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

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

[Out]

1/18*x^18/a/(b*x^2+a)^9

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \[ \frac {x^{18}}{18 a \left (a+b x^2\right )^9} \]

Antiderivative was successfully verified.

[In]

Int[x^17/(a + b*x^2)^10,x]

[Out]

x^18/(18*a*(a + b*x^2)^9)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {x^{17}}{\left (a+b x^2\right )^{10}} \, dx &=\frac {x^{18}}{18 a \left (a+b x^2\right )^9}\\ \end {align*}

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Mathematica [B]  time = 0.02, 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]

-1/18*(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)/(b^9*(a + b*x^2)^9)

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fricas [B]  time = 0.89, size = 190, normalized size = 10.00 \[ -\frac {9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, {\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )}} \]

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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giac [B]  time = 0.64, size = 99, normalized size = 5.21 \[ -\frac {9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, {\left (b x^{2} + a\right )}^{9} b^{9}} \]

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)

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maple [B]  time = 0.01, size = 150, normalized size = 7.89 \[ -\frac {a^{8}}{18 \left (b \,x^{2}+a \right )^{9} b^{9}}+\frac {a^{7}}{2 \left (b \,x^{2}+a \right )^{8} b^{9}}-\frac {2 a^{6}}{\left (b \,x^{2}+a \right )^{7} b^{9}}+\frac {14 a^{5}}{3 \left (b \,x^{2}+a \right )^{6} b^{9}}-\frac {7 a^{4}}{\left (b \,x^{2}+a \right )^{5} b^{9}}+\frac {7 a^{3}}{\left (b \,x^{2}+a \right )^{4} b^{9}}-\frac {14 a^{2}}{3 \left (b \,x^{2}+a \right )^{3} b^{9}}+\frac {2 a}{\left (b \,x^{2}+a \right )^{2} b^{9}}-\frac {1}{2 \left (b \,x^{2}+a \right ) b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17/(b*x^2+a)^10,x)

[Out]

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

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maxima [B]  time = 1.50, size = 190, normalized size = 10.00 \[ -\frac {9 \, b^{8} x^{16} + 36 \, a b^{7} x^{14} + 84 \, a^{2} b^{6} x^{12} + 126 \, a^{3} b^{5} x^{10} + 126 \, a^{4} b^{4} x^{8} + 84 \, a^{5} b^{3} x^{6} + 36 \, a^{6} b^{2} x^{4} + 9 \, a^{7} b x^{2} + a^{8}}{18 \, {\left (b^{18} x^{18} + 9 \, a b^{17} x^{16} + 36 \, a^{2} b^{16} x^{14} + 84 \, a^{3} b^{15} x^{12} + 126 \, a^{4} b^{14} x^{10} + 126 \, a^{5} b^{13} x^{8} + 84 \, a^{6} b^{12} x^{6} + 36 \, a^{7} b^{11} x^{4} + 9 \, a^{8} b^{10} x^{2} + a^{9} b^{9}\right )}} \]

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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mupad [B]  time = 0.12, size = 192, normalized size = 10.11 \[ -\frac {a^8+9\,a^7\,b\,x^2+36\,a^6\,b^2\,x^4+84\,a^5\,b^3\,x^6+126\,a^4\,b^4\,x^8+126\,a^3\,b^5\,x^{10}+84\,a^2\,b^6\,x^{12}+36\,a\,b^7\,x^{14}+9\,b^8\,x^{16}}{18\,a^9\,b^9+162\,a^8\,b^{10}\,x^2+648\,a^7\,b^{11}\,x^4+1512\,a^6\,b^{12}\,x^6+2268\,a^5\,b^{13}\,x^8+2268\,a^4\,b^{14}\,x^{10}+1512\,a^3\,b^{15}\,x^{12}+648\,a^2\,b^{16}\,x^{14}+162\,a\,b^{17}\,x^{16}+18\,b^{18}\,x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^17/(a + b*x^2)^10,x)

[Out]

-(a^8 + 9*b^8*x^16 + 9*a^7*b*x^2 + 36*a*b^7*x^14 + 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)/(18*a^9*b^9 + 18*b^18*x^18 + 162*a*b^17*x^16 + 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)

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sympy [B]  time = 1.70, 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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