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

Optimal. Leaf size=205 \[ \frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{17/2} b^{3/2}}+\frac{715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{x}{288 a b \left (a+b x^2\right )^8}-\frac{x}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x/(18*b*(a + b*x^2)^9) + x/(288*a*b*(a + b*x^2)^8) + (5*x)/(1344*a^2*b*(a + b*x
^2)^7) + (65*x)/(16128*a^3*b*(a + b*x^2)^6) + (143*x)/(32256*a^4*b*(a + b*x^2)^5
) + (143*x)/(28672*a^5*b*(a + b*x^2)^4) + (143*x)/(24576*a^6*b*(a + b*x^2)^3) +
(715*x)/(98304*a^7*b*(a + b*x^2)^2) + (715*x)/(65536*a^8*b*(a + b*x^2)) + (715*A
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(17/2)*b^(3/2))

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Rubi [A]  time = 0.248221, antiderivative size = 205, 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 \[ \frac{715 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{17/2} b^{3/2}}+\frac{715 x}{65536 a^8 b \left (a+b x^2\right )}+\frac{715 x}{98304 a^7 b \left (a+b x^2\right )^2}+\frac{143 x}{24576 a^6 b \left (a+b x^2\right )^3}+\frac{143 x}{28672 a^5 b \left (a+b x^2\right )^4}+\frac{143 x}{32256 a^4 b \left (a+b x^2\right )^5}+\frac{65 x}{16128 a^3 b \left (a+b x^2\right )^6}+\frac{5 x}{1344 a^2 b \left (a+b x^2\right )^7}+\frac{x}{288 a b \left (a+b x^2\right )^8}-\frac{x}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^2)^10,x]

[Out]

-x/(18*b*(a + b*x^2)^9) + x/(288*a*b*(a + b*x^2)^8) + (5*x)/(1344*a^2*b*(a + b*x
^2)^7) + (65*x)/(16128*a^3*b*(a + b*x^2)^6) + (143*x)/(32256*a^4*b*(a + b*x^2)^5
) + (143*x)/(28672*a^5*b*(a + b*x^2)^4) + (143*x)/(24576*a^6*b*(a + b*x^2)^3) +
(715*x)/(98304*a^7*b*(a + b*x^2)^2) + (715*x)/(65536*a^8*b*(a + b*x^2)) + (715*A
rcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(17/2)*b^(3/2))

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Rubi in Sympy [A]  time = 35.1521, size = 184, normalized size = 0.9 \[ - \frac{x}{18 b \left (a + b x^{2}\right )^{9}} + \frac{x}{288 a b \left (a + b x^{2}\right )^{8}} + \frac{5 x}{1344 a^{2} b \left (a + b x^{2}\right )^{7}} + \frac{65 x}{16128 a^{3} b \left (a + b x^{2}\right )^{6}} + \frac{143 x}{32256 a^{4} b \left (a + b x^{2}\right )^{5}} + \frac{143 x}{28672 a^{5} b \left (a + b x^{2}\right )^{4}} + \frac{143 x}{24576 a^{6} b \left (a + b x^{2}\right )^{3}} + \frac{715 x}{98304 a^{7} b \left (a + b x^{2}\right )^{2}} + \frac{715 x}{65536 a^{8} b \left (a + b x^{2}\right )} + \frac{715 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 a^{\frac{17}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(18*b*(a + b*x**2)**9) + x/(288*a*b*(a + b*x**2)**8) + 5*x/(1344*a**2*b*(a +
b*x**2)**7) + 65*x/(16128*a**3*b*(a + b*x**2)**6) + 143*x/(32256*a**4*b*(a + b*x
**2)**5) + 143*x/(28672*a**5*b*(a + b*x**2)**4) + 143*x/(24576*a**6*b*(a + b*x**
2)**3) + 715*x/(98304*a**7*b*(a + b*x**2)**2) + 715*x/(65536*a**8*b*(a + b*x**2)
) + 715*atan(sqrt(b)*x/sqrt(a))/(65536*a**(17/2)*b**(3/2))

