3.217 \(\int \frac{x^8}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=202 \[ \frac{35 x}{65536 a^5 b^4 \left (a+b x^2\right )}+\frac{35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{35 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{11/2} b^{9/2}}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}-\frac{x^7}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^7/(18*b*(a + b*x^2)^9) - (7*x^5)/(288*b^2*(a + b*x^2)^8) - (5*x^3)/(576*b^3*(a + b*x^2)^7) - (5*x)/(2304*b^
4*(a + b*x^2)^6) + x/(4608*a*b^4*(a + b*x^2)^5) + x/(4096*a^2*b^4*(a + b*x^2)^4) + (7*x)/(24576*a^3*b^4*(a + b
*x^2)^3) + (35*x)/(98304*a^4*b^4*(a + b*x^2)^2) + (35*x)/(65536*a^5*b^4*(a + b*x^2)) + (35*ArcTan[(Sqrt[b]*x)/
Sqrt[a]])/(65536*a^(11/2)*b^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 0.115632, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 199, 205} \[ \frac{35 x}{65536 a^5 b^4 \left (a+b x^2\right )}+\frac{35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{35 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{11/2} b^{9/2}}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}-\frac{x^7}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^8/(a + b*x^2)^10,x]

[Out]

-x^7/(18*b*(a + b*x^2)^9) - (7*x^5)/(288*b^2*(a + b*x^2)^8) - (5*x^3)/(576*b^3*(a + b*x^2)^7) - (5*x)/(2304*b^
4*(a + b*x^2)^6) + x/(4608*a*b^4*(a + b*x^2)^5) + x/(4096*a^2*b^4*(a + b*x^2)^4) + (7*x)/(24576*a^3*b^4*(a + b
*x^2)^3) + (35*x)/(98304*a^4*b^4*(a + b*x^2)^2) + (35*x)/(65536*a^5*b^4*(a + b*x^2)) + (35*ArcTan[(Sqrt[b]*x)/
Sqrt[a]])/(65536*a^(11/2)*b^(9/2))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^8}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}+\frac{7 \int \frac{x^6}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}+\frac{35 \int \frac{x^4}{\left (a+b x^2\right )^8} \, dx}{288 b^2}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}+\frac{5 \int \frac{x^2}{\left (a+b x^2\right )^7} \, dx}{192 b^3}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{5 \int \frac{1}{\left (a+b x^2\right )^6} \, dx}{2304 b^4}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac{\int \frac{1}{\left (a+b x^2\right )^5} \, dx}{512 a b^4}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{7 \int \frac{1}{\left (a+b x^2\right )^4} \, dx}{4096 a^2 b^4}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{35 \int \frac{1}{\left (a+b x^2\right )^3} \, dx}{24576 a^3 b^4}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac{35 \int \frac{1}{\left (a+b x^2\right )^2} \, dx}{32768 a^4 b^4}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac{35 x}{65536 a^5 b^4 \left (a+b x^2\right )}+\frac{35 \int \frac{1}{a+b x^2} \, dx}{65536 a^5 b^4}\\ &=-\frac{x^7}{18 b \left (a+b x^2\right )^9}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac{35 x}{65536 a^5 b^4 \left (a+b x^2\right )}+\frac{35 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{11/2} b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0562671, size = 138, normalized size = 0.68 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (10458 a^2 b^6 x^{12}+23202 a^3 b^5 x^{10}+32768 a^4 b^4 x^8-23202 a^5 b^3 x^6-10458 a^6 b^2 x^4-2730 a^7 b x^2-315 a^8+2730 a b^7 x^{14}+315 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+315 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{589824 a^{11/2} b^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-315*a^8 - 2730*a^7*b*x^2 - 10458*a^6*b^2*x^4 - 23202*a^5*b^3*x^6 + 32768*a^4*b^4*x^8 + 2
3202*a^3*b^5*x^10 + 10458*a^2*b^6*x^12 + 2730*a*b^7*x^14 + 315*b^8*x^16))/(a + b*x^2)^9 + 315*ArcTan[(Sqrt[b]*
x)/Sqrt[a]])/(589824*a^(11/2)*b^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 122, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{35\,{a}^{3}x}{65536\,{b}^{4}}}-{\frac{455\,{a}^{2}{x}^{3}}{98304\,{b}^{3}}}-{\frac{581\,a{x}^{5}}{32768\,{b}^{2}}}-{\frac{1289\,{x}^{7}}{32768\,b}}+{\frac{{x}^{9}}{18\,a}}+{\frac{1289\,b{x}^{11}}{32768\,{a}^{2}}}+{\frac{581\,{b}^{2}{x}^{13}}{32768\,{a}^{3}}}+{\frac{455\,{b}^{3}{x}^{15}}{98304\,{a}^{4}}}+{\frac{35\,{b}^{4}{x}^{17}}{65536\,{a}^{5}}} \right ) }+{\frac{35}{65536\,{a}^{5}{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(b*x^2+a)^10,x)

