3.197 \(\int \frac{x^{15}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=39 \[ \frac{x^{16}}{144 a^2 \left (a+b x^2\right )^8}+\frac{x^{16}}{18 a \left (a+b x^2\right )^9} \]

[Out]

x^16/(18*a*(a + b*x^2)^9) + x^16/(144*a^2*(a + b*x^2)^8)

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Rubi [A]  time = 0.0173839, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {266, 45, 37} \[ \frac{x^{16}}{144 a^2 \left (a+b x^2\right )^8}+\frac{x^{16}}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^15/(a + b*x^2)^10,x]

[Out]

x^16/(18*a*(a + b*x^2)^9) + x^16/(144*a^2*(a + b*x^2)^8)

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^{15}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{x^{16}}{18 a \left (a+b x^2\right )^9}+\frac{\operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^9} \, dx,x,x^2\right )}{18 a}\\ &=\frac{x^{16}}{18 a \left (a+b x^2\right )^9}+\frac{x^{16}}{144 a^2 \left (a+b x^2\right )^8}\\ \end{align*}

Mathematica [B]  time = 0.0151744, size = 90, normalized size = 2.31 \[ -\frac{126 a^2 b^5 x^{10}+126 a^3 b^4 x^8+84 a^4 b^3 x^6+36 a^5 b^2 x^4+9 a^6 b x^2+a^7+84 a b^6 x^{12}+36 b^7 x^{14}}{144 b^8 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[x^15/(a + b*x^2)^10,x]

[Out]

-(a^7 + 9*a^6*b*x^2 + 36*a^5*b^2*x^4 + 84*a^4*b^3*x^6 + 126*a^3*b^4*x^8 + 126*a^2*b^5*x^10 + 84*a*b^6*x^12 + 3
6*b^7*x^14)/(144*b^8*(a + b*x^2)^9)

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Maple [B]  time = 0.008, size = 133, normalized size = 3.4 \begin{align*}{\frac{{a}^{7}}{18\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{7\,a}{6\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{7\,{a}^{6}}{16\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{1}{4\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{21\,{a}^{2}}{8\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{4}}}+{\frac{7\,{a}^{3}}{2\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{35\,{a}^{4}}{12\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{3\,{a}^{5}}{2\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^15/(b*x^2+a)^10,x)

[Out]

1/18/b^8*a^7/(b*x^2+a)^9+7/6/b^8*a/(b*x^2+a)^3-7/16*a^6/b^8/(b*x^2+a)^8-1/4/b^8/(b*x^2+a)^2-21/8*a^2/b^8/(b*x^
2+a)^4+7/2*a^3/b^8/(b*x^2+a)^5-35/12*a^4/b^8/(b*x^2+a)^6+3/2*a^5/b^8/(b*x^2+a)^7

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Maxima [B]  time = 2.72399, size = 242, normalized size = 6.21 \begin{align*} -\frac{36 \, b^{7} x^{14} + 84 \, a b^{6} x^{12} + 126 \, a^{2} b^{5} x^{10} + 126 \, a^{3} b^{4} x^{8} + 84 \, a^{4} b^{3} x^{6} + 36 \, a^{5} b^{2} x^{4} + 9 \, a^{6} b x^{2} + a^{7}}{144 \,{\left (b^{17} x^{18} + 9 \, a b^{16} x^{16} + 36 \, a^{2} b^{15} x^{14} + 84 \, a^{3} b^{14} x^{12} + 126 \, a^{4} b^{13} x^{10} + 126 \, a^{5} b^{12} x^{8} + 84 \, a^{6} b^{11} x^{6} + 36 \, a^{7} b^{10} x^{4} + 9 \, a^{8} b^{9} x^{2} + a^{9} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/144*(36*b^7*x^14 + 84*a*b^6*x^12 + 126*a^2*b^5*x^10 + 126*a^3*b^4*x^8 + 84*a^4*b^3*x^6 + 36*a^5*b^2*x^4 + 9
*a^6*b*x^2 + a^7)/(b^17*x^18 + 9*a*b^16*x^16 + 36*a^2*b^15*x^14 + 84*a^3*b^14*x^12 + 126*a^4*b^13*x^10 + 126*a
^5*b^12*x^8 + 84*a^6*b^11*x^6 + 36*a^7*b^10*x^4 + 9*a^8*b^9*x^2 + a^9*b^8)

