\(\int \frac {x^7}{(a+b x)^{10}} \, dx\) [227]

Optimal result

Integrand size = 11, antiderivative size = 35 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=\frac {x^8}{9 a (a+b x)^9}+\frac {x^8}{72 a^2 (a+b x)^8} \]

[Out]

1/9*x^8/a/(b*x+a)^9+1/72*x^8/a^2/(b*x+a)^8

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=\frac {x^8}{72 a^2 (a+b x)^8}+\frac {x^8}{9 a (a+b x)^9} \]

[In]

Int[x^7/(a + b*x)^10,x]

[Out]

x^8/(9*a*(a + b*x)^9) + x^8/(72*a^2*(a + b*x)^8)

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^8}{9 a (a+b x)^9}+\frac {\int \frac {x^7}{(a+b x)^9} \, dx}{9 a} \\ & = \frac {x^8}{9 a (a+b x)^9}+\frac {x^8}{72 a^2 (a+b x)^8} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(35)=70\).

Time = 0.01 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.46 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=-\frac {a^7+9 a^6 b x+36 a^5 b^2 x^2+84 a^4 b^3 x^3+126 a^3 b^4 x^4+126 a^2 b^5 x^5+84 a b^6 x^6+36 b^7 x^7}{72 b^8 (a+b x)^9} \]

[In]

Integrate[x^7/(a + b*x)^10,x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(84\) vs. \(2(31)=62\).

Time = 0.04 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.43

method	result	size
gosper	\(-\frac {36 b^{7} x^{7}+84 a \,b^{6} x^{6}+126 a^{2} b^{5} x^{5}+126 a^{3} b^{4} x^{4}+84 a^{4} b^{3} x^{3}+36 a^{5} b^{2} x^{2}+9 a^{6} b x +a^{7}}{72 \left (b x +a \right )^{9} b^{8}}\)	\(85\)
norman	\(\frac {-\frac {x^{7}}{2 b}-\frac {7 a \,x^{6}}{6 b^{2}}-\frac {7 a^{2} x^{5}}{4 b^{3}}-\frac {7 a^{3} x^{4}}{4 b^{4}}-\frac {7 a^{4} x^{3}}{6 b^{5}}-\frac {a^{5} x^{2}}{2 b^{6}}-\frac {a^{6} x}{8 b^{7}}-\frac {a^{7}}{72 b^{8}}}{\left (b x +a \right )^{9}}\)	\(88\)
risch	\(\frac {-\frac {x^{7}}{2 b}-\frac {7 a \,x^{6}}{6 b^{2}}-\frac {7 a^{2} x^{5}}{4 b^{3}}-\frac {7 a^{3} x^{4}}{4 b^{4}}-\frac {7 a^{4} x^{3}}{6 b^{5}}-\frac {a^{5} x^{2}}{2 b^{6}}-\frac {a^{6} x}{8 b^{7}}-\frac {a^{7}}{72 b^{8}}}{\left (b x +a \right )^{9}}\)	\(88\)
parallelrisch	\(\frac {-36 b^{8} x^{7}-84 a \,b^{7} x^{6}-126 a^{2} b^{6} x^{5}-126 a^{3} b^{5} x^{4}-84 a^{4} b^{4} x^{3}-36 a^{5} b^{3} x^{2}-9 a^{6} b^{2} x -a^{7} b}{72 b^{9} \left (b x +a \right )^{9}}\)	\(90\)
default	\(\frac {a^{7}}{9 b^{8} \left (b x +a \right )^{9}}-\frac {35 a^{4}}{6 b^{8} \left (b x +a \right )^{6}}-\frac {21 a^{2}}{4 b^{8} \left (b x +a \right )^{4}}+\frac {7 a}{3 b^{8} \left (b x +a \right )^{3}}+\frac {3 a^{5}}{b^{8} \left (b x +a \right )^{7}}+\frac {7 a^{3}}{b^{8} \left (b x +a \right )^{5}}-\frac {1}{2 b^{8} \left (b x +a \right )^{2}}-\frac {7 a^{6}}{8 b^{8} \left (b x +a \right )^{8}}\)	\(117\)

[In]

int(x^7/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (31) = 62\).

