3.212 \(\int \frac{x^{18}}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=197 \[ \frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 \sqrt{a} b^{19/2}}-\frac{12155 x}{65536 b^9 \left (a+b x^2\right )}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^{17}}{18 b \left (a+b x^2\right )^9} \]

[Out]

-x^17/(18*b*(a + b*x^2)^9) - (17*x^15)/(288*b^2*(a + b*x^2)^8) - (85*x^13)/(1344
*b^3*(a + b*x^2)^7) - (1105*x^11)/(16128*b^4*(a + b*x^2)^6) - (2431*x^9)/(32256*
b^5*(a + b*x^2)^5) - (2431*x^7)/(28672*b^6*(a + b*x^2)^4) - (2431*x^5)/(24576*b^
7*(a + b*x^2)^3) - (12155*x^3)/(98304*b^8*(a + b*x^2)^2) - (12155*x)/(65536*b^9*
(a + b*x^2)) + (12155*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*Sqrt[a]*b^(19/2))

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Rubi [A]  time = 0.310432, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{12155 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 \sqrt{a} b^{19/2}}-\frac{12155 x}{65536 b^9 \left (a+b x^2\right )}-\frac{12155 x^3}{98304 b^8 \left (a+b x^2\right )^2}-\frac{2431 x^5}{24576 b^7 \left (a+b x^2\right )^3}-\frac{2431 x^7}{28672 b^6 \left (a+b x^2\right )^4}-\frac{2431 x^9}{32256 b^5 \left (a+b x^2\right )^5}-\frac{1105 x^{11}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{85 x^{13}}{1344 b^3 \left (a+b x^2\right )^7}-\frac{17 x^{15}}{288 b^2 \left (a+b x^2\right )^8}-\frac{x^{17}}{18 b \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[x^18/(a + b*x^2)^10,x]

[Out]

-x^17/(18*b*(a + b*x^2)^9) - (17*x^15)/(288*b^2*(a + b*x^2)^8) - (85*x^13)/(1344
*b^3*(a + b*x^2)^7) - (1105*x^11)/(16128*b^4*(a + b*x^2)^6) - (2431*x^9)/(32256*
b^5*(a + b*x^2)^5) - (2431*x^7)/(28672*b^6*(a + b*x^2)^4) - (2431*x^5)/(24576*b^
7*(a + b*x^2)^3) - (12155*x^3)/(98304*b^8*(a + b*x^2)^2) - (12155*x)/(65536*b^9*
(a + b*x^2)) + (12155*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(65536*Sqrt[a]*b^(19/2))

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Rubi in Sympy [A]  time = 50.6253, size = 187, normalized size = 0.95 \[ - \frac{x^{17}}{18 b \left (a + b x^{2}\right )^{9}} - \frac{17 x^{15}}{288 b^{2} \left (a + b x^{2}\right )^{8}} - \frac{85 x^{13}}{1344 b^{3} \left (a + b x^{2}\right )^{7}} - \frac{1105 x^{11}}{16128 b^{4} \left (a + b x^{2}\right )^{6}} - \frac{2431 x^{9}}{32256 b^{5} \left (a + b x^{2}\right )^{5}} - \frac{2431 x^{7}}{28672 b^{6} \left (a + b x^{2}\right )^{4}} - \frac{2431 x^{5}}{24576 b^{7} \left (a + b x^{2}\right )^{3}} - \frac{12155 x^{3}}{98304 b^{8} \left (a + b x^{2}\right )^{2}} - \frac{12155 x}{65536 b^{9} \left (a + b x^{2}\right )} + \frac{12155 \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{65536 \sqrt{a} b^{\frac{19}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**18/(b*x**2+a)**10,x)

[Out]

-x**17/(18*b*(a + b*x**2)**9) - 17*x**15/(288*b**2*(a + b*x**2)**8) - 85*x**13/(
1344*b**3*(a + b*x**2)**7) - 1105*x**11/(16128*b**4*(a + b*x**2)**6) - 2431*x**9
/(32256*b**5*(a + b*x**2)**5) - 2431*x**7/(28672*b**6*(a + b*x**2)**4) - 2431*x*
*5/(24576*b**7*(a + b*x**2)**3) - 12155*x**3/(98304*b**8*(a + b*x**2)**2) - 1215
5*x/(65536*b**9*(a + b*x**2)) + 12155*atan(sqrt(b)*x/sqrt(a))/(65536*sqrt(a)*b**
(19/2))

