3.195 \(\int \frac{x^{19}}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=179 \[ \frac{a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac{9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac{18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac{7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac{63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac{63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac{14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac{9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac{9 a}{2 b^{10} \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^{10}} \]

[Out]

a^9/(18*b^10*(a + b*x^2)^9) - (9*a^8)/(16*b^10*(a + b*x^2)^8) + (18*a^7)/(7*b^10
*(a + b*x^2)^7) - (7*a^6)/(b^10*(a + b*x^2)^6) + (63*a^5)/(5*b^10*(a + b*x^2)^5)
- (63*a^4)/(4*b^10*(a + b*x^2)^4) + (14*a^3)/(b^10*(a + b*x^2)^3) - (9*a^2)/(b^
10*(a + b*x^2)^2) + (9*a)/(2*b^10*(a + b*x^2)) + Log[a + b*x^2]/(2*b^10)

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Rubi [A]  time = 0.378723, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^9}{18 b^{10} \left (a+b x^2\right )^9}-\frac{9 a^8}{16 b^{10} \left (a+b x^2\right )^8}+\frac{18 a^7}{7 b^{10} \left (a+b x^2\right )^7}-\frac{7 a^6}{b^{10} \left (a+b x^2\right )^6}+\frac{63 a^5}{5 b^{10} \left (a+b x^2\right )^5}-\frac{63 a^4}{4 b^{10} \left (a+b x^2\right )^4}+\frac{14 a^3}{b^{10} \left (a+b x^2\right )^3}-\frac{9 a^2}{b^{10} \left (a+b x^2\right )^2}+\frac{9 a}{2 b^{10} \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^{10}} \]

Antiderivative was successfully verified.

[In]  Int[x^19/(a + b*x^2)^10,x]

[Out]

a^9/(18*b^10*(a + b*x^2)^9) - (9*a^8)/(16*b^10*(a + b*x^2)^8) + (18*a^7)/(7*b^10
*(a + b*x^2)^7) - (7*a^6)/(b^10*(a + b*x^2)^6) + (63*a^5)/(5*b^10*(a + b*x^2)^5)
- (63*a^4)/(4*b^10*(a + b*x^2)^4) + (14*a^3)/(b^10*(a + b*x^2)^3) - (9*a^2)/(b^
10*(a + b*x^2)^2) + (9*a)/(2*b^10*(a + b*x^2)) + Log[a + b*x^2]/(2*b^10)

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Rubi in Sympy [A]  time = 51.8683, size = 170, normalized size = 0.95 \[ \frac{a^{9}}{18 b^{10} \left (a + b x^{2}\right )^{9}} - \frac{9 a^{8}}{16 b^{10} \left (a + b x^{2}\right )^{8}} + \frac{18 a^{7}}{7 b^{10} \left (a + b x^{2}\right )^{7}} - \frac{7 a^{6}}{b^{10} \left (a + b x^{2}\right )^{6}} + \frac{63 a^{5}}{5 b^{10} \left (a + b x^{2}\right )^{5}} - \frac{63 a^{4}}{4 b^{10} \left (a + b x^{2}\right )^{4}} + \frac{14 a^{3}}{b^{10} \left (a + b x^{2}\right )^{3}} - \frac{9 a^{2}}{b^{10} \left (a + b x^{2}\right )^{2}} + \frac{9 a}{2 b^{10} \left (a + b x^{2}\right )} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**2+a)**10,x)

[Out]

a**9/(18*b**10*(a + b*x**2)**9) - 9*a**8/(16*b**10*(a + b*x**2)**8) + 18*a**7/(7
*b**10*(a + b*x**2)**7) - 7*a**6/(b**10*(a + b*x**2)**6) + 63*a**5/(5*b**10*(a +
b*x**2)**5) - 63*a**4/(4*b**10*(a + b*x**2)**4) + 14*a**3/(b**10*(a + b*x**2)**
3) - 9*a**2/(b**10*(a + b*x**2)**2) + 9*a/(2*b**10*(a + b*x**2)) + log(a + b*x**
2)/(2*b**10)

