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

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]

-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)/(9
8304*a^4*b^4*(a + b*x^2)^2) + (35*x)/(65536*a^5*b^4*(a + b*x^2)) + (35*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/(65536*a^(11/2)*b^(9/2))

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Rubi [A]  time = 0.287041, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{35 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{11/2} b^{9/2}}+\frac{35 x}{65536 a^5 b^4 \left (a+b x^2\right )}+\frac{35 x}{98304 a^4 b^4 \left (a+b x^2\right )^2}+\frac{7 x}{24576 a^3 b^4 \left (a+b x^2\right )^3}+\frac{x}{4096 a^2 b^4 \left (a+b x^2\right )^4}+\frac{x}{4608 a b^4 \left (a+b x^2\right )^5}-\frac{5 x}{2304 b^4 \left (a+b x^2\right )^6}-\frac{5 x^3}{576 b^3 \left (a+b x^2\right )^7}-\frac{7 x^5}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^7}{18 b \left (a+b x^2\right )^9} \]

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)/(9
8304*a^4*b^4*(a + b*x^2)^2) + (35*x)/(65536*a^5*b^4*(a + b*x^2)) + (35*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/(65536*a^(11/2)*b^(9/2))

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Rubi in Sympy [A]  time = 44.8905, size = 190, normalized size = 0.94 \[ - \frac{x^{7}}{18 b \left (a + b x^{2}\right )^{9}} - \frac{7 x^{5}}{288 b^{2} \left (a + b x^{2}\right )^{8}} - \frac{5 x^{3}}{576 b^{3} \left (a + b x^{2}\right )^{7}} - \frac{5 x}{2304 b^{4} \left (a + b x^{2}\right )^{6}} + \frac{x}{4608 a b^{4} \left (a + b x^{2}\right )^{5}} + \frac{x}{4096 a^{2} b^{4} \left (a + b x^{2}\right )^{4}} + \frac{7 x}{24576 a^{3} b^{4} \left (a + b x^{2}\right )^{3}} + \frac{35 x}{98304 a^{4} b^{4} \left (a + b x^{2}\right )^{2}} + \frac{35 x}{65536 a^{5} b^{4} \left (a + b x^{2}\right )} + \frac{35 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 a^{\frac{11}{2}} b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**2+a)**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*atan(sqrt(b)*x/sqrt(a))/(65536*a**(11/2)*b**(9/2))

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Mathematica [A]  time = 0.134277, 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 + 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))/(a + b*x^2)^9 + 315*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(589824
*a^(11/2)*b^(9/2))

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Maple [A]  time = 0.02, size = 122, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{35\,{a}^{3}x}{65536\,{b}^{4}}}-{\frac{455\,{a}^{2}{x}^{3}}{98304\,{b}^{3}}}-{\frac{581\,a{x}^{5}}{32768\,{b}^{2}}}-{\frac{1289\,{x}^{7}}{32768\,b}}+{\frac{{x}^{9}}{18\,a}}+{\frac{1289\,b{x}^{11}}{32768\,{a}^{2}}}+{\frac{581\,{b}^{2}{x}^{13}}{32768\,{a}^{3}}}+{\frac{455\,{b}^{3}{x}^{15}}{98304\,{a}^{4}}}+{\frac{35\,{b}^{4}{x}^{17}}{65536\,{a}^{5}}} \right ) }+{\frac{35}{65536\,{a}^{5}{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^2+a)^10,x)

[Out]

(-35/65536*a^3*x/b^4-455/98304*a^2*x^3/b^3-581/32768*a*x^5/b^2-1289/32768*x^7/b+
1/18/a*x^9+1289/32768*b/a^2*x^11+581/32768*b^2/a^3*x^13+455/98304*b^3/a^4*x^15+3
5/65536*b^4/a^5*x^17)/(b*x^2+a)^9+35/65536/a^5/b^4/(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^8/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218821, size = 1, normalized size = 0. \[ \left [\frac{315 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x\right )} \sqrt{-a b}}{1179648 \,{\left (a^{5} b^{13} x^{18} + 9 \, a^{6} b^{12} x^{16} + 36 \, a^{7} b^{11} x^{14} + 84 \, a^{8} b^{10} x^{12} + 126 \, a^{9} b^{9} x^{10} + 126 \, a^{10} b^{8} x^{8} + 84 \, a^{11} b^{7} x^{6} + 36 \, a^{12} b^{6} x^{4} + 9 \, a^{13} b^{5} x^{2} + a^{14} b^{4}\right )} \sqrt{-a b}}, \frac{315 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x\right )} \sqrt{a b}}{589824 \,{\left (a^{5} b^{13} x^{18} + 9 \, a^{6} b^{12} x^{16} + 36 \, a^{7} b^{11} x^{14} + 84 \, a^{8} b^{10} x^{12} + 126 \, a^{9} b^{9} x^{10} + 126 \, a^{10} b^{8} x^{8} + 84 \, a^{11} b^{7} x^{6} + 36 \, a^{12} b^{6} x^{4} + 9 \, a^{13} b^{5} x^{2} + a^{14} b^{4}\right )} \sqrt{a b}}\right ] \]

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*(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 + 1
26*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)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(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)*sqrt(-a*b))
/((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)*sqrt(-a*b)), 1/589824*(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)*arctan(sqrt(a*b)*x/a) + (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 - 2320
2*a^5*b^3*x^7 - 10458*a^6*b^2*x^5 - 2730*a^7*b*x^3 - 315*a^8*x)*sqrt(a*b))/((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)*sqrt(a*b))]

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Sympy [A]  time = 34.1086, 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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GIAC/XCAS [A]  time = 0.214016, size = 173, normalized size = 0.86 \[ \frac{35 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{5} b^{4}} + \frac{315 \, b^{8} x^{17} + 2730 \, a b^{7} x^{15} + 10458 \, a^{2} b^{6} x^{13} + 23202 \, a^{3} b^{5} x^{11} + 32768 \, a^{4} b^{4} x^{9} - 23202 \, a^{5} b^{3} x^{7} - 10458 \, a^{6} b^{2} x^{5} - 2730 \, a^{7} b x^{3} - 315 \, a^{8} x}{589824 \,{\left (b x^{2} + a\right )}^{9} a^{5} b^{4}} \]

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 + 27
30*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 - 23
202*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)