3.199 \(\int \frac {x^{11}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=77 \[ \frac {x^{12}}{1008 a^4 \left (a+b x^2\right )^6}+\frac {x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac {x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac {x^{12}}{18 a \left (a+b x^2\right )^9} \]

[Out]

1/18*x^12/a/(b*x^2+a)^9+1/48*x^12/a^2/(b*x^2+a)^8+1/168*x^12/a^3/(b*x^2+a)^7+1/1008*x^12/a^4/(b*x^2+a)^6

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac {x^{12}}{1008 a^4 \left (a+b x^2\right )^6}+\frac {x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac {x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac {x^{12}}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^11/(a + b*x^2)^10,x]

[Out]

x^12/(18*a*(a + b*x^2)^9) + x^12/(48*a^2*(a + b*x^2)^8) + x^12/(168*a^3*(a + b*x^2)^7) + x^12/(1008*a^4*(a + b
*x^2)^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 45

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])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {x^{11}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac {x^{12}}{18 a \left (a+b x^2\right )^9}+\frac {\operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^9} \, dx,x,x^2\right )}{6 a}\\ &=\frac {x^{12}}{18 a \left (a+b x^2\right )^9}+\frac {x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac {\operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^8} \, dx,x,x^2\right )}{24 a^2}\\ &=\frac {x^{12}}{18 a \left (a+b x^2\right )^9}+\frac {x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac {x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac {\operatorname {Subst}\left (\int \frac {x^5}{(a+b x)^7} \, dx,x,x^2\right )}{168 a^3}\\ &=\frac {x^{12}}{18 a \left (a+b x^2\right )^9}+\frac {x^{12}}{48 a^2 \left (a+b x^2\right )^8}+\frac {x^{12}}{168 a^3 \left (a+b x^2\right )^7}+\frac {x^{12}}{1008 a^4 \left (a+b x^2\right )^6}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 68, normalized size = 0.88 \[ -\frac {a^5+9 a^4 b x^2+36 a^3 b^2 x^4+84 a^2 b^3 x^6+126 a b^4 x^8+126 b^5 x^{10}}{1008 b^6 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^2)^10,x]

[Out]

-1/1008*(a^5 + 9*a^4*b*x^2 + 36*a^3*b^2*x^4 + 84*a^2*b^3*x^6 + 126*a*b^4*x^8 + 126*b^5*x^10)/(b^6*(a + b*x^2)^
9)

________________________________________________________________________________________

fricas [B]  time = 0.56, size = 157, normalized size = 2.04 \[ -\frac {126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \, {\left (b^{15} x^{18} + 9 \, a b^{14} x^{16} + 36 \, a^{2} b^{13} x^{14} + 84 \, a^{3} b^{12} x^{12} + 126 \, a^{4} b^{11} x^{10} + 126 \, a^{5} b^{10} x^{8} + 84 \, a^{6} b^{9} x^{6} + 36 \, a^{7} b^{8} x^{4} + 9 \, a^{8} b^{7} x^{2} + a^{9} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/1008*(126*b^5*x^10 + 126*a*b^4*x^8 + 84*a^2*b^3*x^6 + 36*a^3*b^2*x^4 + 9*a^4*b*x^2 + a^5)/(b^15*x^18 + 9*a*
b^14*x^16 + 36*a^2*b^13*x^14 + 84*a^3*b^12*x^12 + 126*a^4*b^11*x^10 + 126*a^5*b^10*x^8 + 84*a^6*b^9*x^6 + 36*a
^7*b^8*x^4 + 9*a^8*b^7*x^2 + a^9*b^6)

________________________________________________________________________________________

giac [A]  time = 0.63, size = 66, normalized size = 0.86 \[ -\frac {126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \, {\left (b x^{2} + a\right )}^{9} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/1008*(126*b^5*x^10 + 126*a*b^4*x^8 + 84*a^2*b^3*x^6 + 36*a^3*b^2*x^4 + 9*a^4*b*x^2 + a^5)/((b*x^2 + a)^9*b^
6)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 99, normalized size = 1.29 \[ \frac {a^{5}}{18 \left (b \,x^{2}+a \right )^{9} b^{6}}-\frac {5 a^{4}}{16 \left (b \,x^{2}+a \right )^{8} b^{6}}+\frac {5 a^{3}}{7 \left (b \,x^{2}+a \right )^{7} b^{6}}-\frac {5 a^{2}}{6 \left (b \,x^{2}+a \right )^{6} b^{6}}+\frac {a}{2 \left (b \,x^{2}+a \right )^{5} b^{6}}-\frac {1}{8 \left (b \,x^{2}+a \right )^{4} b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(b*x^2+a)^10,x)

