3.211 \(\int \frac {x^{20}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=207 \[ -\frac {230945 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{21/2}}-\frac {230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac {46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}+\frac {230945 x}{65536 b^{10}} \]

[Out]

230945/65536*x/b^10-1/18*x^19/b/(b*x^2+a)^9-19/288*x^17/b^2/(b*x^2+a)^8-323/4032*x^15/b^3/(b*x^2+a)^7-1615/161
28*x^13/b^4/(b*x^2+a)^6-4199/32256*x^11/b^5/(b*x^2+a)^5-46189/258048*x^9/b^6/(b*x^2+a)^4-46189/172032*x^7/b^7/
(b*x^2+a)^3-46189/98304*x^5/b^8/(b*x^2+a)^2-230945/196608*x^3/b^9/(b*x^2+a)-230945/65536*arctan(x*b^(1/2)/a^(1
/2))*a^(1/2)/b^(21/2)

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Rubi [A]  time = 0.12, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 321, 205} \[ -\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac {230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac {230945 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{21/2}}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}+\frac {230945 x}{65536 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^20/(a + b*x^2)^10,x]

[Out]

(230945*x)/(65536*b^10) - x^19/(18*b*(a + b*x^2)^9) - (19*x^17)/(288*b^2*(a + b*x^2)^8) - (323*x^15)/(4032*b^3
*(a + b*x^2)^7) - (1615*x^13)/(16128*b^4*(a + b*x^2)^6) - (4199*x^11)/(32256*b^5*(a + b*x^2)^5) - (46189*x^9)/
(258048*b^6*(a + b*x^2)^4) - (46189*x^7)/(172032*b^7*(a + b*x^2)^3) - (46189*x^5)/(98304*b^8*(a + b*x^2)^2) -
(230945*x^3)/(196608*b^9*(a + b*x^2)) - (230945*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*b^(21/2))

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]

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rubi steps

\begin {align*} \int \frac {x^{20}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}+\frac {19 \int \frac {x^{18}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}+\frac {323 \int \frac {x^{16}}{\left (a+b x^2\right )^8} \, dx}{288 b^2}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}+\frac {1615 \int \frac {x^{14}}{\left (a+b x^2\right )^7} \, dx}{1344 b^3}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}+\frac {20995 \int \frac {x^{12}}{\left (a+b x^2\right )^6} \, dx}{16128 b^4}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}+\frac {46189 \int \frac {x^{10}}{\left (a+b x^2\right )^5} \, dx}{32256 b^5}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}+\frac {46189 \int \frac {x^8}{\left (a+b x^2\right )^4} \, dx}{28672 b^6}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}+\frac {46189 \int \frac {x^6}{\left (a+b x^2\right )^3} \, dx}{24576 b^7}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}+\frac {230945 \int \frac {x^4}{\left (a+b x^2\right )^2} \, dx}{98304 b^8}\\ &=-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac {230945 x^3}{196608 b^9 \left (a+b x^2\right )}+\frac {230945 \int \frac {x^2}{a+b x^2} \, dx}{65536 b^9}\\ &=\frac {230945 x}{65536 b^{10}}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac {230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac {(230945 a) \int \frac {1}{a+b x^2} \, dx}{65536 b^{10}}\\ &=\frac {230945 x}{65536 b^{10}}-\frac {x^{19}}{18 b \left (a+b x^2\right )^9}-\frac {19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac {323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac {1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac {4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac {46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac {230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac {230945 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{21/2}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 144, normalized size = 0.70 \[ \frac {\frac {\sqrt {b} x \left (14549535 a^9+126095970 a^8 b x^2+483044562 a^7 b^2 x^4+1071677178 a^6 b^3 x^6+1513521152 a^5 b^4 x^8+1404993798 a^4 b^5 x^{10}[phone] a^3 b^6 x^{12}[phone] a^2 b^7 x^{14}[phone] a b^8 x^{16}+4128768 b^9 x^{18}\right )}{\left (a+b x^2\right )^9}-14549535 \sqrt {a} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{4128768 b^{21/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^20/(a + b*x^2)^10,x]

