3.1975 \(\int \frac{1}{(a+\frac{b}{x^3}) x^5} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt [3]{a} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}-\frac{1}{b x} \]

[Out]

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

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Rubi [A]  time = 0.0646309, antiderivative size = 122, 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, 325, 292, 31, 634, 617, 204, 628} \[ -\frac{\sqrt [3]{a} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 b^{4/3}}+\frac{\sqrt [3]{a} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 b^{4/3}}+\frac{\sqrt [3]{a} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} b^{4/3}}-\frac{1}{b x} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)*x^5),x]

[Out]

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

Mathematica [A]  time = 0.0185458, size = 114, normalized size = 0.93 \[ \frac{-\sqrt [3]{a} x \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+2 \sqrt [3]{a} x \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )+2 \sqrt{3} \sqrt [3]{a} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-6 \sqrt [3]{b}}{6 b^{4/3} x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)*x^5),x]

[Out]

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

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Maple [A]  time = 0.004, size = 99, normalized size = 0.8 \begin{align*}{\frac{1}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ){\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-{\frac{1}{6\,b}\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}}{3\,b}\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}}}}}}-{\frac{1}{bx}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.4481, size = 262, normalized size = 2.15 \begin{align*} -\frac{2 \, \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3} x \left (\frac{a}{b}\right )^{\frac{1}{3}} - \frac{1}{3} \, \sqrt{3}\right ) + x \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a}{b}\right )^{\frac{2}{3}} + b \left (\frac{a}{b}\right )^{\frac{1}{3}}\right ) - 2 \, x \left (\frac{a}{b}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a}{b}\right )^{\frac{2}{3}}\right ) + 6}{6 \, b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.451219, size = 29, normalized size = 0.24 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} b^{4} - a, \left ( t \mapsto t \log{\left (\frac{9 t^{2} b^{3}}{a} + x \right )} \right )\right )} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*b**4 - a, Lambda(_t, _t*log(9*_t**2*b**3/a + x))) - 1/(b*x)

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Giac [A]  time = 1.18712, size = 163, normalized size = 1.34 \begin{align*} \frac{a \left (-\frac{b}{a}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, 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 )}{3 \, a 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 )}{6 \, a b^{2}} - \frac{1}{b x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*(-b/a)^(2/3)*log(abs(x - (-b/a)^(1/3)))/b^2 + 1/3*sqrt(3)*(-a^2*b)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-b/a
)^(1/3))/(-b/a)^(1/3))/(a*b^2) - 1/6*(-a^2*b)^(2/3)*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/(a*b^2) - 1/(b*x)