3.1860 \(\int \frac{x^4}{(a+\frac{b}{x^2})^2} \, dx\)

Optimal. Leaf size=79 \[ \frac{7 b^2 x}{2 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (a x^2+b\right )} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0327662, antiderivative size = 79, 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} \[ \frac{7 b^2 x}{2 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x^2)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0455872, size = 71, normalized size = 0.9 \[ \frac{x \left (6 a^2 x^4+\frac{15 b^3}{a x^2+b}-20 a b x^2+90 b^2\right )}{30 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x^2)^2,x]

[Out]

(x*(90*b^2 - 20*a*b*x^2 + 6*a^2*x^4 + (15*b^3)/(b + a*x^2)))/(30*a^4) - (7*b^(5/2)*ArcTan[(Sqrt[a]*x)/Sqrt[b]]
)/(2*a^(9/2))

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 68, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{a}^{2}}}-{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+3\,{\frac{{b}^{2}x}{{a}^{4}}}+{\frac{{b}^{3}x}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}-{\frac{7\,{b}^{3}}{2\,{a}^{4}}\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^4/(a+1/x^2*b)^2,x)

[Out]

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

________________________________________________________________________________________

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^4/(a+b/x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.48904, size = 409, normalized size = 5.18 \begin{align*} \left [\frac{12 \, a^{3} x^{7} - 28 \, a^{2} b x^{5} + 140 \, a b^{2} x^{3} + 210 \, b^{3} x + 105 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{60 \,{\left (a^{5} x^{2} + a^{4} b\right )}}, \frac{6 \, a^{3} x^{7} - 14 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} x - 105 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{30 \,{\left (a^{5} x^{2} + a^{4} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^2,x, algorithm="fricas")

[Out]

[1/60*(12*a^3*x^7 - 28*a^2*b*x^5 + 140*a*b^2*x^3 + 210*b^3*x + 105*(a*b^2*x^2 + b^3)*sqrt(-b/a)*log((a*x^2 - 2
*a*x*sqrt(-b/a) - b)/(a*x^2 + b)))/(a^5*x^2 + a^4*b), 1/30*(6*a^3*x^7 - 14*a^2*b*x^5 + 70*a*b^2*x^3 + 105*b^3*
x - 105*(a*b^2*x^2 + b^3)*sqrt(b/a)*arctan(a*x*sqrt(b/a)/b))/(a^5*x^2 + a^4*b)]

________________________________________________________________________________________

Sympy [A]  time = 0.580015, size = 124, normalized size = 1.57 \begin{align*} \frac{b^{3} x}{2 a^{5} x^{2} + 2 a^{4} b} + \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{2}} + x \right )}}{4} - \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{2}} + x \right )}}{4} + \frac{x^{5}}{5 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{3 b^{2} x}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b/x**2)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.15071, size = 99, normalized size = 1.25 \begin{align*} -\frac{7 \, b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} + \frac{b^{3} x}{2 \,{\left (a x^{2} + b\right )} a^{4}} + \frac{3 \, a^{8} x^{5} - 10 \, a^{7} b x^{3} + 45 \, a^{6} b^{2} x}{15 \, a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x^2)^2,x, algorithm="giac")

[Out]

-7/2*b^3*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*a^4) + 1/2*b^3*x/((a*x^2 + b)*a^4) + 1/15*(3*a^8*x^5 - 10*a^7*b*x^3
+ 45*a^6*b^2*x)/a^10