3.214 \(\int \frac{x^{14}}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=199 \[ \frac{143 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{5/2} b^{15/2}}+\frac{143 x}{65536 a^2 b^7 \left (a+b x^2\right )}+\frac{143 x}{98304 a b^7 \left (a+b x^2\right )^2}-\frac{143 x}{24576 b^7 \left (a+b x^2\right )^3}-\frac{143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac{143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac{143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac{143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac{13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^{13}}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^13/(18*b*(a + b*x^2)^9) - (13*x^11)/(288*b^2*(a + b*x^2)^8) - (143*x^9)/(4032
*b^3*(a + b*x^2)^7) - (143*x^7)/(5376*b^4*(a + b*x^2)^6) - (143*x^5)/(7680*b^5*(
a + b*x^2)^5) - (143*x^3)/(12288*b^6*(a + b*x^2)^4) - (143*x)/(24576*b^7*(a + b*
x^2)^3) + (143*x)/(98304*a*b^7*(a + b*x^2)^2) + (143*x)/(65536*a^2*b^7*(a + b*x^
2)) + (143*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(5/2)*b^(15/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.311144, antiderivative size = 199, 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 \[ \frac{143 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{5/2} b^{15/2}}+\frac{143 x}{65536 a^2 b^7 \left (a+b x^2\right )}+\frac{143 x}{98304 a b^7 \left (a+b x^2\right )^2}-\frac{143 x}{24576 b^7 \left (a+b x^2\right )^3}-\frac{143 x^3}{12288 b^6 \left (a+b x^2\right )^4}-\frac{143 x^5}{7680 b^5 \left (a+b x^2\right )^5}-\frac{143 x^7}{5376 b^4 \left (a+b x^2\right )^6}-\frac{143 x^9}{4032 b^3 \left (a+b x^2\right )^7}-\frac{13 x^{11}}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^{13}}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[x^14/(a + b*x^2)^10,x]

[Out]

-x^13/(18*b*(a + b*x^2)^9) - (13*x^11)/(288*b^2*(a + b*x^2)^8) - (143*x^9)/(4032
*b^3*(a + b*x^2)^7) - (143*x^7)/(5376*b^4*(a + b*x^2)^6) - (143*x^5)/(7680*b^5*(
a + b*x^2)^5) - (143*x^3)/(12288*b^6*(a + b*x^2)^4) - (143*x)/(24576*b^7*(a + b*
x^2)^3) + (143*x)/(98304*a*b^7*(a + b*x^2)^2) + (143*x)/(65536*a^2*b^7*(a + b*x^
2)) + (143*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*a^(5/2)*b^(15/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.7464, size = 189, normalized size = 0.95 \[ - \frac{x^{13}}{18 b \left (a + b x^{2}\right )^{9}} - \frac{13 x^{11}}{288 b^{2} \left (a + b x^{2}\right )^{8}} - \frac{143 x^{9}}{4032 b^{3} \left (a + b x^{2}\right )^{7}} - \frac{143 x^{7}}{5376 b^{4} \left (a + b x^{2}\right )^{6}} - \frac{143 x^{5}}{7680 b^{5} \left (a + b x^{2}\right )^{5}} - \frac{143 x^{3}}{12288 b^{6} \left (a + b x^{2}\right )^{4}} - \frac{143 x}{24576 b^{7} \left (a + b x^{2}\right )^{3}} + \frac{143 x}{98304 a b^{7} \left (a + b x^{2}\right )^{2}} + \frac{143 x}{65536 a^{2} b^{7} \left (a + b x^{2}\right )} + \frac{143 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 a^{\frac{5}{2}} b^{\frac{15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**2+a)**10,x)

[Out]

-x**13/(18*b*(a + b*x**2)**9) - 13*x**11/(288*b**2*(a + b*x**2)**8) - 143*x**9/(
4032*b**3*(a + b*x**2)**7) - 143*x**7/(5376*b**4*(a + b*x**2)**6) - 143*x**5/(76
80*b**5*(a + b*x**2)**5) - 143*x**3/(12288*b**6*(a + b*x**2)**4) - 143*x/(24576*
b**7*(a + b*x**2)**3) + 143*x/(98304*a*b**7*(a + b*x**2)**2) + 143*x/(65536*a**2
*b**7*(a + b*x**2)) + 143*atan(sqrt(b)*x/sqrt(a))/(65536*a**(5/2)*b**(15/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.122365, size = 138, normalized size = 0.69 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (-45045 a^8-390390 a^7 b x^2-1495494 a^6 b^2 x^4-3317886 a^5 b^3 x^6-4685824 a^4 b^4 x^8-4349826 a^3 b^5 x^{10}-2633274 a^2 b^6 x^{12}+390390 a b^7 x^{14}+45045 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+45045 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{20643840 a^{5/2} b^{15/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(a + b*x^2)^10,x]