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Mathematica [A]  time = 0.145146, size = 134, normalized size = 0.68 \[ \frac{\frac{765765 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}-\frac{\sqrt{b} x \left (765765 a^8+6636630 a^7 b x^2+25423398 a^6 b^2 x^4+56404062 a^5 b^3 x^6+79659008 a^4 b^4 x^8+73947042 a^3 b^5 x^{10}[phone] a^2 b^6 x^{12}[phone] a b^7 x^{14}+3363003 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}}{4128768 b^{19/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^18/(a + b*x^2)^10,x]

[Out]

(-((Sqrt[b]*x*(765765*a^8 + 6636630*a^7*b*x^2 + 25423398*a^6*b^2*x^4 + 56404062*
a^5*b^3*x^6 + 79659008*a^4*b^4*x^8 + 73947042*a^3*b^5*x^10 + 44765658*a^2*b^6*x^
12 + 16759722*a*b^7*x^14 + 3363003*b^8*x^16))/(a + b*x^2)^9) + (765765*ArcTan[(S
qrt[b]*x)/Sqrt[a]])/Sqrt[a])/(4128768*b^(19/2))

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Maple [A]  time = 0.023, size = 124, normalized size = 0.6 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{12155\,{a}^{8}x}{65536\,{b}^{9}}}-{\frac{158015\,{a}^{7}{x}^{3}}{98304\,{b}^{8}}}-{\frac{201773\,{a}^{6}{x}^{5}}{32768\,{b}^{7}}}-{\frac{3133559\,{a}^{5}{x}^{7}}{229376\,{b}^{6}}}-{\frac{2431\,{a}^{4}{x}^{9}}{126\,{b}^{5}}}-{\frac{4108169\,{a}^{3}{x}^{11}}{229376\,{b}^{4}}}-{\frac{355283\,{a}^{2}{x}^{13}}{32768\,{b}^{3}}}-{\frac{399041\,a{x}^{15}}{98304\,{b}^{2}}}-{\frac{53381\,{x}^{17}}{65536\,b}} \right ) }+{\frac{12155}{65536\,{b}^{9}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^18/(b*x^2+a)^10,x)

[Out]

(-12155/65536*a^8/b^9*x-158015/98304*a^7/b^8*x^3-201773/32768*a^6/b^7*x^5-313355
9/229376*a^5/b^6*x^7-2431/126*a^4/b^5*x^9-4108169/229376*a^3/b^4*x^11-355283/327
68*a^2/b^3*x^13-399041/98304*a/b^2*x^15-53381/65536/b*x^17)/(b*x^2+a)^9+12155/65
536/b^9/(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^18/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.256389, size = 1, normalized size = 0.01 \[ \left [\frac{765765 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (3363003 \, b^{8} x^{17} + 16759722 \, a b^{7} x^{15} + 44765658 \, a^{2} b^{6} x^{13} + 73947042 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 56404062 \, a^{5} b^{3} x^{7} + 25423398 \, a^{6} b^{2} x^{5} + 6636630 \, a^{7} b x^{3} + 765765 \, a^{8} x\right )} \sqrt{-a b}}{8257536 \,{\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )} \sqrt{-a b}}, \frac{765765 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (3363003 \, b^{8} x^{17} + 16759722 \, a b^{7} x^{15} + 44765658 \, a^{2} b^{6} x^{13} + 73947042 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 56404062 \, a^{5} b^{3} x^{7} + 25423398 \, a^{6} b^{2} x^{5} + 6636630 \, a^{7} b x^{3} + 765765 \, a^{8} x\right )} \sqrt{a b}}{4128768 \,{\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^18/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

