3.146 \(\int \frac{x^{12}}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=105 \[ \frac{11 a^2 x^5}{10 b^4}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^4 x}{2 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}-\frac{11 a x^7}{14 b^3}-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 x^9}{18 b^2} \]

[Out]

(11*a^4*x)/(2*b^6) - (11*a^3*x^3)/(6*b^5) + (11*a^2*x^5)/(10*b^4) - (11*a*x^7)/(14*b^3) + (11*x^9)/(18*b^2) -
x^11/(2*b*(a + b*x^2)) - (11*a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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Rubi [A]  time = 0.0455382, antiderivative size = 105, 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} \[ \frac{11 a^2 x^5}{10 b^4}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^4 x}{2 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}-\frac{11 a x^7}{14 b^3}-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 x^9}{18 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^12/(a + b*x^2)^2,x]

[Out]

(11*a^4*x)/(2*b^6) - (11*a^3*x^3)/(6*b^5) + (11*a^2*x^5)/(10*b^4) - (11*a*x^7)/(14*b^3) + (11*x^9)/(18*b^2) -
x^11/(2*b*(a + b*x^2)) - (11*a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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^{12}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 \int \frac{x^{10}}{a+b x^2} \, dx}{2 b}\\ &=-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 \int \left (\frac{a^4}{b^5}-\frac{a^3 x^2}{b^4}+\frac{a^2 x^4}{b^3}-\frac{a x^6}{b^2}+\frac{x^8}{b}-\frac{a^5}{b^5 \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=\frac{11 a^4 x}{2 b^6}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^2 x^5}{10 b^4}-\frac{11 a x^7}{14 b^3}+\frac{11 x^9}{18 b^2}-\frac{x^{11}}{2 b \left (a+b x^2\right )}-\frac{\left (11 a^5\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^6}\\ &=\frac{11 a^4 x}{2 b^6}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^2 x^5}{10 b^4}-\frac{11 a x^7}{14 b^3}+\frac{11 x^9}{18 b^2}-\frac{x^{11}}{2 b \left (a+b x^2\right )}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.0588059, size = 93, normalized size = 0.89 \[ \frac{x \left (378 a^2 b^2 x^4-840 a^3 b x^2+\frac{315 a^5}{a+b x^2}+3150 a^4-180 a b^3 x^6+70 b^4 x^8\right )}{630 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(a + b*x^2)^2,x]

[Out]

(x*(3150*a^4 - 840*a^3*b*x^2 + 378*a^2*b^2*x^4 - 180*a*b^3*x^6 + 70*b^4*x^8 + (315*a^5)/(a + b*x^2)))/(630*b^6
) - (11*a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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Maple [A]  time = 0.007, size = 90, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}}{9\,{b}^{2}}}-{\frac{2\,a{x}^{7}}{7\,{b}^{3}}}+{\frac{3\,{a}^{2}{x}^{5}}{5\,{b}^{4}}}-{\frac{4\,{a}^{3}{x}^{3}}{3\,{b}^{5}}}+5\,{\frac{{a}^{4}x}{{b}^{6}}}+{\frac{{a}^{5}x}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}-{\frac{11\,{a}^{5}}{2\,{b}^{6}}\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^12/(b*x^2+a)^2,x)

[Out]

1/9*x^9/b^2-2/7*a*x^7/b^3+3/5*a^2*x^5/b^4-4/3*a^3*x^3/b^5+5*a^4*x/b^6+1/2/b^6*a^5*x/(b*x^2+a)-11/2/b^6*a^5/(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^12/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.27917, size = 528, normalized size = 5.03 \begin{align*} \left [\frac{140 \, b^{5} x^{11} - 220 \, a b^{4} x^{9} + 396 \, a^{2} b^{3} x^{7} - 924 \, a^{3} b^{2} x^{5} + 4620 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \,{\left (a^{4} b x^{2} + a^{5}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{1260 \,{\left (b^{7} x^{2} + a b^{6}\right )}}, \frac{70 \, b^{5} x^{11} - 110 \, a b^{4} x^{9} + 198 \, a^{2} b^{3} x^{7} - 462 \, a^{3} b^{2} x^{5} + 2310 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \,{\left (a^{4} b x^{2} + a^{5}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{630 \,{\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/1260*(140*b^5*x^11 - 220*a*b^4*x^9 + 396*a^2*b^3*x^7 - 924*a^3*b^2*x^5 + 4620*a^4*b*x^3 + 6930*a^5*x + 3465
*(a^4*b*x^2 + a^5)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^7*x^2 + a*b^6), 1/630*(70*b^
5*x^11 - 110*a*b^4*x^9 + 198*a^2*b^3*x^7 - 462*a^3*b^2*x^5 + 2310*a^4*b*x^3 + 3465*a^5*x - 3465*(a^4*b*x^2 + a
^5)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^7*x^2 + a*b^6)]

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Sympy [A]  time = 0.469092, size = 151, normalized size = 1.44 \begin{align*} \frac{a^{5} x}{2 a b^{6} + 2 b^{7} x^{2}} + \frac{5 a^{4} x}{b^{6}} - \frac{4 a^{3} x^{3}}{3 b^{5}} + \frac{3 a^{2} x^{5}}{5 b^{4}} - \frac{2 a x^{7}}{7 b^{3}} + \frac{11 \sqrt{- \frac{a^{9}}{b^{13}}} \log{\left (x - \frac{b^{6} \sqrt{- \frac{a^{9}}{b^{13}}}}{a^{4}} \right )}}{4} - \frac{11 \sqrt{- \frac{a^{9}}{b^{13}}} \log{\left (x + \frac{b^{6} \sqrt{- \frac{a^{9}}{b^{13}}}}{a^{4}} \right )}}{4} + \frac{x^{9}}{9 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(b*x**2+a)**2,x)

[Out]

a**5*x/(2*a*b**6 + 2*b**7*x**2) + 5*a**4*x/b**6 - 4*a**3*x**3/(3*b**5) + 3*a**2*x**5/(5*b**4) - 2*a*x**7/(7*b*
*3) + 11*sqrt(-a**9/b**13)*log(x - b**6*sqrt(-a**9/b**13)/a**4)/4 - 11*sqrt(-a**9/b**13)*log(x + b**6*sqrt(-a*
*9/b**13)/a**4)/4 + x**9/(9*b**2)

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Giac [A]  time = 2.98895, size = 128, normalized size = 1.22 \begin{align*} -\frac{11 \, a^{5} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{6}} + \frac{a^{5} x}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{35 \, b^{16} x^{9} - 90 \, a b^{15} x^{7} + 189 \, a^{2} b^{14} x^{5} - 420 \, a^{3} b^{13} x^{3} + 1575 \, a^{4} b^{12} x}{315 \, b^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-11/2*a^5*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/2*a^5*x/((b*x^2 + a)*b^6) + 1/315*(35*b^16*x^9 - 90*a*b^15
*x^7 + 189*a^2*b^14*x^5 - 420*a^3*b^13*x^3 + 1575*a^4*b^12*x)/b^18