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Mathematica [A]  time = 0.121624, size = 138, normalized size = 0.67 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (-45045 a^8+985866 a^7 b x^2+2633274 a^6 b^2 x^4+4349826 a^5 b^3 x^6+4685824 a^4 b^4 x^8+3317886 a^3 b^5 x^{10}+1495494 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{17/2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 + 985866*a^7*b*x^2 + 2633274*a^6*b^2*x^4 + 43498
26*a^5*b^3*x^6 + 4685824*a^4*b^4*x^8 + 3317886*a^3*b^5*x^10 + 1495494*a^2*b^6*x^
12 + 390390*a*b^7*x^14 + 45045*b^8*x^16))/(a + b*x^2)^9 + 45045*ArcTan[(Sqrt[b]*
x)/Sqrt[a]])/(4128768*a^(17/2)*b^(3/2))

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Maple [A]  time = 0.019, size = 124, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{715\,x}{65536\,b}}+{\frac{23473\,{x}^{3}}{98304\,a}}+{\frac{20899\,b{x}^{5}}{32768\,{a}^{2}}}+{\frac{241657\,{b}^{2}{x}^{7}}{229376\,{a}^{3}}}+{\frac{143\,{b}^{3}{x}^{9}}{126\,{a}^{4}}}+{\frac{184327\,{b}^{4}{x}^{11}}{229376\,{a}^{5}}}+{\frac{11869\,{b}^{5}{x}^{13}}{32768\,{a}^{6}}}+{\frac{9295\,{b}^{6}{x}^{15}}{98304\,{a}^{7}}}+{\frac{715\,{b}^{7}{x}^{17}}{65536\,{a}^{8}}} \right ) }+{\frac{715}{65536\,{a}^{8}b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)^10,x)

[Out]

(-715/65536*x/b+23473/98304/a*x^3+20899/32768*b/a^2*x^5+241657/229376*b^2/a^3*x^
7+143/126*b^3/a^4*x^9+184327/229376*b^4/a^5*x^11+11869/32768/a^6*b^5*x^13+9295/9
8304/a^7*b^6*x^15+715/65536/a^8*b^7*x^17)/(b*x^2+a)^9+715/65536/a^8/b/(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^2/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218748, size = 1, normalized size = 0. \[ \left [\frac{45045 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} + 985866 \, a^{7} b x^{3} - 45045 \, a^{8} x\right )} \sqrt{-a b}}{8257536 \,{\left (a^{8} b^{10} x^{18} + 9 \, a^{9} b^{9} x^{16} + 36 \, a^{10} b^{8} x^{14} + 84 \, a^{11} b^{7} x^{12} + 126 \, a^{12} b^{6} x^{10} + 126 \, a^{13} b^{5} x^{8} + 84 \, a^{14} b^{4} x^{6} + 36 \, a^{15} b^{3} x^{4} + 9 \, a^{16} b^{2} x^{2} + a^{17} b\right )} \sqrt{-a b}}, \frac{45045 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} + 985866 \, a^{7} b x^{3} - 45045 \, a^{8} x\right )} \sqrt{a b}}{4128768 \,{\left (a^{8} b^{10} x^{18} + 9 \, a^{9} b^{9} x^{16} + 36 \, a^{10} b^{8} x^{14} + 84 \, a^{11} b^{7} x^{12} + 126 \, a^{12} b^{6} x^{10} + 126 \, a^{13} b^{5} x^{8} + 84 \, a^{14} b^{4} x^{6} + 36 \, a^{15} b^{3} x^{4} + 9 \, a^{16} b^{2} x^{2} + a^{17} b\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8257536*(45045*(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)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) + 2*(45045*b^8*x^
17 + 390390*a*b^7*x^15 + 1495494*a^2*b^6*x^13 + 3317886*a^3*b^5*x^11 + 4685824*a
^4*b^4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 + 985866*a^7*b*x^3 - 4504
5*a^8*x)*sqrt(-a*b))/((a^8*b^10*x^18 + 9*a^9*b^9*x^16 + 36*a^10*b^8*x^14 + 84*a^
11*b^7*x^12 + 126*a^12*b^6*x^10 + 126*a^13*b^5*x^8 + 84*a^14*b^4*x^6 + 36*a^15*b
^3*x^4 + 9*a^16*b^2*x^2 + a^17*b)*sqrt(-a*b)), 1/4128768*(45045*(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) +
(45045*b^8*x^17 + 390390*a*b^7*x^15 + 1495494*a^2*b^6*x^13 + 3317886*a^3*b^5*x^
11 + 4685824*a^4*b^4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 + 985866*a^
7*b*x^3 - 45045*a^8*x)*sqrt(a*b))/((a^8*b^10*x^18 + 9*a^9*b^9*x^16 + 36*a^10*b^8
*x^14 + 84*a^11*b^7*x^12 + 126*a^12*b^6*x^10 + 126*a^13*b^5*x^8 + 84*a^14*b^4*x^
6 + 36*a^15*b^3*x^4 + 9*a^16*b^2*x^2 + a^17*b)*sqrt(a*b))]

