3.1967 \(\int \frac{x^3}{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=132 \[ -\frac{b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}-\frac{b x}{a^2}+\frac{x^4}{4 a} \]

[Out]

-((b*x)/a^2) + x^4/(4*a) - (b^(4/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(7/3)) + (b^
(4/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(7/3)) - (b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(7
/3))

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Rubi [A]  time = 0.0818292, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {263, 302, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )}{3 a^{7/3}}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}-\frac{b x}{a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x^3),x]

[Out]

-((b*x)/a^2) + x^4/(4*a) - (b^(4/3)*ArcTan[(b^(1/3) - 2*a^(1/3)*x)/(Sqrt[3]*b^(1/3))])/(Sqrt[3]*a^(7/3)) + (b^
(4/3)*Log[b^(1/3) + a^(1/3)*x])/(3*a^(7/3)) - (b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(6*a^(7
/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 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 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*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{x^3}{a+\frac{b}{x^3}} \, dx &=\int \frac{x^6}{b+a x^3} \, dx\\ &=\int \left (-\frac{b}{a^2}+\frac{x^3}{a}+\frac{b^2}{a^2 \left (b+a x^3\right )}\right ) \, dx\\ &=-\frac{b x}{a^2}+\frac{x^4}{4 a}+\frac{b^2 \int \frac{1}{b+a x^3} \, dx}{a^2}\\ &=-\frac{b x}{a^2}+\frac{x^4}{4 a}+\frac{b^{4/3} \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 a^2}+\frac{b^{4/3} \int \frac{2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 a^2}\\ &=-\frac{b x}{a^2}+\frac{x^4}{4 a}+\frac{b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac{b^{4/3} \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 a^{7/3}}+\frac{b^{5/3} \int \frac{1}{b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 a^2}\\ &=-\frac{b x}{a^2}+\frac{x^4}{4 a}+\frac{b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac{b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}}+\frac{b^{4/3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{a^{7/3}}\\ &=-\frac{b x}{a^2}+\frac{x^4}{4 a}-\frac{b^{4/3} \tan ^{-1}\left (\frac{\sqrt [3]{b}-2 \sqrt [3]{a} x}{\sqrt{3} \sqrt [3]{b}}\right )}{\sqrt{3} a^{7/3}}+\frac{b^{4/3} \log \left (\sqrt [3]{b}+\sqrt [3]{a} x\right )}{3 a^{7/3}}-\frac{b^{4/3} \log \left (b^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 a^{7/3}}\\ \end{align*}

Mathematica [A]  time = 0.0224658, size = 120, normalized size = 0.91 \[ \frac{-2 b^{4/3} \log \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )+3 a^{4/3} x^4+4 b^{4/3} \log \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )-4 \sqrt{3} b^{4/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt{3}}\right )-12 \sqrt [3]{a} b x}{12 a^{7/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x^3),x]

[Out]

(-12*a^(1/3)*b*x + 3*a^(4/3)*x^4 - 4*Sqrt[3]*b^(4/3)*ArcTan[(1 - (2*a^(1/3)*x)/b^(1/3))/Sqrt[3]] + 4*b^(4/3)*L
og[b^(1/3) + a^(1/3)*x] - 2*b^(4/3)*Log[b^(2/3) - a^(1/3)*b^(1/3)*x + a^(2/3)*x^2])/(12*a^(7/3))

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Maple [A]  time = 0.003, size = 115, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4\,a}}-{\frac{bx}{{a}^{2}}}+{\frac{{b}^{2}}{3\,{a}^{3}}\ln \left ( x+\sqrt [3]{{\frac{b}{a}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}-{\frac{{b}^{2}}{6\,{a}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{b}{a}}}x+ \left ({\frac{b}{a}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}}+{\frac{{b}^{2}\sqrt{3}}{3\,{a}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{b}{a}}}}}}-1 \right ) } \right ) \left ({\frac{b}{a}} \right ) ^{-{\frac{2}{3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4/a-b*x/a^2+1/3/a^3*b^2/(b/a)^(2/3)*ln(x+(b/a)^(1/3))-1/6/a^3*b^2/(b/a)^(2/3)*ln(x^2-(b/a)^(1/3)*x+(b/a)
^(2/3))+1/3/a^3*b^2/(b/a)^(2/3)*3^(1/2)*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(x^3/(a+b/x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.46038, size = 277, normalized size = 2.1 \begin{align*} \frac{3 \, a x^{4} + 4 \, \sqrt{3} b \left (\frac{b}{a}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (\frac{b}{a}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - 2 \, b \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x^{2} - x \left (\frac{b}{a}\right )^{\frac{1}{3}} + \left (\frac{b}{a}\right )^{\frac{2}{3}}\right ) + 4 \, b \left (\frac{b}{a}\right )^{\frac{1}{3}} \log \left (x + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right ) - 12 \, b x}{12 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*a*x^4 + 4*sqrt(3)*b*(b/a)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(b/a)^(2/3) - sqrt(3)*b)/b) - 2*b*(b/a)^(1/3
)*log(x^2 - x*(b/a)^(1/3) + (b/a)^(2/3)) + 4*b*(b/a)^(1/3)*log(x + (b/a)^(1/3)) - 12*b*x)/a^2

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Sympy [A]  time = 0.457404, size = 37, normalized size = 0.28 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{7} - b^{4}, \left ( t \mapsto t \log{\left (\frac{3 t a^{2}}{b} + x \right )} \right )\right )} + \frac{x^{4}}{4 a} - \frac{b x}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(27*_t**3*a**7 - b**4, Lambda(_t, _t*log(3*_t*a**2/b + x))) + x**4/(4*a) - b*x/a**2

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Giac [A]  time = 1.19911, size = 174, normalized size = 1.32 \begin{align*} -\frac{b \left (-\frac{b}{a}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{b}{a}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a^{2}} + \frac{\sqrt{3} \left (-a^{2} b\right )^{\frac{1}{3}} b \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^{3}} + \frac{\left (-a^{2} b\right )^{\frac{1}{3}} b \log \left (x^{2} + x \left (-\frac{b}{a}\right )^{\frac{1}{3}} + \left (-\frac{b}{a}\right )^{\frac{2}{3}}\right )}{6 \, a^{3}} + \frac{a^{3} x^{4} - 4 \, a^{2} b x}{4 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b*(-b/a)^(1/3)*log(abs(x - (-b/a)^(1/3)))/a^2 + 1/3*sqrt(3)*(-a^2*b)^(1/3)*b*arctan(1/3*sqrt(3)*(2*x + (-
b/a)^(1/3))/(-b/a)^(1/3))/a^3 + 1/6*(-a^2*b)^(1/3)*b*log(x^2 + x*(-b/a)^(1/3) + (-b/a)^(2/3))/a^3 + 1/4*(a^3*x
^4 - 4*a^2*b*x)/a^4