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

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

[Out]

-1/18*x^7/b/(b*x^2+a)^9-7/288*x^5/b^2/(b*x^2+a)^8-5/576*x^3/b^3/(b*x^2+a)^7-5/2304*x/b^4/(b*x^2+a)^6+1/4608*x/
a/b^4/(b*x^2+a)^5+1/4096*x/a^2/b^4/(b*x^2+a)^4+7/24576*x/a^3/b^4/(b*x^2+a)^3+35/98304*x/a^4/b^4/(b*x^2+a)^2+35
/65536*x/a^5/b^4/(b*x^2+a)+35/65536*arctan(x*b^(1/2)/a^(1/2))/a^(11/2)/b^(9/2)

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Rubi [A]  time = 0.12, antiderivative size = 202, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.06, size = 138, normalized size = 0.68 \[ \frac {\frac {\sqrt {a} \sqrt {b} x \left (-315 a^8-2730 a^7 b x^2-10458 a^6 b^2 x^4-23202 a^5 b^3 x^6+32768 a^4 b^4 x^8+23202 a^3 b^5 x^{10}+10458 a^2 b^6 x^{12}+2730 a b^7 x^{14}+315 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+315 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{589824 a^{11/2} b^{9/2}} \]

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))

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fricas [A]  time = 0.86, size = 654, normalized size = 3.24 \[ \left [\frac {630 \, a b^{9} x^{17} + 5460 \, a^{2} b^{8} x^{15} + 20916 \, a^{3} b^{7} x^{13} + 46404 \, a^{4} b^{6} x^{11} + 65536 \, a^{5} b^{5} x^{9} - 46404 \, a^{6} b^{4} x^{7} - 20916 \, a^{7} b^{3} x^{5} - 5460 \, a^{8} b^{2} x^{3} - 630 \, a^{9} b x - 315 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{1179648 \, {\left (a^{6} b^{14} x^{18} + 9 \, a^{7} b^{13} x^{16} + 36 \, a^{8} b^{12} x^{14} + 84 \, a^{9} b^{11} x^{12} + 126 \, a^{10} b^{10} x^{10} + 126 \, a^{11} b^{9} x^{8} + 84 \, a^{12} b^{8} x^{6} + 36 \, a^{13} b^{7} x^{4} + 9 \, a^{14} b^{6} x^{2} + a^{15} b^{5}\right )}}, \frac {315 \, a b^{9} x^{17} + 2730 \, a^{2} b^{8} x^{15} + 10458 \, a^{3} b^{7} x^{13} + 23202 \, a^{4} b^{6} x^{11} + 32768 \, a^{5} b^{5} x^{9} - 23202 \, a^{6} b^{4} x^{7} - 10458 \, a^{7} b^{3} x^{5} - 2730 \, a^{8} b^{2} x^{3} - 315 \, a^{9} b x + 315 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{589824 \, {\left (a^{6} b^{14} x^{18} + 9 \, a^{7} b^{13} x^{16} + 36 \, a^{8} b^{12} x^{14} + 84 \, a^{9} b^{11} x^{12} + 126 \, a^{10} b^{10} x^{10} + 126 \, a^{11} b^{9} x^{8} + 84 \, a^{12} b^{8} x^{6} + 36 \, a^{13} b^{7} x^{4} + 9 \, a^{14} b^{6} x^{2} + a^{15} b^{5}\right )}}\right ] \]

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)]

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giac [A]  time = 0.60, size = 128, normalized size = 0.63 \[ \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{5} b^{4}} + \frac {315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \, {\left (b x^{2} + a\right )}^{9} a^{5} b^{4}} \]

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)

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maple [A]  time = 0.02, size = 122, normalized size = 0.60 \[ \frac {35 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, a^{5} b^{4}}+\frac {\frac {35 b^{4} x^{17}}{65536 a^{5}}+\frac {455 b^{3} x^{15}}{98304 a^{4}}+\frac {581 b^{2} x^{13}}{32768 a^{3}}+\frac {1289 b \,x^{11}}{32768 a^{2}}+\frac {x^{9}}{18 a}-\frac {1289 x^{7}}{32768 b}-\frac {581 a \,x^{5}}{32768 b^{2}}-\frac {455 a^{2} x^{3}}{98304 b^{3}}-\frac {35 a^{3} x}{65536 b^{4}}}{\left (b \,x^{2}+a \right )^{9}} \]