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Fricas [B]  time = 1.22697, size = 400, normalized size = 10.26 \begin{align*} -\frac{36 \, b^{7} x^{14} + 84 \, a b^{6} x^{12} + 126 \, a^{2} b^{5} x^{10} + 126 \, a^{3} b^{4} x^{8} + 84 \, a^{4} b^{3} x^{6} + 36 \, a^{5} b^{2} x^{4} + 9 \, a^{6} b x^{2} + a^{7}}{144 \,{\left (b^{17} x^{18} + 9 \, a b^{16} x^{16} + 36 \, a^{2} b^{15} x^{14} + 84 \, a^{3} b^{14} x^{12} + 126 \, a^{4} b^{13} x^{10} + 126 \, a^{5} b^{12} x^{8} + 84 \, a^{6} b^{11} x^{6} + 36 \, a^{7} b^{10} x^{4} + 9 \, a^{8} b^{9} x^{2} + a^{9} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

-1/144*(36*b^7*x^14 + 84*a*b^6*x^12 + 126*a^2*b^5*x^10 + 126*a^3*b^4*x^8 + 84*a^4*b^3*x^6 + 36*a^5*b^2*x^4 + 9
*a^6*b*x^2 + a^7)/(b^17*x^18 + 9*a*b^16*x^16 + 36*a^2*b^15*x^14 + 84*a^3*b^14*x^12 + 126*a^4*b^13*x^10 + 126*a
^5*b^12*x^8 + 84*a^6*b^11*x^6 + 36*a^7*b^10*x^4 + 9*a^8*b^9*x^2 + a^9*b^8)

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Sympy [B]  time = 7.67083, size = 190, normalized size = 4.87 \begin{align*} - \frac{a^{7} + 9 a^{6} b x^{2} + 36 a^{5} b^{2} x^{4} + 84 a^{4} b^{3} x^{6} + 126 a^{3} b^{4} x^{8} + 126 a^{2} b^{5} x^{10} + 84 a b^{6} x^{12} + 36 b^{7} x^{14}}{144 a^{9} b^{8} + 1296 a^{8} b^{9} x^{2} + 5184 a^{7} b^{10} x^{4} + 12096 a^{6} b^{11} x^{6} + 18144 a^{5} b^{12} x^{8} + 18144 a^{4} b^{13} x^{10} + 12096 a^{3} b^{14} x^{12} + 5184 a^{2} b^{15} x^{14} + 1296 a b^{16} x^{16} + 144 b^{17} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**15/(b*x**2+a)**10,x)

[Out]

-(a**7 + 9*a**6*b*x**2 + 36*a**5*b**2*x**4 + 84*a**4*b**3*x**6 + 126*a**3*b**4*x**8 + 126*a**2*b**5*x**10 + 84
*a*b**6*x**12 + 36*b**7*x**14)/(144*a**9*b**8 + 1296*a**8*b**9*x**2 + 5184*a**7*b**10*x**4 + 12096*a**6*b**11*
x**6 + 18144*a**5*b**12*x**8 + 18144*a**4*b**13*x**10 + 12096*a**3*b**14*x**12 + 5184*a**2*b**15*x**14 + 1296*
a*b**16*x**16 + 144*b**17*x**18)

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Giac [B]  time = 2.31156, size = 119, normalized size = 3.05 \begin{align*} -\frac{36 \, b^{7} x^{14} + 84 \, a b^{6} x^{12} + 126 \, a^{2} b^{5} x^{10} + 126 \, a^{3} b^{4} x^{8} + 84 \, a^{4} b^{3} x^{6} + 36 \, a^{5} b^{2} x^{4} + 9 \, a^{6} b x^{2} + a^{7}}{144 \,{\left (b x^{2} + a\right )}^{9} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^15/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-1/144*(36*b^7*x^14 + 84*a*b^6*x^12 + 126*a^2*b^5*x^10 + 126*a^3*b^4*x^8 + 84*a^4*b^3*x^6 + 36*a^5*b^2*x^4 + 9
*a^6*b*x^2 + a^7)/((b*x^2 + a)^9*b^8)