[1/8257536*(765765*(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*(3363003*b^8
*x^17 + 16759722*a*b^7*x^15 + 44765658*a^2*b^6*x^13 + 73947042*a^3*b^5*x^11 + 79
659008*a^4*b^4*x^9 + 56404062*a^5*b^3*x^7 + 25423398*a^6*b^2*x^5 + 6636630*a^7*b
*x^3 + 765765*a^8*x)*sqrt(-a*b))/((b^18*x^18 + 9*a*b^17*x^16 + 36*a^2*b^16*x^14
+ 84*a^3*b^15*x^12 + 126*a^4*b^14*x^10 + 126*a^5*b^13*x^8 + 84*a^6*b^12*x^6 + 36
*a^7*b^11*x^4 + 9*a^8*b^10*x^2 + a^9*b^9)*sqrt(-a*b)), 1/4128768*(765765*(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) - (3363003*b^8*x^17 + 16759722*a*b^7*x^15 + 44765658*a^2*b^6*x^13 + 7394
7042*a^3*b^5*x^11 + 79659008*a^4*b^4*x^9 + 56404062*a^5*b^3*x^7 + 25423398*a^6*b
^2*x^5 + 6636630*a^7*b*x^3 + 765765*a^8*x)*sqrt(a*b))/((b^18*x^18 + 9*a*b^17*x^1
6 + 36*a^2*b^16*x^14 + 84*a^3*b^15*x^12 + 126*a^4*b^14*x^10 + 126*a^5*b^13*x^8 +
84*a^6*b^12*x^6 + 36*a^7*b^11*x^4 + 9*a^8*b^10*x^2 + a^9*b^9)*sqrt(a*b))]

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Sympy [A]  time = 35.9627, size = 275, normalized size = 1.4 \[ - \frac{12155 \sqrt{- \frac{1}{a b^{19}}} \log{\left (- a b^{9} \sqrt{- \frac{1}{a b^{19}}} + x \right )}}{131072} + \frac{12155 \sqrt{- \frac{1}{a b^{19}}} \log{\left (a b^{9} \sqrt{- \frac{1}{a b^{19}}} + x \right )}}{131072} - \frac{765765 a^{8} x + 6636630 a^{7} b x^{3} + 25423398 a^{6} b^{2} x^{5} + 56404062 a^{5} b^{3} x^{7} + 79659008 a^{4} b^{4} x^{9} + 73947042 a^{3} b^{5} x^{11} + 44765658 a^{2} b^{6} x^{13} + 16759722 a b^{7} x^{15} + 3363003 b^{8} x^{17}}{4128768 a^{9} b^{9} + 37158912 a^{8} b^{10} x^{2} + 148635648 a^{7} b^{11} x^{4} + 346816512 a^{6} b^{12} x^{6} + 520224768 a^{5} b^{13} x^{8} + 520224768 a^{4} b^{14} x^{10} + 346816512 a^{3} b^{15} x^{12} + 148635648 a^{2} b^{16} x^{14} + 37158912 a b^{17} x^{16} + 4128768 b^{18} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**18/(b*x**2+a)**10,x)

[Out]

-12155*sqrt(-1/(a*b**19))*log(-a*b**9*sqrt(-1/(a*b**19)) + x)/131072 + 12155*sqr
t(-1/(a*b**19))*log(a*b**9*sqrt(-1/(a*b**19)) + x)/131072 - (765765*a**8*x + 663
6630*a**7*b*x**3 + 25423398*a**6*b**2*x**5 + 56404062*a**5*b**3*x**7 + 79659008*
a**4*b**4*x**9 + 73947042*a**3*b**5*x**11 + 44765658*a**2*b**6*x**13 + 16759722*
a*b**7*x**15 + 3363003*b**8*x**17)/(4128768*a**9*b**9 + 37158912*a**8*b**10*x**2
+ 148635648*a**7*b**11*x**4 + 346816512*a**6*b**12*x**6 + 520224768*a**5*b**13*
x**8 + 520224768*a**4*b**14*x**10 + 346816512*a**3*b**15*x**12 + 148635648*a**2*
b**16*x**14 + 37158912*a*b**17*x**16 + 4128768*b**18*x**18)

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GIAC/XCAS [A]  time = 0.209883, size = 165, normalized size = 0.84 \[ \frac{12155 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} b^{9}} - \frac{3363003 \, b^{8} x^{17} + 16759722 \, a b^{7} x^{15} + 44765658 \, a^{2} b^{6} x^{13} + 73947042 \, a^{3} b^{5} x^{11} + 79659008 \, a^{4} b^{4} x^{9} + 56404062 \, a^{5} b^{3} x^{7} + 25423398 \, a^{6} b^{2} x^{5} + 6636630 \, a^{7} b x^{3} + 765765 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^18/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

12155/65536*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^9) - 1/4128768*(3363003*b^8*x^17
+ 16759722*a*b^7*x^15 + 44765658*a^2*b^6*x^13 + 73947042*a^3*b^5*x^11 + 79659008
*a^4*b^4*x^9 + 56404062*a^5*b^3*x^7 + 25423398*a^6*b^2*x^5 + 6636630*a^7*b*x^3 +
765765*a^8*x)/((b*x^2 + a)^9*b^9)