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/b^4*x-455/98304*a^2/b^3*x^3-581/32768*a/b^2*x^5-1289/32768/b*x^7+1/18/a*x^9+1289/32768/a^2*b*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(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.08, size = 221, normalized size = 1.09 \[ \frac {315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \, {\left (a^{5} b^{13} x^{18} + 9 \, a^{6} b^{12} x^{16} + 36 \, a^{7} b^{11} x^{14} + 84 \, a^{8} b^{10} x^{12} + 126 \, a^{9} b^{9} x^{10} + 126 \, a^{10} b^{8} x^{8} + 84 \, a^{11} b^{7} x^{6} + 36 \, a^{12} b^{6} x^{4} + 9 \, a^{13} b^{5} x^{2} + a^{14} b^{4}\right )}} + \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} a^{5} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)/(a^5*b^13*x^18 + 9*a^6*b^12*x^16 + 36*a^7*b^11*
x^14 + 84*a^8*b^10*x^12 + 126*a^9*b^9*x^10 + 126*a^10*b^8*x^8 + 84*a^11*b^7*x^6 + 36*a^12*b^6*x^4 + 9*a^13*b^5
*x^2 + a^14*b^4) + 35/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^5*b^4)

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mupad [B]  time = 4.74, size = 204, normalized size = 1.01 \[ \frac {\frac {x^9}{18\,a}-\frac {1289\,x^7}{32768\,b}-\frac {581\,a\,x^5}{32768\,b^2}-\frac {35\,a^3\,x}{65536\,b^4}+\frac {1289\,b\,x^{11}}{32768\,a^2}-\frac {455\,a^2\,x^3}{98304\,b^3}+\frac {581\,b^2\,x^{13}}{32768\,a^3}+\frac {455\,b^3\,x^{15}}{98304\,a^4}+\frac {35\,b^4\,x^{17}}{65536\,a^5}}{a^9+9\,a^8\,b\,x^2+36\,a^7\,b^2\,x^4+84\,a^6\,b^3\,x^6+126\,a^5\,b^4\,x^8+126\,a^4\,b^5\,x^{10}+84\,a^3\,b^6\,x^{12}+36\,a^2\,b^7\,x^{14}+9\,a\,b^8\,x^{16}+b^9\,x^{18}}+\frac {35\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,a^{11/2}\,b^{9/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8/(a + b*x^2)^10,x)

[Out]

(x^9/(18*a) - (1289*x^7)/(32768*b) - (581*a*x^5)/(32768*b^2) - (35*a^3*x)/(65536*b^4) + (1289*b*x^11)/(32768*a
^2) - (455*a^2*x^3)/(98304*b^3) + (581*b^2*x^13)/(32768*a^3) + (455*b^3*x^15)/(98304*a^4) + (35*b^4*x^17)/(655
36*a^5))/(a^9 + b^9*x^18 + 9*a^8*b*x^2 + 9*a*b^8*x^16 + 36*a^7*b^2*x^4 + 84*a^6*b^3*x^6 + 126*a^5*b^4*x^8 + 12
6*a^4*b^5*x^10 + 84*a^3*b^6*x^12 + 36*a^2*b^7*x^14) + (35*atan((b^(1/2)*x)/a^(1/2)))/(65536*a^(11/2)*b^(9/2))

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sympy [A]  time = 1.15, size = 291, normalized size = 1.44 \[ - \frac {35 \sqrt {- \frac {1}{a^{11} b^{9}}} \log {\left (- a^{6} b^{4} \sqrt {- \frac {1}{a^{11} b^{9}}} + x \right )}}{131072} + \frac {35 \sqrt {- \frac {1}{a^{11} b^{9}}} \log {\left (a^{6} b^{4} \sqrt {- \frac {1}{a^{11} b^{9}}} + x \right )}}{131072} + \frac {- 315 a^{8} x - 2730 a^{7} b x^{3} - 10458 a^{6} b^{2} x^{5} - 23202 a^{5} b^{3} x^{7} + 32768 a^{4} b^{4} x^{9} + 23202 a^{3} b^{5} x^{11} + 10458 a^{2} b^{6} x^{13} + 2730 a b^{7} x^{15} + 315 b^{8} x^{17}}{589824 a^{14} b^{4} + 5308416 a^{13} b^{5} x^{2} + 21233664 a^{12} b^{6} x^{4} + 49545216 a^{11} b^{7} x^{6} + 74317824 a^{10} b^{8} x^{8} + 74317824 a^{9} b^{9} x^{10} + 49545216 a^{8} b^{10} x^{12} + 21233664 a^{7} b^{11} x^{14} + 5308416 a^{6} b^{12} x^{16} + 589824 a^{5} b^{13} x^{18}} \]

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)

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