3.210 \(\int \frac {x^{22}}{(a+b x^2)^{10}} \, dx\)

Optimal. Leaf size=218 \[ \frac {1616615 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}}-\frac {1616615 a x}{65536 b^{11}}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}+\frac {1616615 x^3}{196608 b^{10}} \]

[Out]

-1616615/65536*a*x/b^11+1616615/196608*x^3/b^10-1/18*x^21/b/(b*x^2+a)^9-7/96*x^19/b^2/(b*x^2+a)^8-19/192*x^17/
b^3/(b*x^2+a)^7-323/2304*x^15/b^4/(b*x^2+a)^6-323/1536*x^13/b^5/(b*x^2+a)^5-4199/12288*x^11/b^6/(b*x^2+a)^4-46
189/73728*x^9/b^7/(b*x^2+a)^3-46189/32768*x^7/b^8/(b*x^2+a)^2-323323/65536*x^5/b^9/(b*x^2+a)+1616615/65536*a^(
3/2)*arctan(x*b^(1/2)/a^(1/2))/b^(23/2)

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {1616615 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac {1616615 a x}{65536 b^{11}}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}+\frac {1616615 x^3}{196608 b^{10}} \]

Antiderivative was successfully verified.

[In]

Int[x^22/(a + b*x^2)^10,x]

[Out]

(-1616615*a*x)/(65536*b^11) + (1616615*x^3)/(196608*b^10) - x^21/(18*b*(a + b*x^2)^9) - (7*x^19)/(96*b^2*(a +
b*x^2)^8) - (19*x^17)/(192*b^3*(a + b*x^2)^7) - (323*x^15)/(2304*b^4*(a + b*x^2)^6) - (323*x^13)/(1536*b^5*(a
+ b*x^2)^5) - (4199*x^11)/(12288*b^6*(a + b*x^2)^4) - (46189*x^9)/(73728*b^7*(a + b*x^2)^3) - (46189*x^7)/(327
68*b^8*(a + b*x^2)^2) - (323323*x^5)/(65536*b^9*(a + b*x^2)) + (1616615*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(
65536*b^(23/2))

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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps

\begin {align*} \int \frac {x^{22}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}+\frac {7 \int \frac {x^{20}}{\left (a+b x^2\right )^9} \, dx}{6 b}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}+\frac {133 \int \frac {x^{18}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}+\frac {323 \int \frac {x^{16}}{\left (a+b x^2\right )^7} \, dx}{192 b^3}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}+\frac {1615 \int \frac {x^{14}}{\left (a+b x^2\right )^6} \, dx}{768 b^4}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}+\frac {4199 \int \frac {x^{12}}{\left (a+b x^2\right )^5} \, dx}{1536 b^5}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}+\frac {46189 \int \frac {x^{10}}{\left (a+b x^2\right )^4} \, dx}{12288 b^6}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}+\frac {46189 \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx}{8192 b^7}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}+\frac {323323 \int \frac {x^6}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {1616615 \int \frac {x^4}{a+b x^2} \, dx}{65536 b^9}\\ &=-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {1616615 \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{65536 b^9}\\ &=-\frac {1616615 a x}{65536 b^{11}}+\frac {1616615 x^3}{196608 b^{10}}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {\left (1616615 a^2\right ) \int \frac {1}{a+b x^2} \, dx}{65536 b^{11}}\\ &=-\frac {1616615 a x}{65536 b^{11}}+\frac {1616615 x^3}{196608 b^{10}}-\frac {x^{21}}{18 b \left (a+b x^2\right )^9}-\frac {7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac {19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac {323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac {323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac {4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac {46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac {46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac {323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac {1616615 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{65536 b^{23/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 155, normalized size = 0.71 \[ \frac {14549535 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\frac {\sqrt {b} x \left (-14549535 a^{10}-126095970 a^9 b x^2-483044562 a^8 b^2 x^4-1071677178 a^7 b^3 x^6-1513521152 a^6 b^4 x^8-1404993798 a^5 b^5 x^{10}-850547502 a^4 b^6 x^{12}-318434718 a^3 b^7 x^{14}-63897057 a^2 b^8 x^{16}-4128768 a b^9 x^{18}+196608 b^{10} x^{20}\right )}{\left (a+b x^2\right )^9}}{589824 b^{23/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^22/(a + b*x^2)^10,x]

[Out]