[Out]

(-35/65536*a^3*x/b^4-455/98304*a^2*x^3/b^3-581/32768*a*x^5/b^2-1289/32768*x^7/b+1/18/a*x^9+1289/32768*b/a^2*x^
11+581/32768*b^2/a^3*x^13+455/98304*b^3/a^4*x^15+35/65536*b^4/a^5*x^17)/(b*x^2+a)^9+35/65536/a^5/b^4/(a*b)^(1/
2)*arctan(b*x/(a*b)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.25955, size = 1523, normalized size = 7.54 \begin{align*} \left [\frac{630 \, a b^{9} x^{17} + 5460 \, a^{2} b^{8} x^{15} + 20916 \, a^{3} b^{7} x^{13} + 46404 \, a^{4} b^{6} x^{11} + 65536 \, a^{5} b^{5} x^{9} - 46404 \, a^{6} b^{4} x^{7} - 20916 \, a^{7} b^{3} x^{5} - 5460 \, a^{8} b^{2} x^{3} - 630 \, a^{9} b x - 315 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{1179648 \,{\left (a^{6} b^{14} x^{18} + 9 \, a^{7} b^{13} x^{16} + 36 \, a^{8} b^{12} x^{14} + 84 \, a^{9} b^{11} x^{12} + 126 \, a^{10} b^{10} x^{10} + 126 \, a^{11} b^{9} x^{8} + 84 \, a^{12} b^{8} x^{6} + 36 \, a^{13} b^{7} x^{4} + 9 \, a^{14} b^{6} x^{2} + a^{15} b^{5}\right )}}, \frac{315 \, a b^{9} x^{17} + 2730 \, a^{2} b^{8} x^{15} + 10458 \, a^{3} b^{7} x^{13} + 23202 \, a^{4} b^{6} x^{11} + 32768 \, a^{5} b^{5} x^{9} - 23202 \, a^{6} b^{4} x^{7} - 10458 \, a^{7} b^{3} x^{5} - 2730 \, a^{8} b^{2} x^{3} - 315 \, a^{9} b x + 315 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{589824 \,{\left (a^{6} b^{14} x^{18} + 9 \, a^{7} b^{13} x^{16} + 36 \, a^{8} b^{12} x^{14} + 84 \, a^{9} b^{11} x^{12} + 126 \, a^{10} b^{10} x^{10} + 126 \, a^{11} b^{9} x^{8} + 84 \, a^{12} b^{8} x^{6} + 36 \, a^{13} b^{7} x^{4} + 9 \, a^{14} b^{6} x^{2} + a^{15} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/1179648*(630*a*b^9*x^17 + 5460*a^2*b^8*x^15 + 20916*a^3*b^7*x^13 + 46404*a^4*b^6*x^11 + 65536*a^5*b^5*x^9 -
46404*a^6*b^4*x^7 - 20916*a^7*b^3*x^5 - 5460*a^8*b^2*x^3 - 630*a^9*b*x - 315*(b^9*x^18 + 9*a*b^8*x^16 + 36*a^
2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 36*a^7*b^2*x^4 + 9*a^8*b*
x^2 + a^9)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b*x^2 + a)))/(a^6*b^14*x^18 + 9*a^7*b^13*x^16 + 36*a^8
*b^12*x^14 + 84*a^9*b^11*x^12 + 126*a^10*b^10*x^10 + 126*a^11*b^9*x^8 + 84*a^12*b^8*x^6 + 36*a^13*b^7*x^4 + 9*
a^14*b^6*x^2 + a^15*b^5), 1/589824*(315*a*b^9*x^17 + 2730*a^2*b^8*x^15 + 10458*a^3*b^7*x^13 + 23202*a^4*b^6*x^
11 + 32768*a^5*b^5*x^9 - 23202*a^6*b^4*x^7 - 10458*a^7*b^3*x^5 - 2730*a^8*b^2*x^3 - 315*a^9*b*x + 315*(b^9*x^1
8 + 9*a*b^8*x^16 + 36*a^2*b^7*x^14 + 84*a^3*b^6*x^12 + 126*a^4*b^5*x^10 + 126*a^5*b^4*x^8 + 84*a^6*b^3*x^6 + 3
6*a^7*b^2*x^4 + 9*a^8*b*x^2 + a^9)*sqrt(a*b)*arctan(sqrt(a*b)*x/a))/(a^6*b^14*x^18 + 9*a^7*b^13*x^16 + 36*a^8*
b^12*x^14 + 84*a^9*b^11*x^12 + 126*a^10*b^10*x^10 + 126*a^11*b^9*x^8 + 84*a^12*b^8*x^6 + 36*a^13*b^7*x^4 + 9*a
^14*b^6*x^2 + a^15*b^5)]

