3.1858 \(\int \frac{x^6}{(a+\frac{b}{x^2})^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.037117, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 288, 302, 205} \[ \frac{3 b^2 x^3}{2 a^4}-\frac{9 b^3 x}{2 a^5}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{11/2}}-\frac{9 b x^5}{10 a^3}+\frac{9 x^7}{14 a^2}-\frac{x^9}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b/x^2)^2,x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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

Mathematica [A]  time = 0.0522571, size = 82, normalized size = 0.89 \[ \frac{x \left (-28 a^2 b x^4+10 a^3 x^6+70 a b^2 x^2-\frac{35 b^4}{a x^2+b}-280 b^3\right )}{70 a^5}+\frac{9 b^{7/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{11/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b/x^2)^2,x]

[Out]

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

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Maple [A]  time = 0.009, size = 78, normalized size = 0.9 \begin{align*}{\frac{{x}^{7}}{7\,{a}^{2}}}-{\frac{2\,b{x}^{5}}{5\,{a}^{3}}}+{\frac{{b}^{2}{x}^{3}}{{a}^{4}}}-4\,{\frac{{b}^{3}x}{{a}^{5}}}-{\frac{{b}^{4}x}{2\,{a}^{5} \left ( a{x}^{2}+b \right ) }}+{\frac{9\,{b}^{4}}{2\,{a}^{5}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(a+1/x^2*b)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.73171, size = 459, normalized size = 4.99 \begin{align*} \left [\frac{20 \, a^{4} x^{9} - 36 \, a^{3} b x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a b^{3} x^{3} - 630 \, b^{4} x + 315 \,{\left (a b^{3} x^{2} + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{140 \,{\left (a^{6} x^{2} + a^{5} b\right )}}, \frac{10 \, a^{4} x^{9} - 18 \, a^{3} b x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a b^{3} x^{3} - 315 \, b^{4} x + 315 \,{\left (a b^{3} x^{2} + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{70 \,{\left (a^{6} x^{2} + a^{5} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a+b/x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.659533, size = 134, normalized size = 1.46 \begin{align*} - \frac{b^{4} x}{2 a^{6} x^{2} + 2 a^{5} b} - \frac{9 \sqrt{- \frac{b^{7}}{a^{11}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac{9 \sqrt{- \frac{b^{7}}{a^{11}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{7}}{a^{11}}}}{b^{3}} + x \right )}}{4} + \frac{x^{7}}{7 a^{2}} - \frac{2 b x^{5}}{5 a^{3}} + \frac{b^{2} x^{3}}{a^{4}} - \frac{4 b^{3} x}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(a+b/x**2)**2,x)

[Out]

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

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Giac [A]  time = 1.18883, size = 113, normalized size = 1.23 \begin{align*} \frac{9 \, b^{4} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{5}} - \frac{b^{4} x}{2 \,{\left (a x^{2} + b\right )} a^{5}} + \frac{5 \, a^{12} x^{7} - 14 \, a^{11} b x^{5} + 35 \, a^{10} b^{2} x^{3} - 140 \, a^{9} b^{3} x}{35 \, a^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(a+b/x^2)^2,x, algorithm="giac")

[Out]

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