((Sqrt[b]*x*(-14549535*a^10 - 126095970*a^9*b*x^2 - 483044562*a^8*b^2*x^4 - 1071677178*a^7*b^3*x^6 - 151352115
2*a^6*b^4*x^8 - 1404993798*a^5*b^5*x^10 - 850547502*a^4*b^6*x^12 - 318434718*a^3*b^7*x^14 - 63897057*a^2*b^8*x
^16 - 4128768*a*b^9*x^18 + 196608*b^10*x^20))/(a + b*x^2)^9 + 14549535*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(5
89824*b^(23/2))

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fricas [A]  time = 0.84, size = 692, normalized size = 3.17 \[ \left [\frac {393216 \, b^{10} x^{21} - 8257536 \, a b^{9} x^{19} - 127794114 \, a^{2} b^{8} x^{17} - 636869436 \, a^{3} b^{7} x^{15} - 1701095004 \, a^{4} b^{6} x^{13} - 2809987596 \, a^{5} b^{5} x^{11} - 3027042304 \, a^{6} b^{4} x^{9} - 2143354356 \, a^{7} b^{3} x^{7} - 966089124 \, a^{8} b^{2} x^{5} - 252191940 \, a^{9} b x^{3} - 29099070 \, a^{10} x + 14549535 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{1179648 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}, \frac {196608 \, b^{10} x^{21} - 4128768 \, a b^{9} x^{19} - 63897057 \, a^{2} b^{8} x^{17} - 318434718 \, a^{3} b^{7} x^{15} - 850547502 \, a^{4} b^{6} x^{13} - 1404993798 \, a^{5} b^{5} x^{11} - 1513521152 \, a^{6} b^{4} x^{9} - 1071677178 \, a^{7} b^{3} x^{7} - 483044562 \, a^{8} b^{2} x^{5} - 126095970 \, a^{9} b x^{3} - 14549535 \, a^{10} x + 14549535 \, {\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{589824 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^22/(b*x^2+a)^10,x, algorithm="fricas")

[Out]

[1/1179648*(393216*b^10*x^21 - 8257536*a*b^9*x^19 - 127794114*a^2*b^8*x^17 - 636869436*a^3*b^7*x^15 - 17010950
04*a^4*b^6*x^13 - 2809987596*a^5*b^5*x^11 - 3027042304*a^6*b^4*x^9 - 2143354356*a^7*b^3*x^7 - 966089124*a^8*b^
2*x^5 - 252191940*a^9*b*x^3 - 29099070*a^10*x + 14549535*(a*b^9*x^18 + 9*a^2*b^8*x^16 + 36*a^3*b^7*x^14 + 84*a
^4*b^6*x^12 + 126*a^5*b^5*x^10 + 126*a^6*b^4*x^8 + 84*a^7*b^3*x^6 + 36*a^8*b^2*x^4 + 9*a^9*b*x^2 + a^10)*sqrt(
-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x^14 + 84*a^3*
b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^
11), 1/589824*(196608*b^10*x^21 - 4128768*a*b^9*x^19 - 63897057*a^2*b^8*x^17 - 318434718*a^3*b^7*x^15 - 850547
502*a^4*b^6*x^13 - 1404993798*a^5*b^5*x^11 - 1513521152*a^6*b^4*x^9 - 1071677178*a^7*b^3*x^7 - 483044562*a^8*b
^2*x^5 - 126095970*a^9*b*x^3 - 14549535*a^10*x + 14549535*(a*b^9*x^18 + 9*a^2*b^8*x^16 + 36*a^3*b^7*x^14 + 84*
a^4*b^6*x^12 + 126*a^5*b^5*x^10 + 126*a^6*b^4*x^8 + 84*a^7*b^3*x^6 + 36*a^8*b^2*x^4 + 9*a^9*b*x^2 + a^10)*sqrt
(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16
*x^10 + 126*a^5*b^15*x^8 + 84*a^6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^11)]

________________________________________________________________________________________

giac [A]  time = 0.65, size = 150, normalized size = 0.69 \[ \frac {1616615 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{11}} - \frac {17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \, {\left (b x^{2} + a\right )}^{9} b^{11}} + \frac {b^{20} x^{3} - 30 \, a b^{19} x}{3 \, b^{30}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^22/(b*x^2+a)^10,x, algorithm="giac")

[Out]