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Mathematica [A]  time = 0.0638865, size = 116, normalized size = 0.65 \[ \frac{\frac{a \left (7129 a^8+61641 a^7 b x^2+235224 a^6 b^2 x^4+518616 a^5 b^3 x^6+725004 a^4 b^4 x^8+661500 a^3 b^5 x^{10}+388080 a^2 b^6 x^{12}+136080 a b^7 x^{14}+22680 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+2520 \log \left (a+b x^2\right )}{5040 b^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/(a + b*x^2)^10,x]

[Out]

((a*(7129*a^8 + 61641*a^7*b*x^2 + 235224*a^6*b^2*x^4 + 518616*a^5*b^3*x^6 + 7250
04*a^4*b^4*x^8 + 661500*a^3*b^5*x^10 + 388080*a^2*b^6*x^12 + 136080*a*b^7*x^14 +
22680*b^8*x^16))/(a + b*x^2)^9 + 2520*Log[a + b*x^2])/(5040*b^10)

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Maple [A]  time = 0.017, size = 166, normalized size = 0.9 \[{\frac{{a}^{9}}{18\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{9\,{a}^{8}}{16\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{8}}}+{\frac{18\,{a}^{7}}{7\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{7}}}-7\,{\frac{{a}^{6}}{{b}^{10} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{63\,{a}^{5}}{5\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{63\,{a}^{4}}{4\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{4}}}+14\,{\frac{{a}^{3}}{{b}^{10} \left ( b{x}^{2}+a \right ) ^{3}}}-9\,{\frac{{a}^{2}}{{b}^{10} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a}{2\,{b}^{10} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^2+a)^10,x)

[Out]

1/18*a^9/b^10/(b*x^2+a)^9-9/16*a^8/b^10/(b*x^2+a)^8+18/7*a^7/b^10/(b*x^2+a)^7-7*
a^6/b^10/(b*x^2+a)^6+63/5*a^5/b^10/(b*x^2+a)^5-63/4*a^4/b^10/(b*x^2+a)^4+14*a^3/
b^10/(b*x^2+a)^3-9*a^2/b^10/(b*x^2+a)^2+9/2*a/b^10/(b*x^2+a)+1/2*ln(b*x^2+a)/b^1
0

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Maxima [A]  time = 1.36392, size = 282, normalized size = 1.58 \[ \frac{22680 \, a b^{8} x^{16} + 136080 \, a^{2} b^{7} x^{14} + 388080 \, a^{3} b^{6} x^{12} + 661500 \, a^{4} b^{5} x^{10} + 725004 \, a^{5} b^{4} x^{8} + 518616 \, a^{6} b^{3} x^{6} + 235224 \, a^{7} b^{2} x^{4} + 61641 \, a^{8} b x^{2} + 7129 \, a^{9}}{5040 \,{\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

1/5040*(22680*a*b^8*x^16 + 136080*a^2*b^7*x^14 + 388080*a^3*b^6*x^12 + 661500*a^
4*b^5*x^10 + 725004*a^5*b^4*x^8 + 518616*a^6*b^3*x^6 + 235224*a^7*b^2*x^4 + 6164
1*a^8*b*x^2 + 7129*a^9)/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*x^14 + 84*a^3*b
^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6 + 36*a^7*b^12*
x^4 + 9*a^8*b^11*x^2 + a^9*b^10) + 1/2*log(b*x^2 + a)/b^10

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Fricas [A]  time = 0.209447, size = 405, normalized size = 2.26 \[ \frac{22680 \, a b^{8} x^{16} + 136080 \, a^{2} b^{7} x^{14} + 388080 \, a^{3} b^{6} x^{12} + 661500 \, a^{4} b^{5} x^{10} + 725004 \, a^{5} b^{4} x^{8} + 518616 \, a^{6} b^{3} x^{6} + 235224 \, a^{7} b^{2} x^{4} + 61641 \, a^{8} b x^{2} + 7129 \, a^{9} + 2520 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \log \left (b x^{2} + a\right )}{5040 \,{\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

