3.1887 \(\int \frac{1}{(a+\frac{b}{x^2})^3 x^8} \, dx\)

Optimal. Leaf size=76 \[ \frac{5}{8 b^2 x \left (a x^2+b\right )}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}+\frac{1}{4 b x \left (a x^2+b\right )^2}-\frac{15}{8 b^3 x} \]

[Out]

-15/(8*b^3*x) + 1/(4*b*x*(b + a*x^2)^2) + 5/(8*b^2*x*(b + a*x^2)) - (15*Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(
8*b^(7/2))

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Rubi [A]  time = 0.0269846, antiderivative size = 76, 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, 290, 325, 205} \[ \frac{5}{8 b^2 x \left (a x^2+b\right )}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}+\frac{1}{4 b x \left (a x^2+b\right )^2}-\frac{15}{8 b^3 x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^2)^3*x^8),x]

[Out]

-15/(8*b^3*x) + 1/(4*b*x*(b + a*x^2)^2) + 5/(8*b^2*x*(b + a*x^2)) - (15*Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(
8*b^(7/2))

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 290

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

Rule 325

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

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{1}{\left (a+\frac{b}{x^2}\right )^3 x^8} \, dx &=\int \frac{1}{x^2 \left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5 \int \frac{1}{x^2 \left (b+a x^2\right )^2} \, dx}{4 b}\\ &=\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5}{8 b^2 x \left (b+a x^2\right )}+\frac{15 \int \frac{1}{x^2 \left (b+a x^2\right )} \, dx}{8 b^2}\\ &=-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5}{8 b^2 x \left (b+a x^2\right )}-\frac{(15 a) \int \frac{1}{b+a x^2} \, dx}{8 b^3}\\ &=-\frac{15}{8 b^3 x}+\frac{1}{4 b x \left (b+a x^2\right )^2}+\frac{5}{8 b^2 x \left (b+a x^2\right )}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0395061, size = 68, normalized size = 0.89 \[ -\frac{15 a^2 x^4+25 a b x^2+8 b^2}{8 b^3 x \left (a x^2+b\right )^2}-\frac{15 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 b^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^2)^3*x^8),x]

[Out]

-(8*b^2 + 25*a*b*x^2 + 15*a^2*x^4)/(8*b^3*x*(b + a*x^2)^2) - (15*Sqrt[a]*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(8*b^(7/
2))

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Maple [A]  time = 0.01, size = 66, normalized size = 0.9 \begin{align*} -{\frac{7\,{x}^{3}{a}^{2}}{8\,{b}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{9\,ax}{8\,{b}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{15\,a}{8\,{b}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{{b}^{3}x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+1/x^2*b)^3/x^8,x)

[Out]

-7/8/b^3*a^2/(a*x^2+b)^2*x^3-9/8/b^2*a/(a*x^2+b)^2*x-15/8/b^3*a/(a*b)^(1/2)*arctan(a*x/(a*b)^(1/2))-1/b^3/x

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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(1/(a+b/x^2)^3/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.52676, size = 428, normalized size = 5.63 \begin{align*} \left [-\frac{30 \, a^{2} x^{4} + 50 \, a b x^{2} - 15 \,{\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{a x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - b}{a x^{2} + b}\right ) + 16 \, b^{2}}{16 \,{\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}, -\frac{15 \, a^{2} x^{4} + 25 \, a b x^{2} + 15 \,{\left (a^{2} x^{5} + 2 \, a b x^{3} + b^{2} x\right )} \sqrt{\frac{a}{b}} \arctan \left (x \sqrt{\frac{a}{b}}\right ) + 8 \, b^{2}}{8 \,{\left (a^{2} b^{3} x^{5} + 2 \, a b^{4} x^{3} + b^{5} x\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^8,x, algorithm="fricas")

[Out]

[-1/16*(30*a^2*x^4 + 50*a*b*x^2 - 15*(a^2*x^5 + 2*a*b*x^3 + b^2*x)*sqrt(-a/b)*log((a*x^2 - 2*b*x*sqrt(-a/b) -
b)/(a*x^2 + b)) + 16*b^2)/(a^2*b^3*x^5 + 2*a*b^4*x^3 + b^5*x), -1/8*(15*a^2*x^4 + 25*a*b*x^2 + 15*(a^2*x^5 + 2
*a*b*x^3 + b^2*x)*sqrt(a/b)*arctan(x*sqrt(a/b)) + 8*b^2)/(a^2*b^3*x^5 + 2*a*b^4*x^3 + b^5*x)]

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Sympy [A]  time = 0.623701, size = 114, normalized size = 1.5 \begin{align*} \frac{15 \sqrt{- \frac{a}{b^{7}}} \log{\left (x - \frac{b^{4} \sqrt{- \frac{a}{b^{7}}}}{a} \right )}}{16} - \frac{15 \sqrt{- \frac{a}{b^{7}}} \log{\left (x + \frac{b^{4} \sqrt{- \frac{a}{b^{7}}}}{a} \right )}}{16} - \frac{15 a^{2} x^{4} + 25 a b x^{2} + 8 b^{2}}{8 a^{2} b^{3} x^{5} + 16 a b^{4} x^{3} + 8 b^{5} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**2)**3/x**8,x)

[Out]

15*sqrt(-a/b**7)*log(x - b**4*sqrt(-a/b**7)/a)/16 - 15*sqrt(-a/b**7)*log(x + b**4*sqrt(-a/b**7)/a)/16 - (15*a*
*2*x**4 + 25*a*b*x**2 + 8*b**2)/(8*a**2*b**3*x**5 + 16*a*b**4*x**3 + 8*b**5*x)

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Giac [A]  time = 1.15729, size = 77, normalized size = 1.01 \begin{align*} -\frac{15 \, a \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{3}} - \frac{7 \, a^{2} x^{3} + 9 \, a b x}{8 \,{\left (a x^{2} + b\right )}^{2} b^{3}} - \frac{1}{b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^3/x^8,x, algorithm="giac")

[Out]

-15/8*a*arctan(a*x/sqrt(a*b))/(sqrt(a*b)*b^3) - 1/8*(7*a^2*x^3 + 9*a*b*x)/((a*x^2 + b)^2*b^3) - 1/(b^3*x)