3.1988 \(\int \frac{1}{(a+\frac{b}{x^3})^2 x^5} \, dx\)

Optimal. Leaf size=136 \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{2/3} b^{4/3}}-\frac{\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3}}+\frac{x^2}{3 b \left (a x^3+b\right )} \]

[Out]

x^2/(3*b*(b + a*x^3)) - ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))]/(3*Sqrt[3]*a^(2/3)*b^(4/3)) - Log[b^
(1/3) + a^(1/3)*x]/(9*a^(2/3)*b^(4/3)) + Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(18*a^(2/3)*b^(4/3))

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Rubi [A]  time = 0.0693039, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {263, 290, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{18 a^{2/3} b^{4/3}}-\frac{\log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{9 a^{2/3} b^{4/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3}}+\frac{x^2}{3 b \left (a x^3+b\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)^2*x^5),x]

[Out]

x^2/(3*b*(b + a*x^3)) - ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))]/(3*Sqrt[3]*a^(2/3)*b^(4/3)) - Log[b^
(1/3) + a^(1/3)*x]/(9*a^(2/3)*b^(4/3)) + Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/(18*a^(2/3)*b^(4/3))

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 292

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> -Dist[(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right )^2 x^5} \, dx &=\int \frac{x}{\left (b+a x^3\right )^2} \, dx\\ &=\frac{x^2}{3 b \left (b+a x^3\right )}+\frac{\int \frac{x}{b+a x^3} \, dx}{3 b}\\ &=\frac{x^2}{3 b \left (b+a x^3\right )}-\frac{\int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{9 \sqrt [3]{a} b^{4/3}}+\frac{\int \frac{\sqrt [3]{b}+\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{9 \sqrt [3]{a} b^{4/3}}\\ &=\frac{x^2}{3 b \left (b+a x^3\right )}-\frac{\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{2/3} b^{4/3}}+\frac{\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{18 a^{2/3} b^{4/3}}+\frac{\int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b}\\ &=\frac{x^2}{3 b \left (b+a x^3\right )}-\frac{\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{2/3} b^{4/3}}+\frac{\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{3 a^{2/3} b^{4/3}}\\ &=\frac{x^2}{3 b \left (b+a x^3\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{3 \sqrt{3} a^{2/3} b^{4/3}}-\frac{\log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{9 a^{2/3} b^{4/3}}+\frac{\log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{18 a^{2/3} b^{4/3}}\\ \end{align*}

Mathematica [A]  time = 0.0620723, size = 119, normalized size = 0.88 \[ \frac{\frac{\log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{a^{2/3}}-\frac{2 \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{a^{2/3}}-\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )}{a^{2/3}}+\frac{6 \sqrt [3]{b} x^2}{a x^3+b}}{18 b^{4/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)^2*x^5),x]

[Out]

((6*b^(1/3)*x^2)/(b + a*x^3) - (2*Sqrt[3]*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]])/a^(2/3) - (2*Log[b^(1/3
) + a^(1/3)*x])/a^(2/3) + Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2]/a^(2/3))/(18*b^(4/3))

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Maple [A]  time = 0.004, size = 117, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{3\,b \left ( a{x}^{3}+b \right ) }}-{\frac{1}{9\,ab}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{1}{18\,ab}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}+{\frac{\sqrt{3}}{9\,ab}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^2/b/(a*x^3+b)-1/9/b/a/(b/a)^(1/3)*ln(x+(b/a)^(1/3))+1/18/b/a/(b/a)^(1/3)*ln(x^2-(b/a)^(1/3)*x+(b/a)^(2/3
))+1/9/b*3^(1/2)/a/(b/a)^(1/3)*arctan(1/3*3^(1/2)*(2/(b/a)^(1/3)*x-1))