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Sympy [A]  time = 33.8482, size = 286, normalized size = 1.4 \[ - \frac{715 \sqrt{- \frac{1}{a^{17} b^{3}}} \log{\left (- a^{9} b \sqrt{- \frac{1}{a^{17} b^{3}}} + x \right )}}{131072} + \frac{715 \sqrt{- \frac{1}{a^{17} b^{3}}} \log{\left (a^{9} b \sqrt{- \frac{1}{a^{17} b^{3}}} + x \right )}}{131072} + \frac{- 45045 a^{8} x + 985866 a^{7} b x^{3} + 2633274 a^{6} b^{2} x^{5} + 4349826 a^{5} b^{3} x^{7} + 4685824 a^{4} b^{4} x^{9} + 3317886 a^{3} b^{5} x^{11} + 1495494 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{4128768 a^{17} b + 37158912 a^{16} b^{2} x^{2} + 148635648 a^{15} b^{3} x^{4} + 346816512 a^{14} b^{4} x^{6} + 520224768 a^{13} b^{5} x^{8} + 520224768 a^{12} b^{6} x^{10} + 346816512 a^{11} b^{7} x^{12} + 148635648 a^{10} b^{8} x^{14} + 37158912 a^{9} b^{9} x^{16} + 4128768 a^{8} b^{10} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-715*sqrt(-1/(a**17*b**3))*log(-a**9*b*sqrt(-1/(a**17*b**3)) + x)/131072 + 715*s
qrt(-1/(a**17*b**3))*log(a**9*b*sqrt(-1/(a**17*b**3)) + x)/131072 + (-45045*a**8
*x + 985866*a**7*b*x**3 + 2633274*a**6*b**2*x**5 + 4349826*a**5*b**3*x**7 + 4685
824*a**4*b**4*x**9 + 3317886*a**3*b**5*x**11 + 1495494*a**2*b**6*x**13 + 390390*
a*b**7*x**15 + 45045*b**8*x**17)/(4128768*a**17*b + 37158912*a**16*b**2*x**2 + 1
48635648*a**15*b**3*x**4 + 346816512*a**14*b**4*x**6 + 520224768*a**13*b**5*x**8
+ 520224768*a**12*b**6*x**10 + 346816512*a**11*b**7*x**12 + 148635648*a**10*b**
8*x**14 + 37158912*a**9*b**9*x**16 + 4128768*a**8*b**10*x**18)

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GIAC/XCAS [A]  time = 0.210571, size = 173, normalized size = 0.84 \[ \frac{715 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{8} b} + \frac{45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} + 1495494 \, a^{2} b^{6} x^{13} + 3317886 \, a^{3} b^{5} x^{11} + 4685824 \, a^{4} b^{4} x^{9} + 4349826 \, a^{5} b^{3} x^{7} + 2633274 \, a^{6} b^{2} x^{5} + 985866 \, a^{7} b x^{3} - 45045 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{8} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

715/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^8*b) + 1/4128768*(45045*b^8*x^17 +
390390*a*b^7*x^15 + 1495494*a^2*b^6*x^13 + 3317886*a^3*b^5*x^11 + 4685824*a^4*b^
4*x^9 + 4349826*a^5*b^3*x^7 + 2633274*a^6*b^2*x^5 + 985866*a^7*b*x^3 - 45045*a^8
*x)/((b*x^2 + a)^9*a^8*b)