3.223 \(\int \frac{1}{x^4 \left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=220 \[ \frac{1616615 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{23/2}}+\frac{1616615 b}{65536 a^{11} x}-\frac{1616615}{196608 a^{10} x^3}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9} \]

[Out]

-1616615/(196608*a^10*x^3) + (1616615*b)/(65536*a^11*x) + 1/(18*a*x^3*(a + b*x^2
)^9) + 7/(96*a^2*x^3*(a + b*x^2)^8) + 19/(192*a^3*x^3*(a + b*x^2)^7) + 323/(2304
*a^4*x^3*(a + b*x^2)^6) + 323/(1536*a^5*x^3*(a + b*x^2)^5) + 4199/(12288*a^6*x^3
*(a + b*x^2)^4) + 46189/(73728*a^7*x^3*(a + b*x^2)^3) + 46189/(32768*a^8*x^3*(a
+ b*x^2)^2) + 323323/(65536*a^9*x^3*(a + b*x^2)) + (1616615*b^(3/2)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(65536*a^(23/2))

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Rubi [A]  time = 0.353685, antiderivative size = 220, 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 \[ \frac{1616615 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{23/2}}+\frac{1616615 b}{65536 a^{11} x}-\frac{1616615}{196608 a^{10} x^3}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(a + b*x^2)^10),x]

[Out]

-1616615/(196608*a^10*x^3) + (1616615*b)/(65536*a^11*x) + 1/(18*a*x^3*(a + b*x^2
)^9) + 7/(96*a^2*x^3*(a + b*x^2)^8) + 19/(192*a^3*x^3*(a + b*x^2)^7) + 323/(2304
*a^4*x^3*(a + b*x^2)^6) + 323/(1536*a^5*x^3*(a + b*x^2)^5) + 4199/(12288*a^6*x^3
*(a + b*x^2)^4) + 46189/(73728*a^7*x^3*(a + b*x^2)^3) + 46189/(32768*a^8*x^3*(a
+ b*x^2)^2) + 323323/(65536*a^9*x^3*(a + b*x^2)) + (1616615*b^(3/2)*ArcTan[(Sqrt
[b]*x)/Sqrt[a]])/(65536*a^(23/2))

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Rubi in Sympy [A]  time = 61.4104, size = 211, normalized size = 0.96 \[ \frac{1}{18 a x^{3} \left (a + b x^{2}\right )^{9}} + \frac{7}{96 a^{2} x^{3} \left (a + b x^{2}\right )^{8}} + \frac{19}{192 a^{3} x^{3} \left (a + b x^{2}\right )^{7}} + \frac{323}{2304 a^{4} x^{3} \left (a + b x^{2}\right )^{6}} + \frac{323}{1536 a^{5} x^{3} \left (a + b x^{2}\right )^{5}} + \frac{4199}{12288 a^{6} x^{3} \left (a + b x^{2}\right )^{4}} + \frac{46189}{73728 a^{7} x^{3} \left (a + b x^{2}\right )^{3}} + \frac{46189}{32768 a^{8} x^{3} \left (a + b x^{2}\right )^{2}} + \frac{323323}{65536 a^{9} x^{3} \left (a + b x^{2}\right )} - \frac{1616615}{196608 a^{10} x^{3}} + \frac{1616615 b}{65536 a^{11} x} + \frac{1616615 b^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 a^{\frac{23}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**2+a)**10,x)

[Out]

1/(18*a*x**3*(a + b*x**2)**9) + 7/(96*a**2*x**3*(a + b*x**2)**8) + 19/(192*a**3*
x**3*(a + b*x**2)**7) + 323/(2304*a**4*x**3*(a + b*x**2)**6) + 323/(1536*a**5*x*
*3*(a + b*x**2)**5) + 4199/(12288*a**6*x**3*(a + b*x**2)**4) + 46189/(73728*a**7
*x**3*(a + b*x**2)**3) + 46189/(32768*a**8*x**3*(a + b*x**2)**2) + 323323/(65536
*a**9*x**3*(a + b*x**2)) - 1616615/(196608*a**10*x**3) + 1616615*b/(65536*a**11*
x) + 1616615*b**(3/2)*atan(sqrt(b)*x/sqrt(a))/(65536*a**(23/2))

