3.1841 \(\int \frac{(a+\frac{b}{x^2})^3}{x^5} \, dx\)

Optimal. Leaf size=40 \[ \frac{a \left (a x^2+b\right )^4}{40 b^2 x^8}-\frac{\left (a x^2+b\right )^4}{10 b x^{10}} \]

[Out]

-(b + a*x^2)^4/(10*b*x^10) + (a*(b + a*x^2)^4)/(40*b^2*x^8)

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Rubi [A]  time = 0.0182408, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 266, 45, 37} \[ \frac{a \left (a x^2+b\right )^4}{40 b^2 x^8}-\frac{\left (a x^2+b\right )^4}{10 b x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)^3/x^5,x]

[Out]

-(b + a*x^2)^4/(10*b*x^10) + (a*(b + a*x^2)^4)/(40*b^2*x^8)

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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^3}{x^5} \, dx &=\int \frac{\left (b+a x^2\right )^3}{x^{11}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^6} \, dx,x,x^2\right )\\ &=-\frac{\left (b+a x^2\right )^4}{10 b x^{10}}-\frac{a \operatorname{Subst}\left (\int \frac{(b+a x)^3}{x^5} \, dx,x,x^2\right )}{10 b}\\ &=-\frac{\left (b+a x^2\right )^4}{10 b x^{10}}+\frac{a \left (b+a x^2\right )^4}{40 b^2 x^8}\\ \end{align*}

Mathematica [A]  time = 0.0041557, size = 43, normalized size = 1.08 \[ -\frac{a^2 b}{2 x^6}-\frac{a^3}{4 x^4}-\frac{3 a b^2}{8 x^8}-\frac{b^3}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)^3/x^5,x]

[Out]

-b^3/(10*x^10) - (3*a*b^2)/(8*x^8) - (a^2*b)/(2*x^6) - a^3/(4*x^4)

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Maple [A]  time = 0.005, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{{b}^{3}}{10\,{x}^{10}}}-{\frac{3\,{b}^{2}a}{8\,{x}^{8}}}-{\frac{{a}^{2}b}{2\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x^2*b)^3/x^5,x)

[Out]

-1/4*a^3/x^4-1/10*b^3/x^10-3/8*b^2*a/x^8-1/2*a^2*b/x^6

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Maxima [A]  time = 1.03666, size = 50, normalized size = 1.25 \begin{align*} -\frac{10 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} + 4 \, b^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^5,x, algorithm="maxima")

[Out]

-1/40*(10*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2 + 4*b^3)/x^10

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Fricas [A]  time = 1.35295, size = 85, normalized size = 2.12 \begin{align*} -\frac{10 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} + 4 \, b^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^5,x, algorithm="fricas")

[Out]

-1/40*(10*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2 + 4*b^3)/x^10

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Sympy [A]  time = 0.430034, size = 39, normalized size = 0.98 \begin{align*} - \frac{10 a^{3} x^{6} + 20 a^{2} b x^{4} + 15 a b^{2} x^{2} + 4 b^{3}}{40 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)**3/x**5,x)

[Out]

-(10*a**3*x**6 + 20*a**2*b*x**4 + 15*a*b**2*x**2 + 4*b**3)/(40*x**10)

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Giac [A]  time = 1.13822, size = 50, normalized size = 1.25 \begin{align*} -\frac{10 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2} + 4 \, b^{3}}{40 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)^3/x^5,x, algorithm="giac")

[Out]

-1/40*(10*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2 + 4*b^3)/x^10