3.181 \(\int \frac{x^{12}}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=111 \[ \frac{33 a^2 x^3}{8 b^5}-\frac{99 a^3 x}{8 b^6}+\frac{99 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}-\frac{11 x^9}{8 b^2 \left (a+b x^2\right )}-\frac{99 a x^5}{40 b^4}-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}+\frac{99 x^7}{56 b^3} \]

[Out]

(-99*a^3*x)/(8*b^6) + (33*a^2*x^3)/(8*b^5) - (99*a*x^5)/(40*b^4) + (99*x^7)/(56*b^3) - x^11/(4*b*(a + b*x^2)^2
) - (11*x^9)/(8*b^2*(a + b*x^2)) + (99*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Rubi [A]  time = 0.0484199, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac{33 a^2 x^3}{8 b^5}-\frac{99 a^3 x}{8 b^6}+\frac{99 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}-\frac{11 x^9}{8 b^2 \left (a+b x^2\right )}-\frac{99 a x^5}{40 b^4}-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}+\frac{99 x^7}{56 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^12/(a + b*x^2)^3,x]

[Out]

(-99*a^3*x)/(8*b^6) + (33*a^2*x^3)/(8*b^5) - (99*a*x^5)/(40*b^4) + (99*x^7)/(56*b^3) - x^11/(4*b*(a + b*x^2)^2
) - (11*x^9)/(8*b^2*(a + b*x^2)) + (99*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*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 )^3} \, dx &=-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}+\frac{11 \int \frac{x^{10}}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}-\frac{11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac{99 \int \frac{x^8}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}-\frac{11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac{99 \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}{8 b^2}\\ &=-\frac{99 a^3 x}{8 b^6}+\frac{33 a^2 x^3}{8 b^5}-\frac{99 a x^5}{40 b^4}+\frac{99 x^7}{56 b^3}-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}-\frac{11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac{\left (99 a^4\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^6}\\ &=-\frac{99 a^3 x}{8 b^6}+\frac{33 a^2 x^3}{8 b^5}-\frac{99 a x^5}{40 b^4}+\frac{99 x^7}{56 b^3}-\frac{x^{11}}{4 b \left (a+b x^2\right )^2}-\frac{11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac{99 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}\\ \end{align*}

Mathematica [A]  time = 0.0588633, size = 99, normalized size = 0.89 \[ \frac{99 a^{7/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}-\frac{-264 a^2 b^3 x^7+1848 a^3 b^2 x^5+5775 a^4 b x^3+3465 a^5 x+88 a b^4 x^9-40 b^5 x^{11}}{280 b^6 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(a + b*x^2)^3,x]

[Out]

-(3465*a^5*x + 5775*a^4*b*x^3 + 1848*a^3*b^2*x^5 - 264*a^2*b^3*x^7 + 88*a*b^4*x^9 - 40*b^5*x^11)/(280*b^6*(a +
b*x^2)^2) + (99*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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Maple [A]  time = 0.01, size = 99, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,{b}^{3}}}-{\frac{3\,a{x}^{5}}{5\,{b}^{4}}}+2\,{\frac{{a}^{2}{x}^{3}}{{b}^{5}}}-10\,{\frac{{a}^{3}x}{{b}^{6}}}-{\frac{21\,{a}^{4}{x}^{3}}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{19\,{a}^{5}x}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{99\,{a}^{4}}{8\,{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)^3,x)

[Out]

1/7*x^7/b^3-3/5*a*x^5/b^4+2*a^2*x^3/b^5-10*a^3*x/b^6-21/8/b^5*a^4/(b*x^2+a)^2*x^3-19/8/b^6*a^5/(b*x^2+a)^2*x+9
9/8/b^6*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^12/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26598, size = 614, normalized size = 5.53 \begin{align*} \left [\frac{80 \, b^{5} x^{11} - 176 \, a b^{4} x^{9} + 528 \, a^{2} b^{3} x^{7} - 3696 \, a^{3} b^{2} x^{5} - 11550 \, a^{4} b x^{3} - 6930 \, a^{5} x + 3465 \,{\left (a^{3} b^{2} x^{4} + 2 \, 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 )}{560 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac{40 \, b^{5} x^{11} - 88 \, a b^{4} x^{9} + 264 \, a^{2} b^{3} x^{7} - 1848 \, a^{3} b^{2} x^{5} - 5775 \, a^{4} b x^{3} - 3465 \, a^{5} x + 3465 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{280 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/560*(80*b^5*x^11 - 176*a*b^4*x^9 + 528*a^2*b^3*x^7 - 3696*a^3*b^2*x^5 - 11550*a^4*b*x^3 - 6930*a^5*x + 3465
*(a^3*b^2*x^4 + 2*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^8*x^4 + 2*a*
b^7*x^2 + a^2*b^6), 1/280*(40*b^5*x^11 - 88*a*b^4*x^9 + 264*a^2*b^3*x^7 - 1848*a^3*b^2*x^5 - 5775*a^4*b*x^3 -
3465*a^5*x + 3465*(a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^8*x^4 + 2*a*b^7*x^2
+ a^2*b^6)]

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Sympy [A]  time = 0.619371, size = 160, normalized size = 1.44 \begin{align*} - \frac{10 a^{3} x}{b^{6}} + \frac{2 a^{2} x^{3}}{b^{5}} - \frac{3 a x^{5}}{5 b^{4}} - \frac{99 \sqrt{- \frac{a^{7}}{b^{13}}} \log{\left (x - \frac{b^{6} \sqrt{- \frac{a^{7}}{b^{13}}}}{a^{3}} \right )}}{16} + \frac{99 \sqrt{- \frac{a^{7}}{b^{13}}} \log{\left (x + \frac{b^{6} \sqrt{- \frac{a^{7}}{b^{13}}}}{a^{3}} \right )}}{16} - \frac{19 a^{5} x + 21 a^{4} b x^{3}}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac{x^{7}}{7 b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(b*x**2+a)**3,x)

[Out]

-10*a**3*x/b**6 + 2*a**2*x**3/b**5 - 3*a*x**5/(5*b**4) - 99*sqrt(-a**7/b**13)*log(x - b**6*sqrt(-a**7/b**13)/a
**3)/16 + 99*sqrt(-a**7/b**13)*log(x + b**6*sqrt(-a**7/b**13)/a**3)/16 - (19*a**5*x + 21*a**4*b*x**3)/(8*a**2*
b**6 + 16*a*b**7*x**2 + 8*b**8*x**4) + x**7/(7*b**3)

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Giac [A]  time = 2.76754, size = 130, normalized size = 1.17 \begin{align*} \frac{99 \, a^{4} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{6}} - \frac{21 \, a^{4} b x^{3} + 19 \, a^{5} x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{5 \, b^{18} x^{7} - 21 \, a b^{17} x^{5} + 70 \, a^{2} b^{16} x^{3} - 350 \, a^{3} b^{15} x}{35 \, b^{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^3,x, algorithm="giac")

[Out]

99/8*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) - 1/8*(21*a^4*b*x^3 + 19*a^5*x)/((b*x^2 + a)^2*b^6) + 1/35*(5*b
^18*x^7 - 21*a*b^17*x^5 + 70*a^2*b^16*x^3 - 350*a^3*b^15*x)/b^21