[Out]

-5/16*a^4/b^6/(b*x^2+a)^8+5/7*a^3/b^6/(b*x^2+a)^7+1/2*a/b^6/(b*x^2+a)^5-1/8/b^6/(b*x^2+a)^4-5/6*a^2/b^6/(b*x^2
+a)^6+1/18*a^5/b^6/(b*x^2+a)^9

________________________________________________________________________________________

maxima [B]  time = 1.43, size = 157, normalized size = 2.04 \[ -\frac {126 \, b^{5} x^{10} + 126 \, a b^{4} x^{8} + 84 \, a^{2} b^{3} x^{6} + 36 \, a^{3} b^{2} x^{4} + 9 \, a^{4} b x^{2} + a^{5}}{1008 \, {\left (b^{15} x^{18} + 9 \, a b^{14} x^{16} + 36 \, a^{2} b^{13} x^{14} + 84 \, a^{3} b^{12} x^{12} + 126 \, a^{4} b^{11} x^{10} + 126 \, a^{5} b^{10} x^{8} + 84 \, a^{6} b^{9} x^{6} + 36 \, a^{7} b^{8} x^{4} + 9 \, a^{8} b^{7} x^{2} + a^{9} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^11/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/1008*(126*b^5*x^10 + 126*a*b^4*x^8 + 84*a^2*b^3*x^6 + 36*a^3*b^2*x^4 + 9*a^4*b*x^2 + a^5)/(b^15*x^18 + 9*a*
b^14*x^16 + 36*a^2*b^13*x^14 + 84*a^3*b^12*x^12 + 126*a^4*b^11*x^10 + 126*a^5*b^10*x^8 + 84*a^6*b^9*x^6 + 36*a
^7*b^8*x^4 + 9*a^8*b^7*x^2 + a^9*b^6)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 159, normalized size = 2.06 \[ -\frac {a^5+9\,a^4\,b\,x^2+36\,a^3\,b^2\,x^4+84\,a^2\,b^3\,x^6+126\,a\,b^4\,x^8+126\,b^5\,x^{10}}{1008\,a^9\,b^6+9072\,a^8\,b^7\,x^2+36288\,a^7\,b^8\,x^4+84672\,a^6\,b^9\,x^6+127008\,a^5\,b^{10}\,x^8+127008\,a^4\,b^{11}\,x^{10}+84672\,a^3\,b^{12}\,x^{12}+36288\,a^2\,b^{13}\,x^{14}+9072\,a\,b^{14}\,x^{16}+1008\,b^{15}\,x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^11/(a + b*x^2)^10,x)

[Out]

-(a^5 + 126*b^5*x^10 + 9*a^4*b*x^2 + 126*a*b^4*x^8 + 36*a^3*b^2*x^4 + 84*a^2*b^3*x^6)/(1008*a^9*b^6 + 1008*b^1
5*x^18 + 9072*a*b^14*x^16 + 9072*a^8*b^7*x^2 + 36288*a^7*b^8*x^4 + 84672*a^6*b^9*x^6 + 127008*a^5*b^10*x^8 + 1
27008*a^4*b^11*x^10 + 84672*a^3*b^12*x^12 + 36288*a^2*b^13*x^14)

________________________________________________________________________________________

sympy [B]  time = 1.18, size = 167, normalized size = 2.17 \[ \frac {- a^{5} - 9 a^{4} b x^{2} - 36 a^{3} b^{2} x^{4} - 84 a^{2} b^{3} x^{6} - 126 a b^{4} x^{8} - 126 b^{5} x^{10}}{1008 a^{9} b^{6} + 9072 a^{8} b^{7} x^{2} + 36288 a^{7} b^{8} x^{4} + 84672 a^{6} b^{9} x^{6} + 127008 a^{5} b^{10} x^{8} + 127008 a^{4} b^{11} x^{10} + 84672 a^{3} b^{12} x^{12} + 36288 a^{2} b^{13} x^{14} + 9072 a b^{14} x^{16} + 1008 b^{15} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**11/(b*x**2+a)**10,x)

[Out]

(-a**5 - 9*a**4*b*x**2 - 36*a**3*b**2*x**4 - 84*a**2*b**3*x**6 - 126*a*b**4*x**8 - 126*b**5*x**10)/(1008*a**9*
b**6 + 9072*a**8*b**7*x**2 + 36288*a**7*b**8*x**4 + 84672*a**6*b**9*x**6 + 127008*a**5*b**10*x**8 + 127008*a**
4*b**11*x**10 + 84672*a**3*b**12*x**12 + 36288*a**2*b**13*x**14 + 9072*a*b**14*x**16 + 1008*b**15*x**18)

________________________________________________________________________________________