3.196 \(\int \frac{x^{17}}{\left (a+b x^2\right )^{10}} \, dx\)

Optimal. Leaf size=19 \[ \frac{x^{18}}{18 a \left (a+b x^2\right )^9} \]

[Out]

x^18/(18*a*(a + b*x^2)^9)

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Rubi [A]  time = 0.0196809, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x^{18}}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Int[x^17/(a + b*x^2)^10,x]

[Out]

x^18/(18*a*(a + b*x^2)^9)

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Rubi in Sympy [A]  time = 3.29056, size = 14, normalized size = 0.74 \[ \frac{x^{18}}{18 a \left (a + b x^{2}\right )^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**17/(b*x**2+a)**10,x)

[Out]

x**18/(18*a*(a + b*x**2)**9)

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Mathematica [B]  time = 0.0370214, size = 101, normalized size = 5.32 \[ -\frac{a^8+9 a^7 b x^2+36 a^6 b^2 x^4+84 a^5 b^3 x^6+126 a^4 b^4 x^8+126 a^3 b^5 x^{10}+84 a^2 b^6 x^{12}+36 a b^7 x^{14}+9 b^8 x^{16}}{18 b^9 \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^17/(a + b*x^2)^10,x]

[Out]

-(a^8 + 9*a^7*b*x^2 + 36*a^6*b^2*x^4 + 84*a^5*b^3*x^6 + 126*a^4*b^4*x^8 + 126*a^
3*b^5*x^10 + 84*a^2*b^6*x^12 + 36*a*b^7*x^14 + 9*b^8*x^16)/(18*b^9*(a + b*x^2)^9
)

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Maple [B]  time = 0.014, size = 150, normalized size = 7.9 \[{\frac{14\,{a}^{5}}{3\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{6}}}-{\frac{{a}^{8}}{18\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{{a}^{7}}{2\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{8}}}-2\,{\frac{{a}^{6}}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{7}}}-{\frac{14\,{a}^{2}}{3\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{3}}}+2\,{\frac{a}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{2}}}+7\,{\frac{{a}^{3}}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{1}{ \left ( 2\,b{x}^{2}+2\,a \right ){b}^{9}}}-7\,{\frac{{a}^{4}}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^17/(b*x^2+a)^10,x)

[Out]

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

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Maxima [A]  time = 1.36998, size = 257, normalized size = 13.53 \[ -\frac{9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \,{\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/(b*x^2 + a)^10,x, algorithm="maxima")

[Out]

-1/18*(9*b^8*x^16 + 36*a*b^7*x^14 + 84*a^2*b^6*x^12 + 126*a^3*b^5*x^10 + 126*a^4
*b^4*x^8 + 84*a^5*b^3*x^6 + 36*a^6*b^2*x^4 + 9*a^7*b*x^2 + a^8)/(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)

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Fricas [A]  time = 0.203905, size = 257, normalized size = 13.53 \[ -\frac{9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \,{\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/(b*x^2 + a)^10,x, algorithm="fricas")

[Out]

-1/18*(9*b^8*x^16 + 36*a*b^7*x^14 + 84*a^2*b^6*x^12 + 126*a^3*b^5*x^10 + 126*a^4
*b^4*x^8 + 84*a^5*b^3*x^6 + 36*a^6*b^2*x^4 + 9*a^7*b*x^2 + a^8)/(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)

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Sympy [A]  time = 35.8375, size = 202, normalized size = 10.63 \[ - \frac{a^{8} + 9 a^{7} b x^{2} + 36 a^{6} b^{2} x^{4} + 84 a^{5} b^{3} x^{6} + 126 a^{4} b^{4} x^{8} + 126 a^{3} b^{5} x^{10} + 84 a^{2} b^{6} x^{12} + 36 a b^{7} x^{14} + 9 b^{8} x^{16}}{18 a^{9} b^{9} + 162 a^{8} b^{10} x^{2} + 648 a^{7} b^{11} x^{4} + 1512 a^{6} b^{12} x^{6} + 2268 a^{5} b^{13} x^{8} + 2268 a^{4} b^{14} x^{10} + 1512 a^{3} b^{15} x^{12} + 648 a^{2} b^{16} x^{14} + 162 a b^{17} x^{16} + 18 b^{18} x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**17/(b*x**2+a)**10,x)

[Out]

-(a**8 + 9*a**7*b*x**2 + 36*a**6*b**2*x**4 + 84*a**5*b**3*x**6 + 126*a**4*b**4*x
**8 + 126*a**3*b**5*x**10 + 84*a**2*b**6*x**12 + 36*a*b**7*x**14 + 9*b**8*x**16)
/(18*a**9*b**9 + 162*a**8*b**10*x**2 + 648*a**7*b**11*x**4 + 1512*a**6*b**12*x**
6 + 2268*a**5*b**13*x**8 + 2268*a**4*b**14*x**10 + 1512*a**3*b**15*x**12 + 648*a
**2*b**16*x**14 + 162*a*b**17*x**16 + 18*b**18*x**18)

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GIAC/XCAS [A]  time = 0.212008, size = 134, normalized size = 7.05 \[ -\frac{9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \,{\left (b x^{2} + a\right )}^{9} b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/(b*x^2 + a)^10,x, algorithm="giac")

[Out]

-1/18*(9*b^8*x^16 + 36*a*b^7*x^14 + 84*a^2*b^6*x^12 + 126*a^3*b^5*x^10 + 126*a^4
*b^4*x^8 + 84*a^5*b^3*x^6 + 36*a^6*b^2*x^4 + 9*a^7*b*x^2 + a^8)/((b*x^2 + a)^9*b
^9)