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Mathematica [A]  time = 0.168704, size = 157, normalized size = 0.71 \[ \frac{\frac{\sqrt{a} \left (-196608 a^{10}+4128768 a^9 b x^2+63897057 a^8 b^2 x^4+318434718 a^7 b^3 x^6+850547502 a^6 b^4 x^8+1404993798 a^5 b^5 x^{10}[phone] a^4 b^6 x^{12}[phone] a^3 b^7 x^{14}[phone] a^2 b^8 x^{16}[phone] a b^9 x^{18}[phone] b^{10} x^{20}\right )}{x^3 \left (a+b x^2\right )^9}[phone] b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{589824 a^{23/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(a + b*x^2)^10),x]

[Out]

((Sqrt[a]*(-196608*a^10 + 4128768*a^9*b*x^2 + 63897057*a^8*b^2*x^4 + 318434718*a
^7*b^3*x^6 + 850547502*a^6*b^4*x^8 + 1404993798*a^5*b^5*x^10 + 1513521152*a^4*b^
6*x^12 + 1071677178*a^3*b^7*x^14 + 483044562*a^2*b^8*x^16 + 126095970*a*b^9*x^18
+ 14549535*b^10*x^20))/(x^3*(a + b*x^2)^9) + 14549535*b^(3/2)*ArcTan[(Sqrt[b]*x
)/Sqrt[a]])/(589824*a^(23/2))

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Maple [A]  time = 0.03, size = 219, normalized size = 1. \[ -{\frac{1}{3\,{a}^{10}{x}^{3}}}+10\,{\frac{b}{{a}^{11}x}}+{\frac{1987865\,{b}^{2}x}{65536\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{20435525\,{b}^{3}{x}^{3}}{98304\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{21103775\,{b}^{4}{x}^{5}}{32768\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{38143787\,{b}^{5}{x}^{7}}{32768\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{24013\,{b}^{6}{x}^{9}}{18\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{32405717\,{b}^{7}{x}^{11}}{32768\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{15137633\,{b}^{8}{x}^{13}}{32768\,{a}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{12201403\,{b}^{9}{x}^{15}}{98304\,{a}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{961255\,{b}^{10}{x}^{17}}{65536\,{a}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{1616615\,{b}^{2}}{65536\,{a}^{11}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^10,x)

[Out]

-1/3/a^10/x^3+10*b/a^11/x+1987865/65536/a^3*b^2/(b*x^2+a)^9*x+20435525/98304/a^4
*b^3/(b*x^2+a)^9*x^3+21103775/32768/a^5*b^4/(b*x^2+a)^9*x^5+38143787/32768/a^6*b
^5/(b*x^2+a)^9*x^7+24013/18/a^7*b^6/(b*x^2+a)^9*x^9+32405717/32768/a^8*b^7/(b*x^
2+a)^9*x^11+15137633/32768/a^9*b^8/(b*x^2+a)^9*x^13+12201403/98304/a^10*b^9/(b*x
^2+a)^9*x^15+961255/65536/a^11*b^10/(b*x^2+a)^9*x^17+1616615/65536/a^11*b^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(1/((b*x^2 + a)^10*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.250808, size = 1, normalized size = 0. \[ \left [\frac{29099070 \, b^{10} x^{20} + 252191940 \, a b^{9} x^{18} + 966089124 \, a^{2} b^{8} x^{16} + 2143354356 \, a^{3} b^{7} x^{14} + 3027042304 \, a^{4} b^{6} x^{12} + 2809987596 \, a^{5} b^{5} x^{10} + 1701095004 \, a^{6} b^{4} x^{8} + 636869436 \, a^{7} b^{3} x^{6} + 127794114 \, a^{8} b^{2} x^{4} + 8257536 \, a^{9} b x^{2} - 393216 \, a^{10} + 14549535 \,{\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{1179648 \,{\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}, \frac{14549535 \, b^{10} x^{20} + 126095970 \, a b^{9} x^{18} + 483044562 \, a^{2} b^{8} x^{16} + 1071677178 \, a^{3} b^{7} x^{14} + 1513521152 \, a^{4} b^{6} x^{12} + 1404993798 \, a^{5} b^{5} x^{10} + 850547502 \, a^{6} b^{4} x^{8} + 318434718 \, a^{7} b^{3} x^{6} + 63897057 \, a^{8} b^{2} x^{4} + 4128768 \, a^{9} b x^{2} - 196608 \, a^{10} + 14549535 \,{\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{589824 \,{\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^10*x^4),x, algorithm="fricas")