1616615/65536*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^11) - 1/589824*(17890785*a^2*b^8*x^17 + 122613150*a^3*b^7
*x^15 + 379867950*a^4*b^6*x^13 + 686588166*a^5*b^5*x^11 + 786857984*a^6*b^4*x^9 + 583302906*a^7*b^3*x^7 + 2724
77394*a^8*b^2*x^5 + 73208418*a^9*b*x^3 + 8651295*a^10*x)/((b*x^2 + a)^9*b^11) + 1/3*(b^20*x^3 - 30*a*b^19*x)/b
^30

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maple [A]  time = 0.02, size = 217, normalized size = 1.00 \[ -\frac {1987865 a^{2} x^{17}}{65536 \left (b \,x^{2}+a \right )^{9} b^{3}}-\frac {20435525 a^{3} x^{15}}{98304 \left (b \,x^{2}+a \right )^{9} b^{4}}-\frac {21103775 a^{4} x^{13}}{32768 \left (b \,x^{2}+a \right )^{9} b^{5}}-\frac {38143787 a^{5} x^{11}}{32768 \left (b \,x^{2}+a \right )^{9} b^{6}}-\frac {24013 a^{6} x^{9}}{18 \left (b \,x^{2}+a \right )^{9} b^{7}}-\frac {32405717 a^{7} x^{7}}{32768 \left (b \,x^{2}+a \right )^{9} b^{8}}-\frac {15137633 a^{8} x^{5}}{32768 \left (b \,x^{2}+a \right )^{9} b^{9}}-\frac {12201403 a^{9} x^{3}}{98304 \left (b \,x^{2}+a \right )^{9} b^{10}}-\frac {961255 a^{10} x}{65536 \left (b \,x^{2}+a \right )^{9} b^{11}}+\frac {x^{3}}{3 b^{10}}+\frac {1616615 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \sqrt {a b}\, b^{11}}-\frac {10 a x}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^22/(b*x^2+a)^10,x)

[Out]

1/3*x^3/b^10-10*a*x/b^11-961255/65536/b^11*a^10/(b*x^2+a)^9*x-12201403/98304/b^10*a^9/(b*x^2+a)^9*x^3-15137633
/32768/b^9*a^8/(b*x^2+a)^9*x^5-32405717/32768/b^8*a^7/(b*x^2+a)^9*x^7-24013/18/b^7*a^6/(b*x^2+a)^9*x^9-3814378
7/32768/b^6*a^5/(b*x^2+a)^9*x^11-21103775/32768/b^5*a^4/(b*x^2+a)^9*x^13-20435525/98304/b^4*a^3/(b*x^2+a)^9*x^
15-1987865/65536/b^3*a^2/(b*x^2+a)^9*x^17+1616615/65536/b^11*a^2/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 3.25, size = 236, normalized size = 1.08 \[ -\frac {17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \, {\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} + \frac {1616615 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{65536 \, \sqrt {a b} b^{11}} + \frac {b x^{3} - 30 \, a x}{3 \, b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^22/(b*x^2+a)^10,x, algorithm="maxima")

[Out]

-1/589824*(17890785*a^2*b^8*x^17 + 122613150*a^3*b^7*x^15 + 379867950*a^4*b^6*x^13 + 686588166*a^5*b^5*x^11 +
786857984*a^6*b^4*x^9 + 583302906*a^7*b^3*x^7 + 272477394*a^8*b^2*x^5 + 73208418*a^9*b*x^3 + 8651295*a^10*x)/(
b^20*x^18 + 9*a*b^19*x^16 + 36*a^2*b^18*x^14 + 84*a^3*b^17*x^12 + 126*a^4*b^16*x^10 + 126*a^5*b^15*x^8 + 84*a^
6*b^14*x^6 + 36*a^7*b^13*x^4 + 9*a^8*b^12*x^2 + a^9*b^11) + 1616615/65536*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)
*b^11) + 1/3*(b*x^3 - 30*a*x)/b^11