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 5.00 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 126 \, a^{3} b^{4} x^{4} + 84 \, a^{4} b^{3} x^{3} + 36 \, a^{5} b^{2} x^{2} + 9 \, a^{6} b x + a^{7}}{72 \, {\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \]

[In]

integrate(x^7/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/72*(36*b^7*x^7 + 84*a*b^6*x^6 + 126*a^2*b^5*x^5 + 126*a^3*b^4*x^4 + 84*a^4*b^3*x^3 + 36*a^5*b^2*x^2 + 9*a^6
*b*x + a^7)/(b^17*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^5 + 126*a^5*b^12*x^4
+ 84*a^6*b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a^9*b^8)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (27) = 54\).

Time = 0.47 (sec) , antiderivative size = 187, normalized size of antiderivative = 5.34 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=\frac {- a^{7} - 9 a^{6} b x - 36 a^{5} b^{2} x^{2} - 84 a^{4} b^{3} x^{3} - 126 a^{3} b^{4} x^{4} - 126 a^{2} b^{5} x^{5} - 84 a b^{6} x^{6} - 36 b^{7} x^{7}}{72 a^{9} b^{8} + 648 a^{8} b^{9} x + 2592 a^{7} b^{10} x^{2} + 6048 a^{6} b^{11} x^{3} + 9072 a^{5} b^{12} x^{4} + 9072 a^{4} b^{13} x^{5} + 6048 a^{3} b^{14} x^{6} + 2592 a^{2} b^{15} x^{7} + 648 a b^{16} x^{8} + 72 b^{17} x^{9}} \]

[In]

integrate(x**7/(b*x+a)**10,x)

[Out]

(-a**7 - 9*a**6*b*x - 36*a**5*b**2*x**2 - 84*a**4*b**3*x**3 - 126*a**3*b**4*x**4 - 126*a**2*b**5*x**5 - 84*a*b
**6*x**6 - 36*b**7*x**7)/(72*a**9*b**8 + 648*a**8*b**9*x + 2592*a**7*b**10*x**2 + 6048*a**6*b**11*x**3 + 9072*
a**5*b**12*x**4 + 9072*a**4*b**13*x**5 + 6048*a**3*b**14*x**6 + 2592*a**2*b**15*x**7 + 648*a*b**16*x**8 + 72*b
**17*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (31) = 62\).

Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 5.00 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 126 \, a^{3} b^{4} x^{4} + 84 \, a^{4} b^{3} x^{3} + 36 \, a^{5} b^{2} x^{2} + 9 \, a^{6} b x + a^{7}}{72 \, {\left (b^{17} x^{9} + 9 \, a b^{16} x^{8} + 36 \, a^{2} b^{15} x^{7} + 84 \, a^{3} b^{14} x^{6} + 126 \, a^{4} b^{13} x^{5} + 126 \, a^{5} b^{12} x^{4} + 84 \, a^{6} b^{11} x^{3} + 36 \, a^{7} b^{10} x^{2} + 9 \, a^{8} b^{9} x + a^{9} b^{8}\right )}} \]

[In]

integrate(x^7/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/72*(36*b^7*x^7 + 84*a*b^6*x^6 + 126*a^2*b^5*x^5 + 126*a^3*b^4*x^4 + 84*a^4*b^3*x^3 + 36*a^5*b^2*x^2 + 9*a^6
*b*x + a^7)/(b^17*x^9 + 9*a*b^16*x^8 + 36*a^2*b^15*x^7 + 84*a^3*b^14*x^6 + 126*a^4*b^13*x^5 + 126*a^5*b^12*x^4
+ 84*a^6*b^11*x^3 + 36*a^7*b^10*x^2 + 9*a^8*b^9*x + a^9*b^8)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (31) = 62\).

Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.40 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{7} x^{7} + 84 \, a b^{6} x^{6} + 126 \, a^{2} b^{5} x^{5} + 126 \, a^{3} b^{4} x^{4} + 84 \, a^{4} b^{3} x^{3} + 36 \, a^{5} b^{2} x^{2} + 9 \, a^{6} b x + a^{7}}{72 \, {\left (b x + a\right )}^{9} b^{8}} \]

[In]

integrate(x^7/(b*x+a)^10,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.63 \[ \int \frac {x^7}{(a+b x)^{10}} \, dx=\frac {x^8\,\left (9\,a+b\,x\right )}{72\,a^2\,{\left (a+b\,x\right )}^9} \]

[In]

int(x^7/(a + b*x)^10,x)

[Out]

(x^8*(9*a + b*x))/(72*a^2*(a + b*x)^9)