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

Optimal. Leaf size=92 \[ \frac{3 a^2 x^3}{2 b^4}-\frac{9 a^3 x}{2 b^5}+\frac{9 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}-\frac{9 a x^5}{10 b^3}-\frac{x^9}{2 b \left (a+b x^2\right )}+\frac{9 x^7}{14 b^2} \]

[Out]

(-9*a^3*x)/(2*b^5) + (3*a^2*x^3)/(2*b^4) - (9*a*x^5)/(10*b^3) + (9*x^7)/(14*b^2) - x^9/(2*b*(a + b*x^2)) + (9*
a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Rubi [A]  time = 0.0369984, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac{3 a^2 x^3}{2 b^4}-\frac{9 a^3 x}{2 b^5}+\frac{9 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}-\frac{9 a x^5}{10 b^3}-\frac{x^9}{2 b \left (a+b x^2\right )}+\frac{9 x^7}{14 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(a + b*x^2)^2,x]

[Out]

(-9*a^3*x)/(2*b^5) + (3*a^2*x^3)/(2*b^4) - (9*a*x^5)/(10*b^3) + (9*x^7)/(14*b^2) - x^9/(2*b*(a + b*x^2)) + (9*
a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

Rule 288

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{10}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^9}{2 b \left (a+b x^2\right )}+\frac{9 \int \frac{x^8}{a+b x^2} \, dx}{2 b}\\ &=-\frac{x^9}{2 b \left (a+b x^2\right )}+\frac{9 \int \left (-\frac{a^3}{b^4}+\frac{a^2 x^2}{b^3}-\frac{a x^4}{b^2}+\frac{x^6}{b}+\frac{a^4}{b^4 \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=-\frac{9 a^3 x}{2 b^5}+\frac{3 a^2 x^3}{2 b^4}-\frac{9 a x^5}{10 b^3}+\frac{9 x^7}{14 b^2}-\frac{x^9}{2 b \left (a+b x^2\right )}+\frac{\left (9 a^4\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^5}\\ &=-\frac{9 a^3 x}{2 b^5}+\frac{3 a^2 x^3}{2 b^4}-\frac{9 a x^5}{10 b^3}+\frac{9 x^7}{14 b^2}-\frac{x^9}{2 b \left (a+b x^2\right )}+\frac{9 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}}\\ \end{align*}

Mathematica [A]  time = 0.0541899, size = 82, normalized size = 0.89 \[ \frac{x \left (70 a^2 b x^2-\frac{35 a^4}{a+b x^2}-280 a^3-28 a b^2 x^4+10 b^3 x^6\right )}{70 b^5}+\frac{9 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a + b*x^2)^2,x]

[Out]

(x*(-280*a^3 + 70*a^2*b*x^2 - 28*a*b^2*x^4 + 10*b^3*x^6 - (35*a^4)/(a + b*x^2)))/(70*b^5) + (9*a^(7/2)*ArcTan[
(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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Maple [A]  time = 0.008, size = 78, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,{b}^{2}}}-{\frac{2\,a{x}^{5}}{5\,{b}^{3}}}+{\frac{{a}^{2}{x}^{3}}{{b}^{4}}}-4\,{\frac{{a}^{3}x}{{b}^{5}}}-{\frac{{a}^{4}x}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{9\,{a}^{4}}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7/b^2-2/5*a*x^5/b^3+a^2*x^3/b^4-4*a^3*x/b^5-1/2/b^5*a^4*x/(b*x^2+a)+9/2/b^5*a^4/(a*b)^(1/2)*arctan(b*x/(
a*b)^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.28686, size = 459, normalized size = 4.99 \begin{align*} \left [\frac{20 \, b^{4} x^{9} - 36 \, a b^{3} x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a^{3} b x^{3} - 630 \, a^{4} x + 315 \,{\left (a^{3} b x^{2} + a^{4}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{140 \,{\left (b^{6} x^{2} + a b^{5}\right )}}, \frac{10 \, b^{4} x^{9} - 18 \, a b^{3} x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a^{3} b x^{3} - 315 \, a^{4} x + 315 \,{\left (a^{3} b x^{2} + a^{4}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{70 \,{\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/140*(20*b^4*x^9 - 36*a*b^3*x^7 + 84*a^2*b^2*x^5 - 420*a^3*b*x^3 - 630*a^4*x + 315*(a^3*b*x^2 + a^4)*sqrt(-a
/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^6*x^2 + a*b^5), 1/70*(10*b^4*x^9 - 18*a*b^3*x^7 + 42*a
^2*b^2*x^5 - 210*a^3*b*x^3 - 315*a^4*x + 315*(a^3*b*x^2 + a^4)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^6*x^2 + a
*b^5)]

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Sympy [A]  time = 0.479109, size = 134, normalized size = 1.46 \begin{align*} - \frac{a^{4} x}{2 a b^{5} + 2 b^{6} x^{2}} - \frac{4 a^{3} x}{b^{5}} + \frac{a^{2} x^{3}}{b^{4}} - \frac{2 a x^{5}}{5 b^{3}} - \frac{9 \sqrt{- \frac{a^{7}}{b^{11}}} \log{\left (x - \frac{b^{5} \sqrt{- \frac{a^{7}}{b^{11}}}}{a^{3}} \right )}}{4} + \frac{9 \sqrt{- \frac{a^{7}}{b^{11}}} \log{\left (x + \frac{b^{5} \sqrt{- \frac{a^{7}}{b^{11}}}}{a^{3}} \right )}}{4} + \frac{x^{7}}{7 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(b*x**2+a)**2,x)

[Out]

-a**4*x/(2*a*b**5 + 2*b**6*x**2) - 4*a**3*x/b**5 + a**2*x**3/b**4 - 2*a*x**5/(5*b**3) - 9*sqrt(-a**7/b**11)*lo
g(x - b**5*sqrt(-a**7/b**11)/a**3)/4 + 9*sqrt(-a**7/b**11)*log(x + b**5*sqrt(-a**7/b**11)/a**3)/4 + x**7/(7*b*
*2)

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Giac [A]  time = 3.23192, size = 113, normalized size = 1.23 \begin{align*} \frac{9 \, a^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{5}} - \frac{a^{4} x}{2 \,{\left (b x^{2} + a\right )} b^{5}} + \frac{5 \, b^{12} x^{7} - 14 \, a b^{11} x^{5} + 35 \, a^{2} b^{10} x^{3} - 140 \, a^{3} b^{9} x}{35 \, b^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/2*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/2*a^4*x/((b*x^2 + a)*b^5) + 1/35*(5*b^12*x^7 - 14*a*b^11*x^5
+ 35*a^2*b^10*x^3 - 140*a^3*b^9*x)/b^14