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mupad [B]  time = 0.40, size = 231, normalized size = 1.06 \[ \frac {x^3}{3\,b^{10}}-\frac {\frac {961255\,a^{10}\,x}{65536}+\frac {12201403\,a^9\,b\,x^3}{98304}+\frac {15137633\,a^8\,b^2\,x^5}{32768}+\frac {32405717\,a^7\,b^3\,x^7}{32768}+\frac {24013\,a^6\,b^4\,x^9}{18}+\frac {38143787\,a^5\,b^5\,x^{11}}{32768}+\frac {21103775\,a^4\,b^6\,x^{13}}{32768}+\frac {20435525\,a^3\,b^7\,x^{15}}{98304}+\frac {1987865\,a^2\,b^8\,x^{17}}{65536}}{a^9\,b^{11}+9\,a^8\,b^{12}\,x^2+36\,a^7\,b^{13}\,x^4+84\,a^6\,b^{14}\,x^6+126\,a^5\,b^{15}\,x^8+126\,a^4\,b^{16}\,x^{10}+84\,a^3\,b^{17}\,x^{12}+36\,a^2\,b^{18}\,x^{14}+9\,a\,b^{19}\,x^{16}+b^{20}\,x^{18}}+\frac {1616615\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{65536\,b^{23/2}}-\frac {10\,a\,x}{b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^22/(a + b*x^2)^10,x)

[Out]

x^3/(3*b^10) - ((961255*a^10*x)/65536 + (12201403*a^9*b*x^3)/98304 + (15137633*a^8*b^2*x^5)/32768 + (32405717*
a^7*b^3*x^7)/32768 + (24013*a^6*b^4*x^9)/18 + (38143787*a^5*b^5*x^11)/32768 + (21103775*a^4*b^6*x^13)/32768 +
(20435525*a^3*b^7*x^15)/98304 + (1987865*a^2*b^8*x^17)/65536)/(a^9*b^11 + b^20*x^18 + 9*a*b^19*x^16 + 9*a^8*b^
12*x^2 + 36*a^7*b^13*x^4 + 84*a^6*b^14*x^6 + 126*a^5*b^15*x^8 + 126*a^4*b^16*x^10 + 84*a^3*b^17*x^12 + 36*a^2*
b^18*x^14) + (1616615*a^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(65536*b^(23/2)) - (10*a*x)/b^11

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sympy [A]  time = 1.92, size = 299, normalized size = 1.37 \[ - \frac {10 a x}{b^{11}} - \frac {1616615 \sqrt {- \frac {a^{3}}{b^{23}}} \log {\left (x - \frac {b^{11} \sqrt {- \frac {a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac {1616615 \sqrt {- \frac {a^{3}}{b^{23}}} \log {\left (x + \frac {b^{11} \sqrt {- \frac {a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac {- 8651295 a^{10} x - 73208418 a^{9} b x^{3} - 272477394 a^{8} b^{2} x^{5} - 583302906 a^{7} b^{3} x^{7} - 786857984 a^{6} b^{4} x^{9} - 686588166 a^{5} b^{5} x^{11} - 379867950 a^{4} b^{6} x^{13} - 122613150 a^{3} b^{7} x^{15} - 17890785 a^{2} b^{8} x^{17}}{589824 a^{9} b^{11} + 5308416 a^{8} b^{12} x^{2} + 21233664 a^{7} b^{13} x^{4} + 49545216 a^{6} b^{14} x^{6} + 74317824 a^{5} b^{15} x^{8} + 74317824 a^{4} b^{16} x^{10} + 49545216 a^{3} b^{17} x^{12} + 21233664 a^{2} b^{18} x^{14} + 5308416 a b^{19} x^{16} + 589824 b^{20} x^{18}} + \frac {x^{3}}{3 b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**22/(b*x**2+a)**10,x)

[Out]

-10*a*x/b**11 - 1616615*sqrt(-a**3/b**23)*log(x - b**11*sqrt(-a**3/b**23)/a)/131072 + 1616615*sqrt(-a**3/b**23
)*log(x + b**11*sqrt(-a**3/b**23)/a)/131072 + (-8651295*a**10*x - 73208418*a**9*b*x**3 - 272477394*a**8*b**2*x
**5 - 583302906*a**7*b**3*x**7 - 786857984*a**6*b**4*x**9 - 686588166*a**5*b**5*x**11 - 379867950*a**4*b**6*x*
*13 - 122613150*a**3*b**7*x**15 - 17890785*a**2*b**8*x**17)/(589824*a**9*b**11 + 5308416*a**8*b**12*x**2 + 212
33664*a**7*b**13*x**4 + 49545216*a**6*b**14*x**6 + 74317824*a**5*b**15*x**8 + 74317824*a**4*b**16*x**10 + 4954
5216*a**3*b**17*x**12 + 21233664*a**2*b**18*x**14 + 5308416*a*b**19*x**16 + 589824*b**20*x**18) + x**3/(3*b**1
0)

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