[Out]

((Sqrt[b]*x*(14549535*a^9 + 126095970*a^8*b*x^2 + 483044562*a^7*b^2*x^4 + 1071677178*a^6*b^3*x^6 + 1513521152*
a^5*b^4*x^8 + 1404993798*a^4*b^5*x^10 + 850547502*a^3*b^6*x^12 + 318434718*a^2*b^7*x^14 + 63897057*a*b^8*x^16
+ 4128768*b^9*x^18))/(a + b*x^2)^9 - 14549535*Sqrt[a]*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(4128768*b^(21/2))

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fricas [A]  time = 1.05, size = 664, normalized size = 3.21 \[ \left [\frac {8257536 \, b^{9} x^{19} + 127794114 \, a b^{8} x^{17} + 636869436 \, a^{2} b^{7} x^{15} + 1701095004 \, a^{3} b^{6} x^{13} + 2809987596 \, a^{4} b^{5} x^{11} + 3027042304 \, a^{5} b^{4} x^{9} + 2143354356 \, a^{6} b^{3} x^{7} + 966089124 \, a^{7} b^{2} x^{5} + 252191940 \, a^{8} b x^{3} + 29099070 \, a^{9} x + 14549535 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{8257536 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}}, \frac {4128768 \, b^{9} x^{19} + 63897057 \, a b^{8} x^{17} + 318434718 \, a^{2} b^{7} x^{15} + 850547502 \, a^{3} b^{6} x^{13} + 1404993798 \, a^{4} b^{5} x^{11} + 1513521152 \, a^{5} b^{4} x^{9} + 1071677178 \, a^{6} b^{3} x^{7} + 483044562 \, a^{7} b^{2} x^{5} + 126095970 \, a^{8} b x^{3} + 14549535 \, a^{9} x - 14549535 \, {\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{4128768 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/8257536*(8257536*b^9*x^19 + 127794114*a*b^8*x^17 + 636869436*a^2*b^7*x^15 + 1701095004*a^3*b^6*x^13 + 28099
87596*a^4*b^5*x^11 + 3027042304*a^5*b^4*x^9 + 2143354356*a^6*b^3*x^7 + 966089124*a^7*b^2*x^5 + 252191940*a^8*b
*x^3 + 29099070*a^9*x + 14549535*(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*b*x*sqrt
(-a/b) - a)/(b*x^2 + a)))/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10
+ 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10), 1/4128768*(4128768*b^9*x
^19 + 63897057*a*b^8*x^17 + 318434718*a^2*b^7*x^15 + 850547502*a^3*b^6*x^13 + 1404993798*a^4*b^5*x^11 + 151352
1152*a^5*b^4*x^9 + 1071677178*a^6*b^3*x^7 + 483044562*a^7*b^2*x^5 + 126095970*a^8*b*x^3 + 14549535*a^9*x - 145
49535*(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)*arctan(b*x*sqrt(a/b)/a))/(b^19*x^18 + 9*a*b^18*x^16
+ 36*a^2*b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*
x^4 + 9*a^8*b^11*x^2 + a^9*b^10)]

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giac [A]  time = 0.64, size = 131, normalized size = 0.63 \[ -\frac {230945 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{10}} + \frac {x}{b^{10}} + \frac {26738145 \, a b^{8} x^{17} + 169799070 \, a^{2} b^{7} x^{15} + 503730990 \, a^{3} b^{6} x^{13} + 884769030 \, a^{4} b^{5} x^{11} + 993296384 \, a^{5} b^{4} x^{9} + 724860666 \, a^{6} b^{3} x^{7} + 334408914 \, a^{7} b^{2} x^{5} + 88937058 \, a^{8} b x^{3} + 10420767 \, a^{9} x}{4128768 \, {\left (b x^{2} + a\right )}^{9} b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20/(b*x^2+a)^10,x, algorithm="giac")

[Out]

