3.736 \(\int \frac{1}{x (a+b x^3)^{2/3} (c+d x^3)} \, dx\)

Optimal. Leaf size=245 \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} c}-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}-\frac{d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}+\frac{d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.210522, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {446, 86, 57, 617, 204, 31, 58} \[ \frac{\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} c}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} c}-\frac{\log (x)}{2 a^{2/3} c}+\frac{d^{2/3} \log \left (c+d x^3\right )}{6 c (b c-a d)^{2/3}}-\frac{d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 c (b c-a d)^{2/3}}+\frac{d^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt{3}}\right )}{\sqrt{3} c (b c-a d)^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(a + b*x^3)^(2/3)*(c + d*x^3)),x]

[Out]

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 86

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), In
t[(e + f*x)^p/(a + b*x), x], x] - Dist[d/(b*c - a*d), Int[(e + f*x)^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] &&  !IntegerQ[p]

Rule 57

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, -Simp[L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

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 31

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

Rule 58

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-((b*c - a*d)/b), 3]}, -Sim
p[Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (Dist[3/(2*b*q), Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d
*x)^(1/3)], x] + Dist[3/(2*b*q^2), Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x
] && NegQ[(b*c - a*d)/b]

Rubi steps

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

Mathematica [A]  time = 0.278001, size = 308, normalized size = 1.26 \[ -\frac{-\frac{2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{a^{2/3}}+\frac{\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{a^{2/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt{3}}\right )}{a^{2/3}}+\frac{2 d^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{2/3}}-\frac{d^{2/3} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x^3} \sqrt [3]{b c-a d}+(b c-a d)^{2/3}+d^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{2/3}}+\frac{2 \sqrt{3} d^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}-1}{\sqrt{3}}\right )}{(b c-a d)^{2/3}}}{6 c} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(a + b*x^3)^(2/3)*(c + d*x^3)),x]

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x \left ( d{x}^{3}+c \right ) } \left ( b{x}^{3}+a \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{2}{3}}{\left (d x^{3} + c\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(b*x^3+a)^(2/3)/(d*x^3+c),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(2/3)*(d*x^3 + c)*x), x)

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Fricas [B]  time = 1.50184, size = 1115, normalized size = 4.55 \begin{align*} -\frac{2 \, \sqrt{3} a^{2} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} \arctan \left (-\frac{2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c - a d\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{2}{3}} - \sqrt{3} d}{3 \, d}\right ) + a^{2} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} d^{2} +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (b c d - a d^{2}\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} +{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{2}{3}}\right ) - 2 \, a^{2} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} d -{\left (b c - a d\right )} \left (-\frac{d^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac{1}{3}}\right ) + 2 \, \sqrt{3}{\left (a^{2}\right )}^{\frac{1}{6}} a \arctan \left (\frac{{\left (a^{2}\right )}^{\frac{1}{6}}{\left (\sqrt{3}{\left (a^{2}\right )}^{\frac{1}{3}} a + 2 \, \sqrt{3}{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right )}}{3 \, a^{2}}\right ) +{\left (a^{2}\right )}^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} a +{\left (a^{2}\right )}^{\frac{1}{3}} a +{\left (b x^{3} + a\right )}^{\frac{1}{3}}{\left (a^{2}\right )}^{\frac{2}{3}}\right ) - 2 \,{\left (a^{2}\right )}^{\frac{2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{1}{3}} a -{\left (a^{2}\right )}^{\frac{2}{3}}\right )}{6 \, a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \left (a + b x^{3}\right )^{\frac{2}{3}} \left (c + d x^{3}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*(a + b*x**3)**(2/3)*(c + d*x**3)), x)

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Giac [A]  time = 2.61581, size = 462, normalized size = 1.89 \begin{align*} \frac{1}{6} \,{\left (\frac{2 \, d \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} \right |}\right )}{b^{2} c^{2} - a b c d} - \frac{6 \,{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}}}\right )}{\sqrt{3} b^{2} c^{2} - \sqrt{3} a b c d} - \frac{{\left (-b c d^{2} + a d^{3}\right )}^{\frac{1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} \left (-\frac{b c - a d}{d}\right )^{\frac{1}{3}} + \left (-\frac{b c - a d}{d}\right )^{\frac{2}{3}}\right )}{b^{2} c^{2} - a b c d} - \frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (2 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} + a^{\frac{1}{3}}\right )}}{3 \, a^{\frac{1}{3}}}\right )}{a^{\frac{2}{3}} b c} - \frac{\log \left ({\left (b x^{3} + a\right )}^{\frac{2}{3}} +{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{\frac{1}{3}} + a^{\frac{2}{3}}\right )}{a^{\frac{2}{3}} b c} + \frac{2 \, \log \left ({\left |{\left (b x^{3} + a\right )}^{\frac{1}{3}} - a^{\frac{1}{3}} \right |}\right )}{a^{\frac{2}{3}} b c}\right )} b \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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