[Out]

((Sqrt[a]*Sqrt[b]*x*(-45045*a^8 - 390390*a^7*b*x^2 - 1495494*a^6*b^2*x^4 - 33178
86*a^5*b^3*x^6 - 4685824*a^4*b^4*x^8 - 4349826*a^3*b^5*x^10 - 2633274*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]])/(20643840*a^(5/2)*b^(15/2))

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 122, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{143\,{a}^{6}x}{65536\,{b}^{7}}}-{\frac{1859\,{a}^{5}{x}^{3}}{98304\,{b}^{6}}}-{\frac{11869\,{a}^{4}{x}^{5}}{163840\,{b}^{5}}}-{\frac{184327\,{a}^{3}{x}^{7}}{1146880\,{b}^{4}}}-{\frac{143\,{a}^{2}{x}^{9}}{630\,{b}^{3}}}-{\frac{241657\,a{x}^{11}}{1146880\,{b}^{2}}}-{\frac{20899\,{x}^{13}}{163840\,b}}+{\frac{1859\,{x}^{15}}{98304\,a}}+{\frac{143\,b{x}^{17}}{65536\,{a}^{2}}} \right ) }+{\frac{143}{65536\,{a}^{2}{b}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^2+a)^10,x)

[Out]

(-143/65536*a^6/b^7*x-1859/98304*a^5/b^6*x^3-11869/163840*a^4/b^5*x^5-184327/114
6880*a^3/b^4*x^7-143/630*a^2/b^3*x^9-241657/1146880*a/b^2*x^11-20899/163840/b*x^
13+1859/98304/a*x^15+143/65536*b/a^2*x^17)/(b*x^2+a)^9+143/65536/a^2/b^7/(a*b)^(
1/2)*arctan(x*b/(a*b)^(1/2))

_______________________________________________________________________________________

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^14/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.217049, size = 1, normalized size = 0.01 \[ \left [\frac{45045 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x\right )} \sqrt{-a b}}{41287680 \,{\left (a^{2} b^{16} x^{18} + 9 \, a^{3} b^{15} x^{16} + 36 \, a^{4} b^{14} x^{14} + 84 \, a^{5} b^{13} x^{12} + 126 \, a^{6} b^{12} x^{10} + 126 \, a^{7} b^{11} x^{8} + 84 \, a^{8} b^{10} x^{6} + 36 \, a^{9} b^{9} x^{4} + 9 \, a^{10} b^{8} x^{2} + a^{11} b^{7}\right )} \sqrt{-a b}}, \frac{45045 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x\right )} \sqrt{a b}}{20643840 \,{\left (a^{2} b^{16} x^{18} + 9 \, a^{3} b^{15} x^{16} + 36 \, a^{4} b^{14} x^{14} + 84 \, a^{5} b^{13} x^{12} + 126 \, a^{6} b^{12} x^{10} + 126 \, a^{7} b^{11} x^{8} + 84 \, a^{8} b^{10} x^{6} + 36 \, a^{9} b^{9} x^{4} + 9 \, a^{10} b^{8} x^{2} + a^{11} b^{7}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