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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^3)^2/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.51948, size = 936, normalized size = 6.88 \begin{align*} \left [\frac{6 \, a^{2} b x^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a^{2} b\right )^{\frac{2}{3}} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} b\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} - 3 \, \left (-a^{2} b\right )^{\frac{2}{3}} x}{a x^{3} + b}\right ) +{\left (a x^{3} + b\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + \left (-a^{2} b\right )^{\frac{2}{3}}\right ) - 2 \,{\left (a x^{3} + b\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac{1}{3}}\right )}{18 \,{\left (a^{3} b^{2} x^{3} + a^{2} b^{3}\right )}}, \frac{6 \, a^{2} b x^{2} + 6 \, \sqrt{\frac{1}{3}}{\left (a^{2} b x^{3} + a b^{2}\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, a x + \left (-a^{2} b\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{a}\right ) +{\left (a x^{3} + b\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a^{2} x^{2} + \left (-a^{2} b\right )^{\frac{1}{3}} a x + \left (-a^{2} b\right )^{\frac{2}{3}}\right ) - 2 \,{\left (a x^{3} + b\right )} \left (-a^{2} b\right )^{\frac{2}{3}} \log \left (a x - \left (-a^{2} b\right )^{\frac{1}{3}}\right )}{18 \,{\left (a^{3} b^{2} x^{3} + a^{2} b^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^5,x, algorithm="fricas")

[Out]

[1/18*(6*a^2*b*x^2 + 3*sqrt(1/3)*(a^2*b*x^3 + a*b^2)*sqrt((-a^2*b)^(1/3)/b)*log((2*a^2*x^3 - a*b + 3*sqrt(1/3)
*(a*b*x + 2*(-a^2*b)^(2/3)*x^2 + (-a^2*b)^(1/3)*b)*sqrt((-a^2*b)^(1/3)/b) - 3*(-a^2*b)^(2/3)*x)/(a*x^3 + b)) +
(a*x^3 + b)*(-a^2*b)^(2/3)*log(a^2*x^2 + (-a^2*b)^(1/3)*a*x + (-a^2*b)^(2/3)) - 2*(a*x^3 + b)*(-a^2*b)^(2/3)*
log(a*x - (-a^2*b)^(1/3)))/(a^3*b^2*x^3 + a^2*b^3), 1/18*(6*a^2*b*x^2 + 6*sqrt(1/3)*(a^2*b*x^3 + a*b^2)*sqrt(-
(-a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*a*x + (-a^2*b)^(1/3))*sqrt(-(-a^2*b)^(1/3)/b)/a) + (a*x^3 + b)*(-a^2*b)^
(2/3)*log(a^2*x^2 + (-a^2*b)^(1/3)*a*x + (-a^2*b)^(2/3)) - 2*(a*x^3 + b)*(-a^2*b)^(2/3)*log(a*x - (-a^2*b)^(1/
3)))/(a^3*b^2*x^3 + a^2*b^3)]

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Sympy [A]  time = 0.533406, size = 44, normalized size = 0.32 \begin{align*} \frac{x^{2}}{3 a b x^{3} + 3 b^{2}} + \operatorname{RootSum}{\left (729 t^{3} a^{2} b^{4} + 1, \left ( t \mapsto t \log{\left (81 t^{2} a b^{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2/(3*a*b*x**3 + 3*b**2) + RootSum(729*_t**3*a**2*b**4 + 1, Lambda(_t, _t*log(81*_t**2*a*b**3 + x)))

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Giac [A]  time = 1.16053, size = 174, normalized size = 1.28 \begin{align*} \frac{x^{2}}{3 \,{\left (a x^{3} + b\right )} b} - \frac{\left (-\frac{b}{a}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{9 \, b^{2}} - \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{2}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{b}{a}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b}{a}\right )^{\frac{1}{3}}}\right )}{9 \, a^{2} b^{2}} + \frac{\left (-a^{2} b\right )^{\frac{2}{3}} \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{18 \, a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^2/x^5,x, algorithm="giac")

[Out]

1/3*x^2/((a*x^3 + b)*b) - 1/9*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/b^2 - 1/9*sqrt(3)*(-a^2*b)^(2/3)*arctan(
1/3*sqrt(3)*(2*x + (-b/a)^(1/3))/(-b/a)^(1/3))/(a^2*b^2) + 1/18*(-a^2*b)^(2/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/
a)^(2/3))/(a^2*b^2)