[Out]

[1/1179648*(29099070*b^10*x^20 + 252191940*a*b^9*x^18 + 966089124*a^2*b^8*x^16 +
2143354356*a^3*b^7*x^14 + 3027042304*a^4*b^6*x^12 + 2809987596*a^5*b^5*x^10 + 1
701095004*a^6*b^4*x^8 + 636869436*a^7*b^3*x^6 + 127794114*a^8*b^2*x^4 + 8257536*
a^9*b*x^2 - 393216*a^10 + 14549535*(b^10*x^21 + 9*a*b^9*x^19 + 36*a^2*b^8*x^17 +
84*a^3*b^7*x^15 + 126*a^4*b^6*x^13 + 126*a^5*b^5*x^11 + 84*a^6*b^4*x^9 + 36*a^7
*b^3*x^7 + 9*a^8*b^2*x^5 + a^9*b*x^3)*sqrt(-b/a)*log((b*x^2 + 2*a*x*sqrt(-b/a) -
a)/(b*x^2 + a)))/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7*x^17 + 84*a^14*
b^6*x^15 + 126*a^15*b^5*x^13 + 126*a^16*b^4*x^11 + 84*a^17*b^3*x^9 + 36*a^18*b^2
*x^7 + 9*a^19*b*x^5 + a^20*x^3), 1/589824*(14549535*b^10*x^20 + 126095970*a*b^9*
x^18 + 483044562*a^2*b^8*x^16 + 1071677178*a^3*b^7*x^14 + 1513521152*a^4*b^6*x^1
2 + 1404993798*a^5*b^5*x^10 + 850547502*a^6*b^4*x^8 + 318434718*a^7*b^3*x^6 + 63
897057*a^8*b^2*x^4 + 4128768*a^9*b*x^2 - 196608*a^10 + 14549535*(b^10*x^21 + 9*a
*b^9*x^19 + 36*a^2*b^8*x^17 + 84*a^3*b^7*x^15 + 126*a^4*b^6*x^13 + 126*a^5*b^5*x
^11 + 84*a^6*b^4*x^9 + 36*a^7*b^3*x^7 + 9*a^8*b^2*x^5 + a^9*b*x^3)*sqrt(b/a)*arc
tan(b*x/(a*sqrt(b/a))))/(a^11*b^9*x^21 + 9*a^12*b^8*x^19 + 36*a^13*b^7*x^17 + 84
*a^14*b^6*x^15 + 126*a^15*b^5*x^13 + 126*a^16*b^4*x^11 + 84*a^17*b^3*x^9 + 36*a^
18*b^2*x^7 + 9*a^19*b*x^5 + a^20*x^3)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**2+a)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219334, size = 200, normalized size = 0.91 \[ \frac{1616615 \, b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{11}} + \frac{30 \, b x^{2} - a}{3 \, a^{11} x^{3}} + \frac{8651295 \, b^{10} x^{17} + 73208418 \, a b^{9} x^{15} + 272477394 \, a^{2} b^{8} x^{13} + 583302906 \, a^{3} b^{7} x^{11} + 786857984 \, a^{4} b^{6} x^{9} + 686588166 \, a^{5} b^{5} x^{7} + 379867950 \, a^{6} b^{4} x^{5} + 122613150 \, a^{7} b^{3} x^{3} + 17890785 \, a^{8} b^{2} x}{589824 \,{\left (b x^{2} + a\right )}^{9} a^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^10*x^4),x, algorithm="giac")

[Out]

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