b^8*x + a^9*b^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (42) = 84\).

Time = 0.49 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.37 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=\frac {- a^{6} - 9 a^{5} b x - 36 a^{4} b^{2} x^{2} - 84 a^{3} b^{3} x^{3} - 126 a^{2} b^{4} x^{4} - 126 a b^{5} x^{5} - 84 b^{6} x^{6}}{252 a^{9} b^{7} + 2268 a^{8} b^{8} x + 9072 a^{7} b^{9} x^{2} + 21168 a^{6} b^{10} x^{3} + 31752 a^{5} b^{11} x^{4} + 31752 a^{4} b^{12} x^{5} + 21168 a^{3} b^{13} x^{6} + 9072 a^{2} b^{14} x^{7} + 2268 a b^{15} x^{8} + 252 b^{16} x^{9}} \]

[In]

integrate(x**6/(b*x+a)**10,x)

[Out]

(-a**6 - 9*a**5*b*x - 36*a**4*b**2*x**2 - 84*a**3*b**3*x**3 - 126*a**2*b**4*x**4 - 126*a*b**5*x**5 - 84*b**6*x
**6)/(252*a**9*b**7 + 2268*a**8*b**8*x + 9072*a**7*b**9*x**2 + 21168*a**6*b**10*x**3 + 31752*a**5*b**11*x**4 +
31752*a**4*b**12*x**5 + 21168*a**3*b**13*x**6 + 9072*a**2*b**14*x**7 + 2268*a*b**15*x**8 + 252*b**16*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (46) = 92\).

Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 3.15 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=-\frac {84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \, {\left (b^{16} x^{9} + 9 \, a b^{15} x^{8} + 36 \, a^{2} b^{14} x^{7} + 84 \, a^{3} b^{13} x^{6} + 126 \, a^{4} b^{12} x^{5} + 126 \, a^{5} b^{11} x^{4} + 84 \, a^{6} b^{10} x^{3} + 36 \, a^{7} b^{9} x^{2} + 9 \, a^{8} b^{8} x + a^{9} b^{7}\right )}} \]

[In]

integrate(x^6/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/252*(84*b^6*x^6 + 126*a*b^5*x^5 + 126*a^2*b^4*x^4 + 84*a^3*b^3*x^3 + 36*a^4*b^2*x^2 + 9*a^5*b*x + a^6)/(b^1
6*x^9 + 9*a*b^15*x^8 + 36*a^2*b^14*x^7 + 84*a^3*b^13*x^6 + 126*a^4*b^12*x^5 + 126*a^5*b^11*x^4 + 84*a^6*b^10*x
^3 + 36*a^7*b^9*x^2 + 9*a^8*b^8*x + a^9*b^7)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.40 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=-\frac {84 \, b^{6} x^{6} + 126 \, a b^{5} x^{5} + 126 \, a^{2} b^{4} x^{4} + 84 \, a^{3} b^{3} x^{3} + 36 \, a^{4} b^{2} x^{2} + 9 \, a^{5} b x + a^{6}}{252 \, {\left (b x + a\right )}^{9} b^{7}} \]

[In]

integrate(x^6/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6 + 126*a*b^5*x^5 + 126*a^2*b^4*x^4 + 84*a^3*b^3*x^3 + 36*a^4*b^2*x^2 + 9*a^5*b*x + a^6)/((b*
x + a)^9*b^7)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.63 \[ \int \frac {x^6}{(a+b x)^{10}} \, dx=-\frac {\frac {1}{3\,{\left (a+b\,x\right )}^3}-\frac {3\,a}{2\,{\left (a+b\,x\right )}^4}+\frac {3\,a^2}{{\left (a+b\,x\right )}^5}-\frac {10\,a^3}{3\,{\left (a+b\,x\right )}^6}+\frac {15\,a^4}{7\,{\left (a+b\,x\right )}^7}-\frac {3\,a^5}{4\,{\left (a+b\,x\right )}^8}+\frac {a^6}{9\,{\left (a+b\,x\right )}^9}}{b^7} \]

[In]

int(x^6/(a + b*x)^10,x)

[Out]

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