3.198 \(\int \frac{x^{13}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=58 \[ \frac{x^{14}}{504 a^3 \left (a+b x^2\right )^7}+\frac{x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac{x^{14}}{18 a \left (a+b x^2\right )^9} \]

[Out]

x^14/(18*a*(a + b*x^2)^9) + x^14/(72*a^2*(a + b*x^2)^8) + x^14/(504*a^3*(a + b*x^2)^7)

________________________________________________________________________________________

Rubi [A]  time = 0.0269605, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{x^{14}}{504 a^3 \left (a+b x^2\right )^7}+\frac{x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac{x^{14}}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^13/(a + b*x^2)^10,x]

[Out]

x^14/(18*a*(a + b*x^2)^9) + x^14/(72*a^2*(a + b*x^2)^8) + x^14/(504*a^3*(a + b*x^2)^7)

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^{13}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^6}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{x^{14}}{18 a \left (a+b x^2\right )^9}+\frac{\operatorname{Subst}\left (\int \frac{x^6}{(a+b x)^9} \, dx,x,x^2\right )}{9 a}\\ &=\frac{x^{14}}{18 a \left (a+b x^2\right )^9}+\frac{x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac{\operatorname{Subst}\left (\int \frac{x^6}{(a+b x)^8} \, dx,x,x^2\right )}{72 a^2}\\ &=\frac{x^{14}}{18 a \left (a+b x^2\right )^9}+\frac{x^{14}}{72 a^2 \left (a+b x^2\right )^8}+\frac{x^{14}}{504 a^3 \left (a+b x^2\right )^7}\\ \end{align*}

Mathematica [A]  time = 0.017492, size = 79, normalized size = 1.36 \[ -\frac{126 a^2 b^4 x^8+84 a^3 b^3 x^6+36 a^4 b^2 x^4+9 a^5 b x^2+a^6+126 a b^5 x^{10}+84 b^6 x^{12}}{504 b^7 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13/(a + b*x^2)^10,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 116, normalized size = 2. \begin{align*} -{\frac{{a}^{6}}{18\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{5\,{a}^{3}}{3\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{3\,{a}^{5}}{8\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{1}{6\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{3\,{a}^{2}}{2\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3\,a}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{15\,{a}^{4}}{14\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13/(b*x^2+a)^10,x)

[Out]

-1/18*a^6/b^7/(b*x^2+a)^9+5/3*a^3/b^7/(b*x^2+a)^6+3/8*a^5/b^7/(b*x^2+a)^8-1/6/b^7/(b*x^2+a)^3-3/2*a^2/b^7/(b*x
^2+a)^5+3/4/b^7*a/(b*x^2+a)^4-15/14*a^4/b^7/(b*x^2+a)^7

________________________________________________________________________________________

Maxima [B]  time = 2.76735, size = 227, normalized size = 3.91 \begin{align*} -\frac{84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \,{\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/
(b^16*x^18 + 9*a*b^15*x^16 + 36*a^2*b^14*x^14 + 84*a^3*b^13*x^12 + 126*a^4*b^12*x^10 + 126*a^5*b^11*x^8 + 84*a
^6*b^10*x^6 + 36*a^7*b^9*x^4 + 9*a^8*b^8*x^2 + a^9*b^7)

________________________________________________________________________________________

Fricas [B]  time = 1.16262, size = 374, normalized size = 6.45 \begin{align*} -\frac{84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \,{\left (b^{16} x^{18} + 9 \, a b^{15} x^{16} + 36 \, a^{2} b^{14} x^{14} + 84 \, a^{3} b^{13} x^{12} + 126 \, a^{4} b^{12} x^{10} + 126 \, a^{5} b^{11} x^{8} + 84 \, a^{6} b^{10} x^{6} + 36 \, a^{7} b^{9} x^{4} + 9 \, a^{8} b^{8} x^{2} + a^{9} b^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/504*(84*b^6*x^12 + 126*a*b^5*x^10 + 126*a^2*b^4*x^8 + 84*a^3*b^3*x^6 + 36*a^4*b^2*x^4 + 9*a^5*b*x^2 + a^6)/
(b^16*x^18 + 9*a*b^15*x^16 + 36*a^2*b^14*x^14 + 84*a^3*b^13*x^12 + 126*a^4*b^12*x^10 + 126*a^5*b^11*x^8 + 84*a
^6*b^10*x^6 + 36*a^7*b^9*x^4 + 9*a^8*b^8*x^2 + a^9*b^7)

________________________________________________________________________________________

Sympy [B]  time = 7.59069, size = 178, normalized size = 3.07 \begin{align*} - \frac{a^{6} + 9 a^{5} b x^{2} + 36 a^{4} b^{2} x^{4} + 84 a^{3} b^{3} x^{6} + 126 a^{2} b^{4} x^{8} + 126 a b^{5} x^{10} + 84 b^{6} x^{12}}{504 a^{9} b^{7} + 4536 a^{8} b^{8} x^{2} + 18144 a^{7} b^{9} x^{4} + 42336 a^{6} b^{10} x^{6} + 63504 a^{5} b^{11} x^{8} + 63504 a^{4} b^{12} x^{10} + 42336 a^{3} b^{13} x^{12} + 18144 a^{2} b^{14} x^{14} + 4536 a b^{15} x^{16} + 504 b^{16} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13/(b*x**2+a)**10,x)

[Out]

-(a**6 + 9*a**5*b*x**2 + 36*a**4*b**2*x**4 + 84*a**3*b**3*x**6 + 126*a**2*b**4*x**8 + 126*a*b**5*x**10 + 84*b*
*6*x**12)/(504*a**9*b**7 + 4536*a**8*b**8*x**2 + 18144*a**7*b**9*x**4 + 42336*a**6*b**10*x**6 + 63504*a**5*b**
11*x**8 + 63504*a**4*b**12*x**10 + 42336*a**3*b**13*x**12 + 18144*a**2*b**14*x**14 + 4536*a*b**15*x**16 + 504*
b**16*x**18)

________________________________________________________________________________________

Giac [A]  time = 1.84436, size = 104, normalized size = 1.79 \begin{align*} -\frac{84 \, b^{6} x^{12} + 126 \, a b^{5} x^{10} + 126 \, a^{2} b^{4} x^{8} + 84 \, a^{3} b^{3} x^{6} + 36 \, a^{4} b^{2} x^{4} + 9 \, a^{5} b x^{2} + a^{6}}{504 \,{\left (b x^{2} + a\right )}^{9} b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13/(b*x^2+a)^10,x, algorithm="giac")

[Out]

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