________________________________________________________________________________________

Sympy [A]  time = 7.20997, size = 291, normalized size = 1.44 \begin{align*} - \frac{35 \sqrt{- \frac{1}{a^{11} b^{9}}} \log{\left (- a^{6} b^{4} \sqrt{- \frac{1}{a^{11} b^{9}}} + x \right )}}{131072} + \frac{35 \sqrt{- \frac{1}{a^{11} b^{9}}} \log{\left (a^{6} b^{4} \sqrt{- \frac{1}{a^{11} b^{9}}} + x \right )}}{131072} + \frac{- 315 a^{8} x - 2730 a^{7} b x^{3} - 10458 a^{6} b^{2} x^{5} - 23202 a^{5} b^{3} x^{7} + 32768 a^{4} b^{4} x^{9} + 23202 a^{3} b^{5} x^{11} + 10458 a^{2} b^{6} x^{13} + 2730 a b^{7} x^{15} + 315 b^{8} x^{17}}{589824 a^{14} b^{4} + 5308416 a^{13} b^{5} x^{2} + 21233664 a^{12} b^{6} x^{4} + 49545216 a^{11} b^{7} x^{6} + 74317824 a^{10} b^{8} x^{8} + 74317824 a^{9} b^{9} x^{10} + 49545216 a^{8} b^{10} x^{12} + 21233664 a^{7} b^{11} x^{14} + 5308416 a^{6} b^{12} x^{16} + 589824 a^{5} b^{13} x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8/(b*x**2+a)**10,x)

[Out]

-35*sqrt(-1/(a**11*b**9))*log(-a**6*b**4*sqrt(-1/(a**11*b**9)) + x)/131072 + 35*sqrt(-1/(a**11*b**9))*log(a**6
*b**4*sqrt(-1/(a**11*b**9)) + x)/131072 + (-315*a**8*x - 2730*a**7*b*x**3 - 10458*a**6*b**2*x**5 - 23202*a**5*
b**3*x**7 + 32768*a**4*b**4*x**9 + 23202*a**3*b**5*x**11 + 10458*a**2*b**6*x**13 + 2730*a*b**7*x**15 + 315*b**
8*x**17)/(589824*a**14*b**4 + 5308416*a**13*b**5*x**2 + 21233664*a**12*b**6*x**4 + 49545216*a**11*b**7*x**6 +
74317824*a**10*b**8*x**8 + 74317824*a**9*b**9*x**10 + 49545216*a**8*b**10*x**12 + 21233664*a**7*b**11*x**14 +
5308416*a**6*b**12*x**16 + 589824*a**5*b**13*x**18)

________________________________________________________________________________________

Giac [A]  time = 1.457, size = 173, normalized size = 0.86 \begin{align*} \frac{35 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{5} b^{4}} + \frac{315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \,{\left (b x^{2} + a\right )}^{9} a^{5} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/(b*x^2+a)^10,x, algorithm="giac")

[Out]

35/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5*b^4) + 1/589824*(315*b^8*x^17 + 2730*a*b^7*x^15 + 10458*a^2*b^6*
x^13 + 23202*a^3*b^5*x^11 + 32768*a^4*b^4*x^9 - 23202*a^5*b^3*x^7 - 10458*a^6*b^2*x^5 - 2730*a^7*b*x^3 - 315*a
^8*x)/((b*x^2 + a)^9*a^5*b^4)