[1/41287680*(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 - 2633274*a^2*b^6*x^13 - 4349826*a^3*b^5*x^11 - 4685824*
a^4*b^4*x^9 - 3317886*a^5*b^3*x^7 - 1495494*a^6*b^2*x^5 - 390390*a^7*b*x^3 - 450
45*a^8*x)*sqrt(-a*b))/((a^2*b^16*x^18 + 9*a^3*b^15*x^16 + 36*a^4*b^14*x^14 + 84*
a^5*b^13*x^12 + 126*a^6*b^12*x^10 + 126*a^7*b^11*x^8 + 84*a^8*b^10*x^6 + 36*a^9*
b^9*x^4 + 9*a^10*b^8*x^2 + a^11*b^7)*sqrt(-a*b)), 1/20643840*(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 - 2633274*a^2*b^6*x^13 - 4349826*a^3*b^
5*x^11 - 4685824*a^4*b^4*x^9 - 3317886*a^5*b^3*x^7 - 1495494*a^6*b^2*x^5 - 39039
0*a^7*b*x^3 - 45045*a^8*x)*sqrt(a*b))/((a^2*b^16*x^18 + 9*a^3*b^15*x^16 + 36*a^4
*b^14*x^14 + 84*a^5*b^13*x^12 + 126*a^6*b^12*x^10 + 126*a^7*b^11*x^8 + 84*a^8*b^
10*x^6 + 36*a^9*b^9*x^4 + 9*a^10*b^8*x^2 + a^11*b^7)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [A]  time = 35.2115, size = 291, normalized size = 1.46 \[ - \frac{143 \sqrt{- \frac{1}{a^{5} b^{15}}} \log{\left (- a^{3} b^{7} \sqrt{- \frac{1}{a^{5} b^{15}}} + x \right )}}{131072} + \frac{143 \sqrt{- \frac{1}{a^{5} b^{15}}} \log{\left (a^{3} b^{7} \sqrt{- \frac{1}{a^{5} b^{15}}} + x \right )}}{131072} + \frac{- 45045 a^{8} x - 390390 a^{7} b x^{3} - 1495494 a^{6} b^{2} x^{5} - 3317886 a^{5} b^{3} x^{7} - 4685824 a^{4} b^{4} x^{9} - 4349826 a^{3} b^{5} x^{11} - 2633274 a^{2} b^{6} x^{13} + 390390 a b^{7} x^{15} + 45045 b^{8} x^{17}}{20643840 a^{11} b^{7} + 185794560 a^{10} b^{8} x^{2} + 743178240 a^{9} b^{9} x^{4} + 1734082560 a^{8} b^{10} x^{6} + 2601123840 a^{7} b^{11} x^{8} + 2601123840 a^{6} b^{12} x^{10} + 1734082560 a^{5} b^{13} x^{12} + 743178240 a^{4} b^{14} x^{14} + 185794560 a^{3} b^{15} x^{16} + 20643840 a^{2} b^{16} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b*x**2+a)**10,x)

[Out]

-143*sqrt(-1/(a**5*b**15))*log(-a**3*b**7*sqrt(-1/(a**5*b**15)) + x)/131072 + 14
3*sqrt(-1/(a**5*b**15))*log(a**3*b**7*sqrt(-1/(a**5*b**15)) + x)/131072 + (-4504
5*a**8*x - 390390*a**7*b*x**3 - 1495494*a**6*b**2*x**5 - 3317886*a**5*b**3*x**7
- 4685824*a**4*b**4*x**9 - 4349826*a**3*b**5*x**11 - 2633274*a**2*b**6*x**13 + 3
90390*a*b**7*x**15 + 45045*b**8*x**17)/(20643840*a**11*b**7 + 185794560*a**10*b*
*8*x**2 + 743178240*a**9*b**9*x**4 + 1734082560*a**8*b**10*x**6 + 2601123840*a**
7*b**11*x**8 + 2601123840*a**6*b**12*x**10 + 1734082560*a**5*b**13*x**12 + 74317
8240*a**4*b**14*x**14 + 185794560*a**3*b**15*x**16 + 20643840*a**2*b**16*x**18)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210483, size = 173, normalized size = 0.87 \[ \frac{143 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{2} b^{7}} + \frac{45045 \, b^{8} x^{17} + 390390 \, a b^{7} x^{15} - 2633274 \, a^{2} b^{6} x^{13} - 4349826 \, a^{3} b^{5} x^{11} - 4685824 \, a^{4} b^{4} x^{9} - 3317886 \, a^{5} b^{3} x^{7} - 1495494 \, a^{6} b^{2} x^{5} - 390390 \, a^{7} b x^{3} - 45045 \, a^{8} x}{20643840 \,{\left (b x^{2} + a\right )}^{9} a^{2} b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

143/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^2*b^7) + 1/20643840*(45045*b^8*x^17
+ 390390*a*b^7*x^15 - 2633274*a^2*b^6*x^13 - 4349826*a^3*b^5*x^11 - 4685824*a^4
*b^4*x^9 - 3317886*a^5*b^3*x^7 - 1495494*a^6*b^2*x^5 - 390390*a^7*b*x^3 - 45045*
a^8*x)/((b*x^2 + a)^9*a^2*b^7)