1/5040*(22680*a*b^8*x^16 + 136080*a^2*b^7*x^14 + 388080*a^3*b^6*x^12 + 661500*a^
4*b^5*x^10 + 725004*a^5*b^4*x^8 + 518616*a^6*b^3*x^6 + 235224*a^7*b^2*x^4 + 6164
1*a^8*b*x^2 + 7129*a^9 + 2520*(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(b*x^2 + a))/(b^19*x^18 + 9*a*b^18*x^16 + 36*a^2*b^17*
x^14 + 84*a^3*b^16*x^12 + 126*a^4*b^15*x^10 + 126*a^5*b^14*x^8 + 84*a^6*b^13*x^6
+ 36*a^7*b^12*x^4 + 9*a^8*b^11*x^2 + a^9*b^10)

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Sympy [A]  time = 36.144, size = 219, normalized size = 1.22 \[ \frac{7129 a^{9} + 61641 a^{8} b x^{2} + 235224 a^{7} b^{2} x^{4} + 518616 a^{6} b^{3} x^{6} + 725004 a^{5} b^{4} x^{8} + 661500 a^{4} b^{5} x^{10} + 388080 a^{3} b^{6} x^{12} + 136080 a^{2} b^{7} x^{14} + 22680 a b^{8} x^{16}}{5040 a^{9} b^{10} + 45360 a^{8} b^{11} x^{2} + 181440 a^{7} b^{12} x^{4} + 423360 a^{6} b^{13} x^{6} + 635040 a^{5} b^{14} x^{8} + 635040 a^{4} b^{15} x^{10} + 423360 a^{3} b^{16} x^{12} + 181440 a^{2} b^{17} x^{14} + 45360 a b^{18} x^{16} + 5040 b^{19} x^{18}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(b*x**2+a)**10,x)

[Out]

(7129*a**9 + 61641*a**8*b*x**2 + 235224*a**7*b**2*x**4 + 518616*a**6*b**3*x**6 +
725004*a**5*b**4*x**8 + 661500*a**4*b**5*x**10 + 388080*a**3*b**6*x**12 + 13608
0*a**2*b**7*x**14 + 22680*a*b**8*x**16)/(5040*a**9*b**10 + 45360*a**8*b**11*x**2
+ 181440*a**7*b**12*x**4 + 423360*a**6*b**13*x**6 + 635040*a**5*b**14*x**8 + 63
5040*a**4*b**15*x**10 + 423360*a**3*b**16*x**12 + 181440*a**2*b**17*x**14 + 4536
0*a*b**18*x**16 + 5040*b**19*x**18) + log(a + b*x**2)/(2*b**10)

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GIAC/XCAS [A]  time = 0.211642, size = 161, normalized size = 0.9 \[ \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{10}} - \frac{7129 \, b^{8} x^{18} + 41481 \, a b^{7} x^{16} + 120564 \, a^{2} b^{6} x^{14} + 210756 \, a^{3} b^{5} x^{12} + 236754 \, a^{4} b^{4} x^{10} + 173250 \, a^{5} b^{3} x^{8} + 80220 \, a^{6} b^{2} x^{6} + 21420 \, a^{7} b x^{4} + 2520 \, a^{8} x^{2}}{5040 \,{\left (b x^{2} + a\right )}^{9} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

1/2*ln(abs(b*x^2 + a))/b^10 - 1/5040*(7129*b^8*x^18 + 41481*a*b^7*x^16 + 120564*
a^2*b^6*x^14 + 210756*a^3*b^5*x^12 + 236754*a^4*b^4*x^10 + 173250*a^5*b^3*x^8 +
80220*a^6*b^2*x^6 + 21420*a^7*b*x^4 + 2520*a^8*x^2)/((b*x^2 + a)^9*b^9)