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]

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)

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Rubi [A]  time = 0.0389887, antiderivative size = 77, normalized size of antiderivative = 1., 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 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]]

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 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]

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 = 0.021162, size = 68, normalized size = 0.88 \[ -\frac{84 a^2 b^3 x^6+36 a^3 b^2 x^4+9 a^4 b x^2+a^5+126 a b^4 x^8+126 b^5 x^{10}}{1008 b^6 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^11/(a + b*x^2)^10,x]

[Out]

-(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*(a + b*x^2)^9)

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Maple [A]  time = 0.009, size = 99, normalized size = 1.3 \begin{align*} -{\frac{5\,{a}^{4}}{16\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{5\,{a}^{2}}{6\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{{a}^{5}}{18\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{a}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{5\,{a}^{3}}{7\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}

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

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Maxima [B]  time = 2.36448, size = 212, normalized size = 2.75 \begin{align*} -\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 )}} \end{align*}

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)

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Fricas [B]  time = 1.24011, size = 350, normalized size = 4.55 \begin{align*} -\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 )}} \end{align*}

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)

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Sympy [B]  time = 7.2542, size = 167, normalized size = 2.17 \begin{align*} - \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}} \end{align*}

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)

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Giac [A]  time = 2.57667, size = 89, normalized size = 1.16 \begin{align*} -\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}} \end{align*}

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)