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

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*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) - (3
23*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)/(32768*b^8*(a + b*x^2)^2) - (323323*x^5)/(65536*b^9*(a + b*x^2)) + (161661
5*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*b^(23/2))

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Rubi [A]  time = 0.372654, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1616615 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{23/2}}-\frac{1616615 a x}{65536 b^{11}}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}+\frac{1616615 x^3}{196608 b^{10}} \]

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) - (3
23*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)/(32768*b^8*(a + b*x^2)^2) - (323323*x^5)/(65536*b^9*(a + b*x^2)) + (161661
5*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*b^(23/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{1616615 a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 b^{\frac{23}{2}}} - \frac{x^{21}}{18 b \left (a + b x^{2}\right )^{9}} - \frac{7 x^{19}}{96 b^{2} \left (a + b x^{2}\right )^{8}} - \frac{19 x^{17}}{192 b^{3} \left (a + b x^{2}\right )^{7}} - \frac{323 x^{15}}{2304 b^{4} \left (a + b x^{2}\right )^{6}} - \frac{323 x^{13}}{1536 b^{5} \left (a + b x^{2}\right )^{5}} - \frac{4199 x^{11}}{12288 b^{6} \left (a + b x^{2}\right )^{4}} - \frac{46189 x^{9}}{73728 b^{7} \left (a + b x^{2}\right )^{3}} - \frac{46189 x^{7}}{32768 b^{8} \left (a + b x^{2}\right )^{2}} - \frac{323323 x^{5}}{65536 b^{9} \left (a + b x^{2}\right )} + \frac{1616615 x^{3}}{196608 b^{10}} - \frac{1616615 \int a\, dx}{65536 b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**22/(b*x**2+a)**10,x)

[Out]

1616615*a**(3/2)*atan(sqrt(b)*x/sqrt(a))/(65536*b**(23/2)) - 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/(32768*b**8*(a + b*x**2)**2) - 323323*x**5/(65536*b**9*(a +
b*x**2)) + 1616615*x**3/(196608*b**10) - 1616615*Integral(a, x)/(65536*b**11)

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Mathematica [A]  time = 0.156047, 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 - 1071
677178*a^7*b^3*x^6 - 1513521152*a^6*b^4*x^8 - 1404993798*a^5*b^5*x^10 - 85054750
2*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)/Sq
rt[a]])/(589824*b^(23/2))

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Maple [A]  time = 0.028, size = 217, normalized size = 1. \[{\frac{{x}^{3}}{3\,{b}^{10}}}-10\,{\frac{ax}{{b}^{11}}}-{\frac{961255\,{a}^{10}x}{65536\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{12201403\,{a}^{9}{x}^{3}}{98304\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{15137633\,{a}^{8}{x}^{5}}{32768\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{32405717\,{a}^{7}{x}^{7}}{32768\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{24013\,{a}^{6}{x}^{9}}{18\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{38143787\,{a}^{5}{x}^{11}}{32768\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{21103775\,{a}^{4}{x}^{13}}{32768\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{20435525\,{a}^{3}{x}^{15}}{98304\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{1987865\,{a}^{2}{x}^{17}}{65536\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{1616615\,{a}^{2}}{65536\,{b}^{11}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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^1
0*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^9-38143787/32768/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(x*b/(a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217931, size = 1, normalized size = 0. \[ \left [\frac{393216 \, b^{10} x^{21} - 8257536 \, a b^{9} x^{19} - 127794114 \, a^{2} b^{8} x^{17} - 636869436 \, a^{3} b^{7} x^{15} - 1701095004 \, a^{4} b^{6} x^{13} - 2809987596 \, a^{5} b^{5} x^{11} - 3027042304 \, a^{6} b^{4} x^{9} - 2143354356 \, a^{7} b^{3} x^{7} - 966089124 \, a^{8} b^{2} x^{5} - 252191940 \, a^{9} b x^{3} - 29099070 \, a^{10} x + 14549535 \,{\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{1179648 \,{\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}, \frac{196608 \, b^{10} x^{21} - 4128768 \, a b^{9} x^{19} - 63897057 \, a^{2} b^{8} x^{17} - 318434718 \, a^{3} b^{7} x^{15} - 850547502 \, a^{4} b^{6} x^{13} - 1404993798 \, a^{5} b^{5} x^{11} - 1513521152 \, a^{6} b^{4} x^{9} - 1071677178 \, a^{7} b^{3} x^{7} - 483044562 \, a^{8} b^{2} x^{5} - 126095970 \, a^{9} b x^{3} - 14549535 \, a^{10} x + 14549535 \,{\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right )}{589824 \,{\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}}\right ] \]

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 - 636
869436*a^3*b^7*x^15 - 1701095004*a^4*b^6*x^13 - 2809987596*a^5*b^5*x^11 - 302704
2304*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 - 63897
057*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*(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(x/sqrt(
a/b)))/(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)]

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Sympy [A]  time = 37.1602, size = 298, 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)/131
072 + 1616615*sqrt(-a**3/b**23)*log(x + b**11*sqrt(-a**3/b**23)/a)/131072 - (865
1295*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) + x**3/(3*b**10)

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GIAC/XCAS [A]  time = 0.215538, size = 203, normalized size = 0.93 \[ \frac{1616615 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} b^{11}} - \frac{17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \,{\left (b x^{2} + a\right )}^{9} b^{11}} + \frac{b^{20} x^{3} - 30 \, a b^{19} x}{3 \, b^{30}} \]

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 + 272477394*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