-230945/65536*a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^10) + x/b^10 + 1/4128768*(26738145*a*b^8*x^17 + 169799070*a
^2*b^7*x^15 + 503730990*a^3*b^6*x^13 + 884769030*a^4*b^5*x^11 + 993296384*a^5*b^4*x^9 + 724860666*a^6*b^3*x^7
+ 334408914*a^7*b^2*x^5 + 88937058*a^8*b*x^3 + 10420767*a^9*x)/((b*x^2 + a)^9*b^10)

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maple [A]  time = 0.02, size = 203, normalized size = 0.98 \[ \frac {424415 a \,x^{17}}{65536 \left (b \,x^{2}+a \right )^{9} b^{2}}+\frac {4042835 a^{2} x^{15}}{98304 \left (b \,x^{2}+a \right )^{9} b^{3}}+\frac {3997865 a^{3} x^{13}}{32768 \left (b \,x^{2}+a \right )^{9} b^{4}}+\frac {49153835 a^{4} x^{11}}{229376 \left (b \,x^{2}+a \right )^{9} b^{5}}+\frac {30313 a^{5} x^{9}}{126 \left (b \,x^{2}+a \right )^{9} b^{6}}+\frac {40270037 a^{6} x^{7}}{229376 \left (b \,x^{2}+a \right )^{9} b^{7}}+\frac {2654039 a^{7} x^{5}}{32768 \left (b \,x^{2}+a \right )^{9} b^{8}}+\frac {2117549 a^{8} x^{3}}{98304 \left (b \,x^{2}+a \right )^{9} b^{9}}+\frac {165409 a^{9} x}{65536 \left (b \,x^{2}+a \right )^{9} b^{10}}-\frac {230945 a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, b^{10}}+\frac {x}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^20/(b*x^2+a)^10,x)

[Out]

x/b^10+165409/65536/b^10*a^9/(b*x^2+a)^9*x+2117549/98304/b^9*a^8/(b*x^2+a)^9*x^3+2654039/32768/b^8*a^7/(b*x^2+
a)^9*x^5+40270037/229376/b^7*a^6/(b*x^2+a)^9*x^7+30313/126/b^6*a^5/(b*x^2+a)^9*x^9+49153835/229376/b^5*a^4/(b*
x^2+a)^9*x^11+3997865/32768/b^4*a^3/(b*x^2+a)^9*x^13+4042835/98304/b^3*a^2/(b*x^2+a)^9*x^15+424415/65536/b^2*a
/(b*x^2+a)^9*x^17-230945/65536/b^10*a/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.17, size = 222, normalized size = 1.07 \[ \frac {26738145 \, a b^{8} x^{17} + 169799070 \, a^{2} b^{7} x^{15} + 503730990 \, a^{3} b^{6} x^{13} + 884769030 \, a^{4} b^{5} x^{11} + 993296384 \, a^{5} b^{4} x^{9} + 724860666 \, a^{6} b^{3} x^{7} + 334408914 \, a^{7} b^{2} x^{5} + 88937058 \, a^{8} b x^{3} + 10420767 \, a^{9} x}{4128768 \, {\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} - \frac {230945 \, a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{10}} + \frac {x}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^20/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

1/4128768*(26738145*a*b^8*x^17 + 169799070*a^2*b^7*x^15 + 503730990*a^3*b^6*x^13 + 884769030*a^4*b^5*x^11 + 99
3296384*a^5*b^4*x^9 + 724860666*a^6*b^3*x^7 + 334408914*a^7*b^2*x^5 + 88937058*a^8*b*x^3 + 10420767*a^9*x)/(b^
19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*
b^13*x^6 + 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10) - 230945/65536*a*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^10
) + x/b^10

