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]

x^18/(18*a*(a + b*x^2)^9)

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Rubi [A]  time = 0.003412, 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, 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*}

Mathematica [B]  time = 0.0193256, size = 101, normalized size = 5.32 \[ -\frac{84 a^2 b^6 x^{12}+126 a^3 b^5 x^{10}+126 a^4 b^4 x^8+84 a^5 b^3 x^6+36 a^6 b^2 x^4+9 a^7 b x^2+a^8+36 a b^7 x^{14}+9 b^8 x^{16}}{18 b^9 \left (a+b x^2\right )^9} \]

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 = 0.01, size = 150, normalized size = 7.9 \begin{align*} -{\frac{{a}^{8}}{18\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}+7\,{\frac{{a}^{3}}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{4}}}-{\frac{14\,{a}^{2}}{3\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{14\,{a}^{5}}{3\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{6}}}+{\frac{{a}^{7}}{2\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{8}}}+2\,{\frac{a}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{2}}}-7\,{\frac{{a}^{4}}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1}{2\,{b}^{9} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{{a}^{6}}{{b}^{9} \left ( b{x}^{2}+a \right ) ^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.01939, size = 257, normalized size = 13.53 \begin{align*} -\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 )}} \end{align*}

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 [B]  time = 1.20712, size = 423, normalized size = 22.26 \begin{align*} -\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 )}} \end{align*}

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 [B]  time = 7.60824, size = 202, normalized size = 10.63 \begin{align*} - \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}} \end{align*}

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 [B]  time = 2.71458, size = 134, normalized size = 7.05 \begin{align*} -\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}} \end{align*}

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)