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mupad [B]  time = 0.44, size = 218, normalized size = 1.05 \[ \frac {\frac {165409\,a^9\,x}{65536}+\frac {2117549\,a^8\,b\,x^3}{98304}+\frac {2654039\,a^7\,b^2\,x^5}{32768}+\frac {40270037\,a^6\,b^3\,x^7}{229376}+\frac {30313\,a^5\,b^4\,x^9}{126}+\frac {49153835\,a^4\,b^5\,x^{11}}{229376}+\frac {3997865\,a^3\,b^6\,x^{13}}{32768}+\frac {4042835\,a^2\,b^7\,x^{15}}{98304}+\frac {424415\,a\,b^8\,x^{17}}{65536}}{a^9\,b^{10}+9\,a^8\,b^{11}\,x^2+36\,a^7\,b^{12}\,x^4+84\,a^6\,b^{13}\,x^6+126\,a^5\,b^{14}\,x^8+126\,a^4\,b^{15}\,x^{10}+84\,a^3\,b^{16}\,x^{12}+36\,a^2\,b^{17}\,x^{14}+9\,a\,b^{18}\,x^{16}+b^{19}\,x^{18}}+\frac {x}{b^{10}}-\frac {230945\,\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{21/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^20/(a + b*x^2)^10,x)

[Out]

((165409*a^9*x)/65536 + (2117549*a^8*b*x^3)/98304 + (424415*a*b^8*x^17)/65536 + (2654039*a^7*b^2*x^5)/32768 +
(40270037*a^6*b^3*x^7)/229376 + (30313*a^5*b^4*x^9)/126 + (49153835*a^4*b^5*x^11)/229376 + (3997865*a^3*b^6*x^
13)/32768 + (4042835*a^2*b^7*x^15)/98304)/(a^9*b^10 + b^19*x^18 + 9*a*b^18*x^16 + 9*a^8*b^11*x^2 + 36*a^7*b^12
*x^4 + 84*a^6*b^13*x^6 + 126*a^5*b^14*x^8 + 126*a^4*b^15*x^10 + 84*a^3*b^16*x^12 + 36*a^2*b^17*x^14) + x/b^10
- (230945*a^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*b^(21/2))

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sympy [A]  time = 1.82, size = 274, normalized size = 1.32 \[ \frac {230945 \sqrt {- \frac {a}{b^{21}}} \log {\left (- b^{10} \sqrt {- \frac {a}{b^{21}}} + x \right )}}{131072} - \frac {230945 \sqrt {- \frac {a}{b^{21}}} \log {\left (b^{10} \sqrt {- \frac {a}{b^{21}}} + x \right )}}{131072} + \frac {10420767 a^{9} x + 88937058 a^{8} b x^{3} + 334408914 a^{7} b^{2} x^{5} + 724860666 a^{6} b^{3} x^{7} + 993296384 a^{5} b^{4} x^{9} + 884769030 a^{4} b^{5} x^{11} + 503730990 a^{3} b^{6} x^{13} + 169799070 a^{2} b^{7} x^{15} + 26738145 a b^{8} x^{17}}{4128768 a^{9} b^{10} + 37158912 a^{8} b^{11} x^{2} + 148635648 a^{7} b^{12} x^{4} + 346816512 a^{6} b^{13} x^{6} + 520224768 a^{5} b^{14} x^{8} + 520224768 a^{4} b^{15} x^{10} + 346816512 a^{3} b^{16} x^{12} + 148635648 a^{2} b^{17} x^{14} + 37158912 a b^{18} x^{16} + 4128768 b^{19} x^{18}} + \frac {x}{b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**20/(b*x**2+a)**10,x)

[Out]

230945*sqrt(-a/b**21)*log(-b**10*sqrt(-a/b**21) + x)/131072 - 230945*sqrt(-a/b**21)*log(b**10*sqrt(-a/b**21) +
x)/131072 + (10420767*a**9*x + 88937058*a**8*b*x**3 + 334408914*a**7*b**2*x**5 + 724860666*a**6*b**3*x**7 + 9
93296384*a**5*b**4*x**9 + 884769030*a**4*b**5*x**11 + 503730990*a**3*b**6*x**13 + 169799070*a**2*b**7*x**15 +
26738145*a*b**8*x**17)/(4128768*a**9*b**10 + 37158912*a**8*b**11*x**2 + 148635648*a**7*b**12*x**4 + 346816512*
a**6*b**13*x**6 + 520224768*a**5*b**14*x**8 + 520224768*a**4*b**15*x**10 + 346816512*a**3*b**16*x**12 + 148635
648*a**2*b**17*x**14 + 37158912*a*b**18*x**16 + 4128768*b**19*x**18) + x/b**10

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